Spin-phonon interaction in transition-metal difluoride antiferromagnets: Theory and experiment
An overall comparative study is made of the spin-phonon interactions in several rutile-structure transition-metal difluorides, specifically FeF₂, MnF₂, NiF₂, and CoF₂, in terms of recent developments obtained experimentally using inelastic light scattering spectroscopy and theoretically using a modi...
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nasplib_isofts_kiev_ua-123456789-1754702025-02-23T17:12:28Z Spin-phonon interaction in transition-metal difluoride antiferromagnets: Theory and experiment Cottam, M.G. Lockwood, D.J. Низькотемпературний магнетизм An overall comparative study is made of the spin-phonon interactions in several rutile-structure transition-metal difluorides, specifically FeF₂, MnF₂, NiF₂, and CoF₂, in terms of recent developments obtained experimentally using inelastic light scattering spectroscopy and theoretically using a modified mean-field approach to estimate spin-pair correlation functions. New experimental data are presented here and interpreted within an extended and comprehensive theoretical treatment to yield estimates for the spin-phonon coupling coefficients and the relative magnitudes of the magneto-optical coupling coefficients. С использованием последних результатов экспериментально с применением спектроскопии неупругого рассеяния света и теоретически методом модифицированного среднеполевого подхода для оценки корреляционных функций спиновой пары проведено полное сравнительное исследование спин-фононных взаимодействий в нескольких дифлуоридах переходных металлов рутиловой структуры, в частности FeF₂, MnF₂, NiF₂ и CoF₂. С целью получения оценок коэффициентов спин-фононной связи и относительных величин коэффициентов магнитооптической связи представлены новые экспериментальные данные, которые интерпретируются в рамках всестороннего расширенного теоретического описания З використанням останніх результатів експериментально зі застосуванням спектроскопії непружного розсіяння світла та теоретично методом модифікованого середньопольового підходу для оцінки кореляційних функцій спінової пари проведено повне порівняльне дослідження спін-фононних взаємодій в декількох діфлуоридах перехідних металів рутилової структури, зокрема FeF₂, MnF₂, NiF₂ та CoF₂. З метою отримання оцінок коефіцієнтів спін-фононного зв’язку та відносних величин коефіцієнтів магнітооптичного зв’язку представлено нові експериментальні данні, які інтерпретуються в рамках всебічного розширеного теоретичного опису. This paper is dedicated to the memory of Professor Victor V. Eremenko (1932-2016), formerly of the B. Verkin Institute of Low Temperature Physics and Engineering and Editor in Chief of this journal (Low Temperature Physics), whose pioneering work on the measurement and comprehension of the optical properties of magnetic systems continues to be of great value to us all. MGC gratefully acknowledges partial support from the Natural Sciences and Engineering Research Council (NSERC) of Canada through Discovery Grant RGPIN-2017-04429. We thank H.J. Labbé for the crystal sample preparation and R. Radomski and J. Johnson for curve fitting the CoF₂ Raman spectra. 2019 Article Spin-phonon interaction in transition-metal difluoride antiferromagnets: Theory and experiment / M.G. Cottam, D.J. Lockwood // Физика низких температур. — 2019. — Т. 45, № 1. — С. 90-103. — Бібліогр.: 35 назв. — англ. 0132-6414 https://nasplib.isofts.kiev.ua/handle/123456789/175470 en Физика низких температур application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Низькотемпературний магнетизм Низькотемпературний магнетизм |
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Низькотемпературний магнетизм Низькотемпературний магнетизм Cottam, M.G. Lockwood, D.J. Spin-phonon interaction in transition-metal difluoride antiferromagnets: Theory and experiment Физика низких температур |
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An overall comparative study is made of the spin-phonon interactions in several rutile-structure transition-metal difluorides, specifically FeF₂, MnF₂, NiF₂, and CoF₂, in terms of recent developments obtained experimentally using inelastic light scattering spectroscopy and theoretically using a modified mean-field approach to estimate spin-pair correlation functions. New experimental data are presented here and interpreted within an extended and comprehensive theoretical treatment to yield estimates for the spin-phonon coupling coefficients and the relative magnitudes of the magneto-optical coupling coefficients. |
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Cottam, M.G. Lockwood, D.J. |
| author_facet |
Cottam, M.G. Lockwood, D.J. |
| author_sort |
Cottam, M.G. |
| title |
Spin-phonon interaction in transition-metal difluoride antiferromagnets: Theory and experiment |
| title_short |
Spin-phonon interaction in transition-metal difluoride antiferromagnets: Theory and experiment |
| title_full |
Spin-phonon interaction in transition-metal difluoride antiferromagnets: Theory and experiment |
| title_fullStr |
Spin-phonon interaction in transition-metal difluoride antiferromagnets: Theory and experiment |
| title_full_unstemmed |
Spin-phonon interaction in transition-metal difluoride antiferromagnets: Theory and experiment |
| title_sort |
spin-phonon interaction in transition-metal difluoride antiferromagnets: theory and experiment |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Низькотемпературний магнетизм |
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Spin-phonon interaction in transition-metal difluoride antiferromagnets: Theory and experiment / M.G. Cottam, D.J. Lockwood // Физика низких температур. — 2019. — Т. 45, № 1. — С. 90-103. — Бібліогр.: 35 назв. — англ. |
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2025-11-24T02:52:51Z |
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| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 1, pp. 90–103
Spin-phonon interaction in transition-metal difluoride
antiferromagnets: Theory and experiment
M.G. Cottam
Department of Physics and Astronomy, University of Western Ontario, London, Ontario, Canada N6A 3K7
E-mail: cottam@uwo.ca
D.J. Lockwood
Measurement Science and Standards, National Research Council, Ottawa, Ontario, Canada K1A 0R6
E-mail: david.lockwood@nrc-cnrc.gc.ca
Received July 26, 2018, published online November 26, 2018
An overall comparative study is made of the spin-phonon interactions in several rutile-structure transition-
metal difluorides, specifically FeF2, MnF2, NiF2, and CoF2, in terms of recent developments obtained experimen-
tally using inelastic light scattering spectroscopy and theoretically using a modified mean-field approach to esti-
mate spin-pair correlation functions. New experimental data are presented here and interpreted within an extend-
ed and comprehensive theoretical treatment to yield estimates for the spin-phonon coupling coefficients and the
relative magnitudes of the magneto-optical coupling coefficients.
Keywords: antiferromagnets, spin-phonon coupling, rutile structure, magneto-optical coupling, Raman spectroscopy.
1. Introduction
The invention of the laser led to a revolution in the appli-
cations of inelastic light scattering spectroscopy. In the
1960s, laser Raman scattering spectroscopy was widely used
to study the electronic and magnetic properties of transition-
metal and rare-earth compounds. Many interesting and novel
features of the effects of magnetic ordering in such solids
were reported [1]. These early studies focused primarily on
the fundamental magnetic excitation, the magnon, but also
investigated associated excitons at higher energies [2].
Nowadays, Raman scattering has become a standard tech-
nique for characterizing such excitations. However, much
less was known of the spin-phonon interaction in these
compounds from light scattering measurements, which
probe the excitations close to the Brillouin zone center (i.e.,
near zero wave vector). This is because direct interactions of
magnons, or higher-lying excitons, with phonons are less
likely in the range of wave vectors near the Brillouin zone
center accessed in Raman spectroscopy. Nevertheless, such
strong interactions have been observed, for example, in
FeCl2⋅2H2O, CsCoCl3, and RbCoCl3, where there is an ac-
cidental near-degeneracy of the magnon/exciton and phonon
frequencies [1,3,4].
More generally, spin-phonon interactions manifest
themselves through modifications to the normal tempera-
ture dependences of the optic phonons. The exchange cou-
pling between magnetic ions influences the phonon fre-
quency, integrated intensity, and linewidth. Such spin-
dependent effects have been reported earlier on in the pho-
non Raman spectra of, for example, KCoF3, VI2, CsCoBr3,
EuSe, EuTe, EuO, EuS, CdCr2S4, and CdCr2Se4 [1,5]. The
transition metal halides including the perovskite (cubic)
structure fluorides, rutile (tetragonal) structure fluorides,
and trigonal structure chlorides, bromides, and iodides
have produced a rich vein of spin-phonon interactions
amenable to detailed study [1]. The observation through
Raman scattering of such magnon-phonon interactions via
the phonon behavior as a function of temperature has be-
come ubiquitous today.
The basic mechanism assumed for the spin-phonon
coupling in the rutile-structure antiferromagnets arises
from a spatial modulation of the dominant exchange pa-
rameter due to the relative displacements of the atoms
corresponding to the phonon modes, as first proposed by
Akhiezer [6]. This effect can lead to a modification of
both the phonon and magnon dispersion relations, and has
been studied theoretically for ferromagnets [7–9] and
antiferromagnets [10,11]. In this work an extended version
of the approach used in our previous spin-phonon calcula-
tions [12,13], and applied primarily to FeF2 and MnF2, will
be followed. Other forms of spin-phonon coupling, such as
those due to magnetoelastic interactions [14,15], are not
relevant to the present studies.
© M.G. Cottam and D.J. Lockwood, 2019
Spin-phonon interaction in transition-metal difluoride antiferromagnets: Theory and experiment
For the phonons in transition metal fluorides with the
rutile structure, which are the main concern of this work,
experimental results and comparisons with theory have
previously been reported for the antiferromagnets FeF2,
MnF2, and NiF2 [12,13,16–19], and the diluted antiferro-
magnet Fe1–xZnxF2 [20]. These compounds possess four
zone-center Raman-active optic phonons, all of different
symmetries, and because of their simple magnetic struc-
tures are ideal candidates for investigating the spin-phonon
interaction in antiferromagnets. The Raman-active phonon
frequencies, linewidths, and intensities of these materials
are affected to varying extents by the antiferromagnetic
ordering, and from the extensive experimental data spin-
phonon coupling coefficients have been deduced for FeF2,
MnF2, and NiF2 [13,19]. It is of interest to determine the
magnitudes of such effects, because they can also influence
the magnon Raman scattering [10].
Here, in this overview of the spin-phonon effects in the-
se four relatively simple antiferromagnets, we augment this
previous work with additional experimental data for FeF2,
MnF2, and CoF2 together with a comprehensive theoretical
interpretation and analysis, as pledged in [13]. Significant
difference are highlighted (and accounted for) in the cases
of CoF2 and NiF2 compared with FeF2 and MnF2. Our
study of diverse rutile-structure antiferromagnets provides
new insights into the processes and mechanisms involved in
spin-phonon effects for different phonons, and different spin
values, spin alignments, and spin-orbit coupling strengths.
2. Experimental results
2.1. FeF2, MnF2, and NiF2
Detailed experimental investigations of spin-phonon in-
teractions in FeF2, MnF2, and NiF2 have been reported
previously, as mentioned above [12,13,19]. Here we sum-
marize the results obtained. There are four Raman-active
vibrational modes in these transition-metal difluorides, and
the reported phonon frequencies at low temperature along
with their symmetry classifications are given in Table 1. In
particular, new results have been obtained for the B2g pho-
non mode in FeF2 at low temperatures, and some repre-
sentative Raman spectra recorded under the same experi-
mental conditions as reported in our earlier work [17]
using laser excitation at 514.5 nm are shown in Fig. 1. The
B2g mode frequency, which is much higher than for the
other three Raman-active modes in rutile-structure com-
pounds (see Table 1), occurs at 505.6 cm–1 in FeF2 at low
temperatures. Figure 1 exhibits another weaker, and
asymmetric, Raman peak at ~ 530 cm–1 that becomes even
weaker compared to the B2g mode at higher temperatures.
This peak is assigned to a phonon combination band asso-
ciated with the Eg mode. At 10 K the overtone of the Eg
mode at zero wave vector is expected (using the datum in
Table 1) to be near 2×261.7 = 523.4 cm–1, which is above
the frequency of the B2g mode but lower than that of the
additional peak at 530 cm–1. This indicates that the latter
peak arises from pairs of phonons with equal and opposite,
but non-zero, wave vector arising from the Eg mode dis-
persion. Results obtained for the temperature dependences
of the Raman-active phonon mode frequencies, linewidths,
and integrated intensities, respectively, of these three com-
pounds are illustrated in Figs. 2–4 for FeF2, in Figs. 5 and 6
for MnF2, and in Figs. 7–9 for NiF2.
2.2. CoF2
Although general studies of the Raman scattering from
phonons in CoF2 have been reported previously [21–23],
the new experimental results presented here were needed
for a detailed investigation of the spin-phonon interaction.
The purplish-red-colored sample of CoF2 was prepared
from a single crystal grown at the Clarendon Laboratory,
Oxford University, especially for these spin-phonon stud-
ies and our earlier one- and two-magnon studies [23,24].
The cuboid sample of dimensions 3.2×2.0×1.7 mm was cut
to expose (001) [Z axis direction], (110) [X], and (110) [Y]
faces, respectively, and these faces were highly polished with
1 µm diamond powder. The Raman spectrum was excited
Table 1. Frequencies (in cm–1) and symmetries of the Raman-
active phonon modes at zero wave vector in transition-metal
difluorides at low temperature (~10 K)
Mode
symmetry
Frequency, cm–1
FeF2 [13] MnF2 [13] NiF2 [19] CoF2
A1g 346.8 346.9 414.5 371.2
B1g 68.8 55.9 68.7 65.2
B2g 505.6 480.4 541.0 517.4
Eg 261.7 246.6 308.0 255.3
Fig. 1. Raman spectrum of the B2g phonon mode in FeF2 recorded at
several different temperatures below and above its Néel temperature
(TN) of 78 K. The curves are offset vertically for clarity.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 1 91
M.G. Cottam and D.J. Lockwood
with 500–600 mW of Ti:sapphire laser light at 800 nm,
which avoided any optical absorption [25], analyzed with
a Spex 14018 double monochromator at a spectral resolu-
tion of (3.2 ± 0.1) cm–1 unless otherwise indicated, and
detected by a cooled RCA 31034A photomultiplier. The
sample was mounted in the helium exchange-gas space of
a Thor S500 continuous flow cryostat, where the tem-
perature could be controlled to within 0.1 K and was
measured with a gold–iron/chromel thermocouple clam-
ped to the sample. Spectra were recorded in the 90° scat-
tering geometry. The phonon features were investigated
at temperatures up to room temperature.
Fig. 2. (Color online) Temperature dependences of the frequen-
cies of the A1g, B1g, B2g and Eg Raman-active phonons in FeF2
(following [13]). For the Eg mode, the data points marked with
a cross are taken from Ref. 16. The black solid lines (1) and the
green chain lines (2) are guides to the eye for the experimental
data points and for the expected behavior in the absence of
spin-phonon coupling (see later theory), respectively.
Fig. 3. Temperature dependences of the linewidths (full width
at half maximum) of the B1g, B2g and Eg Raman-active pho-
nons in FeF2. The spectrometer resolution was 2.5 cm–1. For
the Eg mode, the data points marked with a cross are taken
from Ref. 16.
92 Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 1
Spin-phonon interaction in transition-metal difluoride antiferromagnets: Theory and experiment
The polarization dependence of the Raman spectrum of
antiferromagnetic CoF2 at low temperature is shown in
Fig. 10. The spectra show that depolarization effects due to
imperfect experimental conditions are small; see, for ex-
ample, the extreme weakness of the A1g mode appearing in
the ‘forbidden’ X(ZX)Y-polarized spectrum. Note also that
there is no sign of the B2g mode in the X(YX)Y-polarized
spectrum. This is because of the orientation of the crystal a
and b axes, which are rotated by 45○ in the X–Y plane away
from the X and Y axes. It implies that the B2g mode is now
to be expected in (XX) and (YY) polarizations (see Fig. 11).
Likewise, the B1g mode is expected in (YX) and (XY) polar-
izations. These spectra exhibit a sharp peak at 37.0 cm–1
that is the lowest lying exciton (conventionally referred to
as the magnon) of the ground state multiplet of the Co2+
ion in the exchange field: other excitons can be seen at
higher frequencies [23,24]. The one-magnon scattering is
observed only in off-diagonal polarizations. The tempera-
Fig. 4. Temperature dependences of the integrated intensities of
the A1g, B2g and Eg Raman-active phonons in FeF2. The intensi-
ties are normalized to the highest temperature in each case. For
the A1g mode, the Raman scattering intensities are shown for
(XX) [○] and (ZZ) [+] polarizations.
Fig. 5. Temperature dependences of the frequencies of the A1g,
B1g, B2g and Eg Raman-active phonons in MnF2 (following [13]).
The lines have the same meaning as in Fig. 2.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 1 93
M.G. Cottam and D.J. Lockwood
ture dependences of the spectra for these same polariza-
tions are next given in Fig. 11, where it can be seen that the
phonon peaks vary in frequency and increase in width with
increasing temperature in a range extending from well be-
low TN up to and above TN, while their peak intensities
also vary considerably.
The phonon spectra as a function of temperature and po-
larization were all readily fitted with a Gaussian/Lorentzian
line shape model [26], as shown for example in Fig. 12, and
the results obtained for the line parameters in Stokes scatter-
ing versus temperature are given in Figs. 13 to 15. In the
case of the B2g mode spectral region (see Fig. 10), an addi-
tional weak Raman band is seen at ~ 525 cm–1 at low tem-
perature and needs to be accommodated by adding an extra
band to the fit, as shown in Fig. 12. As was mentioned
earlier for the case for FeF2, this additional band is a se-
cond-order combination band arising from the Eg mode
and it is hardly noticeable for temperatures well above TN.
In this case, at 10 K the zone-center overtone of the Eg
mode is expected (from Table 1) at 2×255.3 = 510.6 cm–1,
which implies that the band at 525 cm–1 must arise from
pairs of non-zero wave vector phonons like in FeF2.
It is noticeable from Figs. 13–15 that the results obtained
for CoF2 differ in several aspects from those of the other
three rutile-structure antiferromagnets, presumably because
of the strong effect of the spin-orbit interaction in CoF2 and
through the presence of other electronic excitations (higher-
lying excitons) [22], as will be discussed later. However, the
B1g mode softening with decreasing temperature from room
temperature down to the Néel temperature TN, as shown in
Fig. 13, is analogous to that found earlier in the other transi-
tion metal fluorides (see, for example, Figs. 2 and 7) and is
ascribed to the anisotropic thermal contraction of the crystal
lattice on cooling [19,27].
Fig. 6. Temperature dependences of the integrated intensities of the
A1g [in (XX) polarization] and Eg Raman-active phonons in MnF2.
Fig. 7. Temperature dependences of the frequencies of the A1g,
B1g, B2g, and Eg Raman-active phonons in NiF2 (following [19]).
The lines have the same meaning as in Fig. 2.
94 Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 1
Spin-phonon interaction in transition-metal difluoride antiferromagnets: Theory and experiment
The phonon linewidths for the A1g, B1g, and B2g
modes (see Fig. 14) show little variation with temperature
below TN, similar to what was found for the these modes
in FeF2 and MnF2. It should be noted, however, that the
B1g mode linewidth is resolution limited at low tempera-
tures. In that regard, high-resolution Raman scattering
measurements have revealed that the true linewidth of the
Eg mode in FeF2 (NiF2) at 0 K is 0.6 (0.5) cm–1 [28]. The
Eg mode exhibits a striking increase in linewidth below
TN (note also that this line becomes very sharp at TN) that
has not been observed for the other fluorides. This anom-
alous behavior is attributed to a resonant increase in an-
Fig. 8. Temperature dependences of the linewidths of the A1g,
B1g, B2g, and Eg Raman-active phonons in NiF2 (following [19]).
The spectrometer resolution was 2.5 cm–1.
Fig. 9. Temperature dependences of the integrated intensities of
the A1g [in (XX) polarization], B1g, B2g, and Eg Raman-active
phonons in NiF2 (following [19]).
Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 1 95
M.G. Cottam and D.J. Lockwood
harmonic coupling with other lower-frequency phonons
and/or excitons consequent upon the continued sharp
increase in the phonon frequency with decreasing tem-
perature (see Fig. 13).
The integrated intensities of the various modes (see
Fig. 15) either decrease with decreasing temperature
below TN (for the A1g and Eg modes) or slightly increase
(for the B1g and B2g modes).
Fig. 10. Polarization dependences of the Raman spectrum of anti-
ferromagnetic CoF2 at 10 K. The spectral resolution was 2.4 cm–1.
The first-order Raman-active phonons are identified by their mode
symmetries.
Fig. 11. The temperature dependences at low temperatures of the
four phonon modes in the CoF2 Raman spectrum recorded in the
polarizations shown in the figures.
96 Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 1
Spin-phonon interaction in transition-metal difluoride antiferromagnets: Theory and experiment
3. Theoretical methods
Here we start with a theoretical approach similar to that
used in our earlier spin-phonon studies of the rutile
antiferromagnets [12,13,19], which all have the structure
shown in Fig. 16. For FeF2, MnF2 and CoF2 the spins are
aligned parallel and antiparallel to the crystal c axis (as
shown), but in NiF2 the spin alignment is in the ab plane
with a very slight canting away from the principal axes. In
all cases we may write the dominant inter-sublattice ex-
change part of the magnetic Hamiltonian as
Fig. 12. Fits to the CoF2 phonon spectra recorded at ~ 10 K for
various polarizations. See text for details.
Fig. 13. Temperature dependences of the frequencies of the A1g,
B1g, B2g, and Eg Raman-active phonons in CoF2. The lines have
the same meaning as in Fig. 2.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 1 97
M.G. Cottam and D.J. Lockwood
ex 1 2 3 4
,
( , , , )ij i j
i j
H J= ⋅∑ r r r r S S , (1)
where i and j denote the neighboring magnetic sites on
opposite sublattices. As well as the dominant exchange
interaction being dependent on i and j, its value is also af-
fected by the positions rn (n = 1, 2, 3, 4) of the four non-
magnetic F atoms in the unit cell (see Fig. 16) through an
Fig. 14. Temperature dependences of the linewidths of the A1g,
B1g, B2g, and Eg Raman-active phonons in CoF2. The spectrome-
ter resolution was 3.2 cm–1.
Fig. 15. Temperature dependences of the integrated intensities of
the A1g {recorded in Y(XX)Z [+], and X(ZZ)Y [○] polarizations},
B1g {recorded in X(YX)Y [○] polarization}, B2g {recorded in
Y(XX)Z [○] polarization}, and Eg {recorded in Y(ZX)Z [+] and
X(YZ)Y [○] polarizations} Raman-active phonons in CoF2.
98 Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 1
Spin-phonon interaction in transition-metal difluoride antiferromagnets: Theory and experiment
indirect process of superexchange [1,29]. In rutile-
structure materials the F atoms become displaced due to
the phonon modes, hence modulating the exchange term.
In fact, the four Raman-active modes leave the magnetic
atoms stationary, while producing displacements in rn
that can be written down following [29]. Then a Taylor-
series expansion in terms of these displacements yields a
leading-order spin-phonon coupling Hamiltonian that is
quadratic in the spin operators and linear in the phonon
operators (see, e.g., [6–11]). The effects of the weaker in-
tra-sublattice exchange [1] on the phonon modes in the
rutile antiferromagnets will be neglected here, but it is im-
portant to include the magnetic single-ion anisotropy terms
in the full Hamiltonian, since these are relatively large for
some of the materials studied here.
3.1. Phonon frequency shifts
In lowest order of perturbation theory the result for the
renormalized phonon frequency phω for any mode can be
expressed as [12,13]
0
ph ph i jω = ω + λ ⋅S S . (2)
Here 0
phω denotes the phonon frequency in the absence of
the spin-phonon coupling, and i j⋅S S is a statistical av-
erage involving the neighboring spins on opposite
sublattices. The factor λ , which may be positive or nega-
tive, takes a different value for each phonon mode and is of
second order in the spin-phonon interaction parameters
(spatial derivatives of the exchange terms) [10,11]. It is
convenient to write the frequency shift as
0 2
ph phph( ) ( ) ( )T S T∆ω ≡ ω −ω = −λ Φ , (3)
where we define an antiferromagnetic short-range order
parameter by
2
1( ) 0.i jT
S
Φ = − ⋅ >S S (4)
We next discuss a procedure to evaluate approximately
the above quantity, so that a comparison can be made be-
tween theory and the experimental data. Results are re-
quired for Φ(T) at temperatures both below and above TN,
as well as for different values of the spin S. First we note
that in a mean-field approximation Φ(T) varies from unity
at T = 0 and decreases monotonically with increasing tem-
perature as 2( / )zS S〉〈 , where zS〈 〉 is the longitudinal spin
average (proportional to the sublattice magnetization) cal-
culated using the appropriate Brillouin function. This result
is applicable also for NiF2 provided the canting effects
through an angle of order 0.4o at low temperatures (see,
e.g., [1]) are ignored and we now take the superscript z to
refer to the new direction of spin alignment.
The above approximation is good for T < TN only, be-
cause it ignores the short-range order in the antiferro-
magnet that becomes important in the vicinity of TN and
above. For this latter temperature region we make use of a
two-spin cluster method (see, e.g., Ref. 30 for a general
account), following in particular the modified form of the
constant coupling approximation for antiferromagnets due
to Elliott [31]. In the present application this allows us to
deduce that
0( 1)/( 1)i j S S z= − +⋅ −S S (5)
as T tends to TN from above. Here 0z denotes the number
of nearest neighbors on the opposite sublattice for any
magnetic site. Hence for rutile-structure antiferromagnets
where 0z = 8, we have the remarkably simple approximate
result that
Φ( ( 1)/7)NT S S= + . (6)
Thus it is predicted that the ratio )/ 0)Φ( Φ(NT decreases
with increasing spin quantum number, ranging in the
present applications from approximately 0.43 for CoF2
(S = 1/2), to 0.29 for NiF2 (S = 1), to 0.21 for FeF2 (S = 2),
and to 0.20 for MnF2 (S = 5/2). The latter two ratios (for
the larger S values) agree accurately with the values com-
puted in [12,13] without the use of Eq. (6), and thus vali-
date this new method of calculating ( )TΦ at TN and above.
Using the same approach, it is found that the temperature
dependence for the magnetic susceptibility above TN is
given by
( )
2
2
Φ N
N
T
T
T T
∝ +
. (7)
Therefore we conclude that ( )TΦ decreases monotonically
with temperature, falling at T = 2TN (for example) to a
fraction 4/9 of its value at TN.
Fig. 16. (Color online) Schematic of the tetragonal rutile unit cell
of XF2 (X = Fe, Mn, Ni, or Co), showing the dominant inter-
sublattice exchange Jij and the labeling of four F atoms relative to
the body-centered magnetic site. We have a = b ≠ c. In the case of
NiF2 the spin alignments are different (see text).
Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 1 99
M.G. Cottam and D.J. Lockwood
The general results obtained on calculating ( )TΦ from
zero temperature up to twice TN are shown in Fig. 17. We
employed mean-field theory for the estimates at T < TN, as
for [12,13], and the new two-spin cluster approach above
TN. It is evident, however, that the new results for S equal
to 1/2 and 1 are quite distinct from those for the higher
spin values, where there is a larger high-temperature “tail”.
The consequences of this for interpretation of the spin-
phonon experimental data will be discussed in Sec. 4.
3.2. Phonon Raman intensities
The analytical and numerical results for ( )TΦ are also
useful for the interpretation of the Raman scattering inte-
grated intensities. For this application the conventional
approach (see, e.g., [1,32]) is to expand the polarizability
tensor in terms of both the spin operators for the magnetic
sites and the displacement operators for the F atoms. By
analogy with studies for ferromagnetic materials [16,33],
the spin-dependent Stokes intensity due to the phonons in
the present case can be expressed as [12,19]
22 2
ph( ) ( 1) z
i jI T n A B C S = + +
⋅ +
S S . (8)
Here phn is the Bose thermal population factor for pho-
nons, while A, B, and C are magneto-optical constants.
Coefficient A provides the sole contribution to the phonon
scattering intensity in the absence of spin-phonon coupling
and the extra terms involving B and C involve the spin
averages.
At temperatures below TN in the mean-field approxima-
tion the above result can be rewritten as
( ) ( )
22 2 2
ph( ) ( 1) Φ ΦI T n A BS T C S T = + − +
, T < TN,
(9)
and so the numerical values obtained for Φ(T) as in Fig. 17
may be employed. On the other hand, at temperatures
above TN we have zS = 0, so the term proportion to C2 in
Eq. (8) vanishes.
3.3. Phonon linewidths
Finally we remark that the phonon linewidths due to
damping of the modes are more complicated to interpret
because there are various mechanisms that may occur. In
general, phonon damping in crystalline solids may arise
from three-phonon and four-phonon processes due to an-
harmonic interactions between atoms. For example, in the
three-phonon case a zone-center phonon may split into two
other phonons with equal and opposite wave vectors, giv-
ing rise to a linewidth. Phonon damping may occur also as
a consequence of scattering from any static impurities or
lattice defects in the crystal.
Calculations of the phonon damping contributions in
antiferromagnets that were specifically due to the spin-
phonon interactions were made by Cottam [11]. Processes
whereby a phonon undergoes either splitting (into two spin
waves) or confluence (by absorbing a spin wave and scat-
tering into another spin-wave state) were studied, subject to
conservation of overall energy and wave vector. Estimates
of the importance of these damping mechanisms were made
for a cubic material, but the results were mainly for acoustic
phonons. Other phonon damping calculations in magnetic
materials were reported by Wakamura [34] as part of an
experimental investigation using infrared reflectivity of the
ferrimagnet FeCr2S4. A large anomaly in the phonon damp-
ing was observed close to the Curie temperature of the mate-
rial and was attributed to spin-phonon effects near the
Brillouin-zone boundary for the phonon. This latter mecha-
nism, however, is not applicable to our case.
4. Comparison of experiment and theory
4.1. FeF2 and MnF2
The compounds FeF2 and MnF2 with spins of S = 2 and
5/2, respectively, are expected on the basis of numerical
results shown in Fig. 17 to exhibit similar variations in their
phonon band parameters with temperature. The experi-
mental results given in Figs. 2–6 show that this is indeed
generally the case.
As noted earlier, the new theoretical results obtained
here for the behavior of Φ(T) for temperatures T > TN are
identical to those found earlier following a different ap-
proach [12,13]. We have carried out a re-analysis of the
experimental data for the phonon frequencies, paying par-
ticular attention to the high-temperature “tail” (leading to
only slight differences for these two compounds), and the
estimated values for the spin-phonon interaction parameter
are given in Table 2. Here it has been convenient (in view
of comparisons to be made with the other compounds) for
the results to be scaled as λS2, as in Eq. (3).
Fig. 17. Numerical estimates for the normalized short-range
order parameter Φ(T) as a function of reduced temperature.
Curves are shown for rutile-structure antiferromagnets with spin
S = 1/2, 1, 2, and 5/2.
100 Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 1
Spin-phonon interaction in transition-metal difluoride antiferromagnets: Theory and experiment
As can be seen in Fig. 3, the phonon linewidths in FeF2 are
scarcely, if at all, affected by the antiferromagnetic ordering
for the B1g, B2g, and Eg modes. The linewidth is essentially
constant at temperatures below TN, and the same is true for
the A1g mode [12,13,16]. The results for MnF2 also show no
obvious effect of magnetic ordering on the phonon linewidths
[12,13]. A note of caution needs to be added here for the B1g
mode, which is so sharp that the data are resolution limited at
low temperatures (see Fig. 3) and thus some small spin-
phonon interaction effect might be observable at a much high-
er instrumental resolution than the one used here.
The integrated intensities of the phonon modes, howev-
er, do display variations at low temperatures consistent
with a spin-phonon coupling effect, as shown in Figs. 4
and 6. These figures show that the intensities of the A1g
and Eg modes of both FeF2 and MnF2 are considerably
enhanced by spin-phonon coupling.
4.2. CoF2
The magnetic behavior of cobalt fluoride is quite distinct
from FeF2 and MnF2, mainly because of its strong spin-orbit
interaction and the existence of a ground state multiplet of
Co2+ ion electronic states (or excitons for T < TN). The latter
property usually leads to a simplified theoretical model for
CoF2 in terms of it being a Heisenberg antiferromagnet
with effective spin S = 1/2 (see, e.g., [1,23,24]. Corre-
sponding to this lowest nonzero value for S we have the
case where the largest spin-phonon “tail” persists at T > TN
(as seen by the behavior predicted for ( )TΦ in Fig. 17).
This property makes the spin-phonon effects less obvious
to detect in CoF2, which is what we find experimentally.
Also, the estimates for the coupling coefficients λ given in
Table 2 are likely to be less precise for CoF2.
4.3. NiF2
Nickel fluoride with S = 1 is also different from FeF2 and
MnF2, but in this case it is because of its different spin align-
ment as well as the spin canting that occurs for T < TN. Instead
of lying along the crystal c axis, the spins in NiF2 lie in the ab
plane and are tilted slightly away from the principal axes. This
spin canting modifies the magnetic properties of NiF2, giving
rise to a lower “ferromagnetic” spin wave branch as well as
the usual antiferromagnetic branch found in isomorphous
FeF2 and MnF2 [35]. The spin canting is small (~0.4○ away
from the a axis), and can be ignored in our spin-phonon
analysis since it has negligible effect on the calculation of
the pairwise spin average Φ(T). However, in our earlier
work [19] we used approximate numerical values for Φ(T)
as a function of temperature, similar to those used for FeF2
and MnF2. In the present case, given the novel definitive
result for Φ(T) presented here for other values of S, new
results have been obtained for the spin-phonon coupling
constants, based on the experimental frequency dependences
on temperature shown in Fig. 7. These values are given for
the four phonons in Table 2.
Generally, we find that the phonon linewidth increases
steadily with increasing temperature above ~50 K (see
Fig. 8), as is observed in other rutile compounds. However,
for the A1g, B2g, and Eg modes a sharp rise in the linewidth
of about 20% of the 10 K value is evident at temperatures
near TN. These linewidth anomalies have not been seen in
MnF2 and FeF2. The B1g line is very sharp at all tempera-
tures, and no anomaly was detected in the linewidth at
temperatures near TN within the experimental uncertainty.
Usually, when considering anharmonic phonon decay, the
higher the phonon frequency the broader the Raman line,
as observed here for NiF2 (see Fig. 8), owing to the higher
number of available decay paths into phonons with equal
and opposite wave vector. The low frequency B1g mode
can only decay into acoustic phonon modes and hence its
extreme sharpness even at room temperature. For the pho-
nons in antiferromagnets below TN, an additional decay
channel is possible into pairs of magnons of equal and op-
posite wave vector [9,11,12]. However, in NiF2 a low-
frequency “ferromagnetic” magnetic branch exists, as men-
tioned earlier. It is thus possible that phonon decays into
this branch are responsible for the observed changes in
linewidth of the A1g, B2g, and Eg modes in NiF2 near TN.
On the other hand, the B1g mode is too low in energy to
avail itself of this extra decay path and thus exhibits no
observable anomaly.
4.4. Discussion of optical coupling coefficients
From the general expression in Eq. (8) it is apparent
that separating out the two spin-dependent contributions to
the Raman intensity is not straightforward, as the relative
weights of coefficients B and C are not known. For a par-
tial analysis we may proceed as follows. In the limit of
taking T → 0 (which implies Φ → 1) in Eq. (9), which is
valid for all T < TN, we have
( )
22 2 20I A BS C S= − + , T = 0. (10)
In the absence of any spin-phonon coupling the result
would simply be
( ) 2
0 0I A= , T = 0. (11)
Table 2. Scaled spin-phonon coupling coefficients, λS2 (in
cm–1), for each Raman-active phonon mode in the rutile-structure
antiferromagnets as deduced from the phonon frequency versus
temperature behavior
Compound Spin S λS2
for A1g
λS2
for B1g
λS2
for B2g
λS2
for Eg
MnF2 5/2 –1.9 –1.9 –1.9 1.2
FeF2 2 –1.2 –0.3 2.0 2.0
NiF2 1 –1.8 –1.8 –0.7 1.0
CoF2 1/2 1.0 –0.3 0.9 2.0
Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 1 101
M.G. Cottam and D.J. Lockwood
We obtain a third relationship by considering the form
taken by Eq. (9) when T = TN. Since Φ is given by Eq. (6)
in this case and <S z > = 0 we have
( ) ( ) ( )
2
ph
11 1
7NI T n A S S B= + − + , T = TN. (12)
Here it is understood that the Bose–Einstein thermal popu-
lation factor nph is evaluated with the relevant phonon fre-
quency at T = TN.
Of the four Raman-active vibrational modes, only the
A1g and Eg modes exhibit a strong variation with tempera-
ture below TN and thus we choose these two modes for
further analysis here. In both cases, the mode intensity for
MnF2, FeF2, and NiF2 increases with decreasing tempera-
ture below TN, while for CoF2 it decreases in intensity. On
applying the same general procedure as used for the mode
frequencies, but in this case utilizing Eqs. (10)–(12), we
find for each compound the approximate relative values of
the A, B, and C coefficients with the numerical results for
|B/A| and |C/A| being presented in Table 3.
These approximate results (which are estimated to be re-
liable to ±0.1) allow us to make several general observations
regarding the magnitudes of the coefficients and (in some
cases) their signs. First, we see that |B/A| is generally smaller
than |C/A| for MnF2, FeF2, and NiF2. This behavior indi-
cates that the stronger effect of the spin-phonon coupling
comes via the <Si·Sj> term in Eq. (8). By contrast, the B and
C coefficients for CoF2 are approximately equal for both
modes. The reason for this different behavior for CoF2 re-
mains an open question, but it may possibly be related to the
strong spin-orbit and angular-momentum effects mentioned
earlier, which are absent in the other compounds. Second,
since the mode intensity increases below TN we can say that
A and B have opposite signs for MnF2, FeF2, and NiF2, alt-
hough the sign of C remains uncertain. Third, the relative
magnitudes of the coefficient ratios for each compound are
remarkably similar for these two modes, including the rela-
tively high value found for |C/A| in NiF2.
5. Conclusions
A comprehensive theoretical analysis of the spin-
phonon coupling in rutile-structure antiferromagnets has
allowed a detailed interpretation of existing experimental
work on the four Raman-active phonons found in MnF2,
FeF2, and NiF2 and new experimental work on CoF2. Alt-
hough the absolute value for the spin-phonon coupling
coefficients can have a wide range of values between pho-
nons and as well as between compounds, it is discovered
that the spin value has a prominent scaling effect that,
when taken into account, results in parameter values that
are much closer together. From an analysis of the variation
in temperature of the A1g and Eg phonon Raman intensi-
ties, it is revealed that MnF2, FeF2, and NiF2 have very
similar behaviors for the ratios |B/A| and |C/A| of the mag-
neto-optic coefficients, while in CoF2 these ratios are ap-
proximately equal for both phonon modes. The theoretical
approach developed here can be applied to most antiferro-
magnets and it would be informative to investigate the
phonon behavior below the magnetic ordering temperature
in other series of antiferromagnetic systems to see if the
results obtained here for rutile structure compounds are
universal.
Acknowledgments
This paper is dedicated to the memory of Professor Victor
V. Eremenko (1932–2016), formerly of the B. Verkin Insti-
tute of Low Temperature Physics and Engineering and Editor
in Chief of this journal (Low Temperature Physics), whose
pioneering work on the measurement and comprehension of
the optical properties of magnetic systems continues to be of
great value to us all. MGC gratefully acknowledges partial
support from the Natural Sciences and Engineering Research
Council (NSERC) of Canada through Discovery Grant
RGPIN-2017-04429. We thank H.J. Labbé for the crystal
sample preparation and R. Radomski and J. Johnson for
curve fitting the CoF2 Raman spectra.
________
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the polarizations are as indicated earlier in the text
Compound A1g mode Eg mode
|B/A| |C/A| |B/A| |C/A|
MnF2 <0.1 0.3 <0.1 0.3
FeF2 (XX)
FeF2 (ZZ)
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<0.1
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<0.1 0.7
NiF2 0.1 0.8 0.2 0.8
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___________________________
Спін-фононна взаємодія в діфлуоридах
перехідних металів: теорія та експеримент
M.G. Cottam, D.J. Lockwood
З використанням останніх результатів експериментально
зі застосуванням спектроскопії непружного розсіяння світла
та теоретично методом модифікованого середньопольового
підходу для оцінки кореляційних функцій спінової пари про-
ведено повне порівняльне дослідження спін-фононних взає-
модій в декількох діфлуоридах перехідних металів рутилової
структури, зокрема FeF2, MnF2, NiF2 та CoF2. З метою отри-
мання оцінок коефіцієнтів спін-фононного зв’язку та віднос-
них величин коефіцієнтів магнітооптичного зв’язку предста-
влено нові експериментальні данні, які інтерпретуються в
рамках всебічного розширеного теоретичного опису.
Ключові слова: антиферомагнетики, спін-фононний зв’язок,
структура рутилу, магнітооптичний зв'язок, раманівська спект-
роскопія.
Спин-фононное взаимодействие в дифлуоридах
переходных металлов: теория и експеримент
M.G. Cottam, D.J. Lockwood
С использованием последних результатов эксперимен-
тально с применением спектроскопии неупругого рассеяния
света и теоретически методом модифицированного средне-
полевого подхода для оценки корреляционных функций спи-
новой пары проведено полное сравнительное исследование
спин-фононных взаимодействий в нескольких дифлуоридах
переходных металлов рутиловой структуры, в частности FeF2,
MnF2, NiF2 и CoF2. С целью получения оценок коэффициентов
спин-фононной связи и относительных величин коэффициен-
тов магнитооптической связи представлены новые экспери-
ментальные данные, которые интерпретируются в рамках все-
стороннего расширенного теоретического описания.
Ключевые слова: антиферромагнетики, спин-фононная связь,
структура рутила, магнитооптическая связь, рамановская
спектроскопия.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 1 103
https://doi.org/10.1002/pssb.2220660229
https://doi.org/10.1002/pssb.2220670105
https://doi.org/10.1063/1.342186
https://doi.org/10.1103/PhysRev.110.836
https://doi.org/10.1103/PhysRevB.28.1983
https://doi.org/10.1063/1.1496657
https://doi.org/10.1063/1.1496657
https://doi.org/10.1016/0038-1098(70)90264-4
https://doi.org/10.1063/1.4865560
https://doi.org/10.1103/PhysRevB.76.104406
https://doi.org/10.1063/1.1671779
https://doi.org/10.1103/PhysRevB.70.155202
https://doi.org/10.1103/PhysRevB.59.775
https://doi.org/10.1016/0038-1098(83)90835-9
https://doi.org/10.1016/0038-1098(83)90835-9
https://doi.org/10.1016/0022-3697(60)90144-X
https://doi.org/10.1016/0038-1098(89)90585-1
https://doi.org/10.1103/PhysRevB.2.1362
1. Introduction
2. Experimental results
2.1. FeF2, MnF2, and NiF2
2.2. CoF2
3. Theoretical methods
3.1. Phonon frequency shifts
3.2. Phonon Raman intensities
3.3. Phonon linewidths
4. Comparison of experiment and theory
4.1. FeF2 and MnF2
4.2. CoF2
4.3. NiF2
4.4. Discussion of optical coupling coefficients
5. Conclusions
Acknowledgments
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