Circular dichroism and Raman optical activity in antiferromagnetic transition metal fluorides
The Raman optical activity (ROA) of magnons in rutile-structure antiferromagnetic FeF₂ (TN = 78 K) has been studied as a function of temperature and applied magnetic field. For exciting light incident along the c axis, ROA is observed for magnons but not for phonons. In zero field, a small splitt...
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nasplib_isofts_kiev_ua-123456789-1216982025-06-03T16:29:05Z Circular dichroism and Raman optical activity in antiferromagnetic transition metal fluorides Hoffman, K.R. Lockwood, D.J. Yen, W.M. К семидесятилетию антиферромагнетизма The Raman optical activity (ROA) of magnons in rutile-structure antiferromagnetic FeF₂ (TN = 78 K) has been studied as a function of temperature and applied magnetic field. For exciting light incident along the c axis, ROA is observed for magnons but not for phonons. In zero field, a small splitting (0.09 cm⁻¹) of the two acoustic–magnon branches is observed for the first time by inelastic light scattering. The splitting in applied magnetic field is found to reduce with increasing temperature in accordance with theory. No ROA was detected for two-magnon excitations. In optical absorption measurements performed over thirty years ago, a very small circular dichroism (CD) was observed in the magnon sidebands of other simple rutile antiferromagnetic fluorides (MnF₂ and CoF₂). The origin of this CD was not understood at the time. The Raman studies of the one-magnon Raman scattering in FeF₂ have demonstrated that in zero field the degeneracy of the antiferromagnetic magnon branches is lifted by a weak magnetic dipole–dipole interaction, as predicted by Pincus and Loudon and by White four decades ago. The source of the observed CD in the magnon sidebands can now be traced to this same magnetic–dipole induced splitting. The precise experimental work of Professor F.L. Scarpace and Dr. Y.H. (Russ) Wong from thirty years ago is noted with appreciation. We thank Professors R. M. White and M. G. Cottam for useful discussions and the National Science Foundation for continued support spanning these three decades. 2005 Article Circular dichroism and Raman optical activity in antiferromagnetic transition metal fluorides / K.R. Hoffman, D.J. Lockwood, W.M. Yen // Физика низких температур. — 2005. — Т. 31, № 8-9. — С. 1032-1041. — Бібліогр.: 32 назв. — англ. 0132-6414 PACS: 75.30.Ds, 75.30.Gw, 75.50.Ee, 78.20.Ls, 78.30.Hv https://nasplib.isofts.kiev.ua/handle/123456789/121698 en Физика низких температур application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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К семидесятилетию антиферромагнетизма К семидесятилетию антиферромагнетизма |
| spellingShingle |
К семидесятилетию антиферромагнетизма К семидесятилетию антиферромагнетизма Hoffman, K.R. Lockwood, D.J. Yen, W.M. Circular dichroism and Raman optical activity in antiferromagnetic transition metal fluorides Физика низких температур |
| description |
The Raman optical activity (ROA) of magnons in rutile-structure antiferromagnetic FeF₂
(TN = 78 K) has been studied as a function of temperature and applied magnetic field. For exciting
light incident along the c axis, ROA is observed for magnons but not for phonons. In zero field, a
small splitting (0.09 cm⁻¹) of the two acoustic–magnon branches is observed for the first time by
inelastic light scattering. The splitting in applied magnetic field is found to reduce with increasing
temperature in accordance with theory. No ROA was detected for two-magnon excitations. In optical
absorption measurements performed over thirty years ago, a very small circular dichroism
(CD) was observed in the magnon sidebands of other simple rutile antiferromagnetic fluorides
(MnF₂ and CoF₂). The origin of this CD was not understood at the time. The Raman studies of the
one-magnon Raman scattering in FeF₂ have demonstrated that in zero field the degeneracy of the
antiferromagnetic magnon branches is lifted by a weak magnetic dipole–dipole interaction, as predicted
by Pincus and Loudon and by White four decades ago. The source of the observed CD in the
magnon sidebands can now be traced to this same magnetic–dipole induced splitting. |
| format |
Article |
| author |
Hoffman, K.R. Lockwood, D.J. Yen, W.M. |
| author_facet |
Hoffman, K.R. Lockwood, D.J. Yen, W.M. |
| author_sort |
Hoffman, K.R. |
| title |
Circular dichroism and Raman optical activity in antiferromagnetic transition metal fluorides |
| title_short |
Circular dichroism and Raman optical activity in antiferromagnetic transition metal fluorides |
| title_full |
Circular dichroism and Raman optical activity in antiferromagnetic transition metal fluorides |
| title_fullStr |
Circular dichroism and Raman optical activity in antiferromagnetic transition metal fluorides |
| title_full_unstemmed |
Circular dichroism and Raman optical activity in antiferromagnetic transition metal fluorides |
| title_sort |
circular dichroism and raman optical activity in antiferromagnetic transition metal fluorides |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2005 |
| topic_facet |
К семидесятилетию антиферромагнетизма |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/121698 |
| citation_txt |
Circular dichroism and Raman optical activity in antiferromagnetic transition metal fluorides / K.R. Hoffman, D.J. Lockwood, W.M. Yen // Физика низких температур. — 2005. — Т. 31, № 8-9. — С. 1032-1041. — Бібліогр.: 32 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT hoffmankr circulardichroismandramanopticalactivityinantiferromagnetictransitionmetalfluorides AT lockwooddj circulardichroismandramanopticalactivityinantiferromagnetictransitionmetalfluorides AT yenwm circulardichroismandramanopticalactivityinantiferromagnetictransitionmetalfluorides |
| first_indexed |
2025-11-24T02:25:34Z |
| last_indexed |
2025-11-24T02:25:34Z |
| _version_ |
1849636824624922624 |
| fulltext |
Fizika Nizkikh Temperatur, 2005, v. 31, Nos. 8/9, p. 1032–1041
Circular dichroism and Raman optical activity in
antiferromagnetic transition metal fluorides
K.R. Hoffman
Department of Physics, Whitman College, Walla Walla, WA 99362, USA
D.J. Lockwood
Institute for Microstructural Sciences, National Research Council, Ottawa, ON
K1A 0R6, Canada
E-mail: david.lockwood@nrc.ca
W.M. Yen
Department of Physics and Astronomy, University of Georgia, Athens, GA 30602, USA
Received February 17, 2005
The Raman optical activity (ROA) of magnons in rutile-structure antiferromagnetic FeF2
(TN = 78 K) has been studied as a function of temperature and applied magnetic field. For exciting
light incident along the c axis, ROA is observed for magnons but not for phonons. In zero field, a
small splitting (0.09 cm–1) of the two acoustic–magnon branches is observed for the first time by
inelastic light scattering. The splitting in applied magnetic field is found to reduce with increasing
temperature in accordance with theory. No ROA was detected for two-magnon excitations. In opti-
cal absorption measurements performed over thirty years ago, a very small circular dichroism
(CD) was observed in the magnon sidebands of other simple rutile antiferromagnetic fluorides
(MnF2 and CoF2). The origin of this CD was not understood at the time. The Raman studies of the
one-magnon Raman scattering in FeF2 have demonstrated that in zero field the degeneracy of the
antiferromagnetic magnon branches is lifted by a weak magnetic dipole–dipole interaction, as pre-
dicted by Pincus and Loudon and by White four decades ago. The source of the observed CD in the
magnon sidebands can now be traced to this same magnetic–dipole induced splitting.
PACS: 75.30.Ds, 75.30.Gw, 75.50.Ee, 78.20.Ls, 78.30.Hv
1. Introduction
In 1965, the discovery of the one-magnon sideband
in the visible absorption spectrum of MnF2 [1] and
the two-magnon absorption of the far-infrared spec-
trum of FeF2 [2] spurred many investigations into the
optical properties of antiferromagnetic insulators. The
theoretical framework regarding these two processes
was developed by Tanabe et al. [3] and Allen et al. [4]
In these materials, coupling between the various in-
trinsic excitations of the lattice, i.e., excitons, mag-
nons and phonons, leads to the formation of coopera-
tive transitions which can account for much of the
spectral structure. An article discussing this work ap-
pears elsewhere in this special issue of Low Tempera-
ture Physics. In addition to absorption and lumines-
cence measurements, magnon lines were also observed
in Raman scattering measurements [5,6]. Raman scat-
tering measurements permit the physical properties of
magnons to be studied without the additional contri-
butions due to electronic states associated with optical
emission or absorption. A comprehensive review of
Raman theory and experimental results for antifer-
romagnets is presented in the book by Cottam and
Lockwood [7].
The magnetic character of the magnon transitions
led to the utilization of circularly polarized light to
explore the dichroic properties of these materials. In
the early 1970’s, a series of papers reported on their
magnetic circular dichroism (MCD) in an applied
magnetic field [8–10]. The authors reported at that
time the observation of circular dichroism (CD) in
© K.R. Hoffman, D.J. Lockwood, and W.M. Yen, 2005
certain magnon sidebands of MnF2 [8] and CoF2 [10]
but not in FeF2 [9]; CD occurs in the absence of an
applied magnetic field and corresponds to zero field
MCD though the origins of the dichroism may arise
from different sources. The authors were able to elimi-
nate experimental artifacts as the source of the ob-
served CD, but were not able at that time to identify
the physical basis for the unexpected observation. Re-
cently, we revisited this problem and showed that the
source of the CD is to be found in the dipolar interac-
tions that exist between the antiferromagnetic sub-
lattices, which render them inequivalent [11].
Raman optical activity (ROA) characterizes the
difference in the Raman line intensity when excited
with right and left circularly polarized light. The ear-
liest experiments to combine Raman spectroscopy
with chirality in solids were performed in the 1970’s
on CdS under resonance excitation conditions [12].
Attempts were also made to observe a similar effect in
antiferromagnetic fluoride materials but the experi-
mental sensitivity was insufficient. At about the same
time ROA was also measured in a chiral molecule
[13]. The chiral structure of some molecules makes
them naturally suited to couple more strongly to ei-
ther right or left circularly polarized light. The theo-
retical [14] and experimental developments in this
field have been reviewed in the literature [15,16]. A
series of Raman papers in the 1980’s on the magnetic
properties of dilute magnetic semiconductors employed
the use of circular polarization to isolate spin flip tran-
sitions of Mn ions [17,18]. The improvement in detec-
tor sensitivity over a period of twenty years afforded us
the opportunity in the 1990’s to revisit the question of
ROA in antiferromagnetic insulators. We were able to
observe ROA in the one-magnon line of FeF2 [19–21]
and several phonon lines in rutile structure materials
[22,23]. In addition, we demonstrated for the first time
that Raman scattering could be used to measure a zero
field splitting of the magnon branches in antifer-
romagnetic FeF2 [24]. These results were then utilized
to reinterpret the CD measurements completed in the
early 1970’s that were alluded to above [11].
In the following Sections, we present our current
understanding of the dichroic properties of antifer-
romagnetic insulators possessing the rutile structure.
The focus of this review will be on ROA measurements
to explore the magnetic properties of these materials.
We start with a summary of the zero field dichroism
measurements in MnF2, CoF2, and FeF2 because the
CD studies prompted one of us (WMY) to explore the
use of ROA in antiferromagnetic fluorides. We then
present a more thorough review of the ROA measure-
ments, techniques, and analysis in FeF2. Here we
mention first the work on phonons before focusing on
the one- and two-magnon scattering results, including a
revised assessment of the results from the temperature
dependence and zero magnetic field studies of FeF2.
2. Optical dichroism in rutile antiferromagnets
Dichroism in a material is defined as the difference
in the absorption coefficient of a medium for two re-
lated polarizations; thus CD arises from the difference
in the absorption for right- and left-circularly polar-
ized radiation. When these differences are induced by
the application of an external magnetic field, the ef-
fect is termed MCD. In both CD and MCD, a unique
direction is necessary in order to define the sense of
circulation in the circularly polarized light; this is
provided by some unique crystalline axis or by the di-
rection of the applied magnetic field. Materials show-
ing CD are normally called «optically active» and
such crystals are relatively rare, whereas MCD is
quite common in paramagnetic systems possessing a
unique axis. MCD is conjugated to Faraday rotation
by the Kramers–Kronig relations and constitutes the
resonant rather than the dispersive phenomena in the
transformation.
The compounds that we wish to discuss here are the
common difluorides, MnF2, CoF2, and FeF2, all of
which possess the rutile crystal structure. Below the
Neel temperature, TN, these materials are describable
as simple two sublattice antiferromagnets and in prin-
ciple they are ideal systems on which to study magnet-
ically dichroic phenomena. The ground states of ions
in each sublattice as well as the collective magnetic
excitations (magnons) are identical to each other in
every respect except their polarization. The sole re-
maining energy degeneracy, which will respond to an
applied magnetic field, is that of the two sublattices;
hence, the origin of magnetic dichroism in these sys-
tems is directly related to the lifting of this energy
equivalence both in the electronic (exciton), the spin
(magnon) or the combined (magnon sideband) systems.
For simple antiferromagnets in their ordered state,
the ground state of each sublattice is non-degenerate
and is identical to the other except for their sense of
magnetization with respect to the unique crystalline
axis. If the state for the «up» or A sublattice is given
by |L,M� then the corresponding «down» or B
sublattice is represented by |L,–M�. For pure elec-
tronic transitions, the appropriate dipolar transition
operators for circular polarized radiation transform as
L±+ 2S± where the (+) and (–) subscripts refer to
right and left circular polarizations, respectively. If
ions in one of the sublattices have a non-vanishing
transition probability between two states for one of
these operators, then the ions in the opposite sub-
lattice will have an identical element for the corre-
Circular dichroism and Raman optical activity in antiferromagnetic transition metal fluorides
Fizika Nizkikh Temperatur, 2005, v. 31, Nos. 8/9 1033
sponding conjugated operator. In the absence of a
magnetic field, the sublattices are degenerate in en-
ergy and no CD should be observed. However, weak
interactions between the sublattices, such as magnetic
dipole–dipole interactions, can produce a small aniso-
tropy between the two sublattices that results in CD.
At low temperatures, a very small zero-field CD signal
was observed in two of the three compounds investi-
gated and it is the origin of these signals that will be
addressed in this paper.
3. Experimental results for circular dichroism
measurements
A finite CD was observed at low temperatures in
two of the three difluorides studied. Signals were
observed in the magnon sidebands of the 6A1g ground
state to 4T1g and ( 4
1A g, 4Eg) absorptions of MnF2
and of the 4T1 to 2A, 2T1 (2P) of CoF2; no CD signals
were observed in FeF2. As noted earlier, the origins of
CD and MCD are different; as a consequence, CD can
be subtracted from MCD in order to determine the
magnetic field induced splittings. When this is done
the, the residual or pure MCD signal can be fitted
accurately with a differential line shape; this indicates
that in our cases the MCD arise simply from the lifting
of the sublattice degeneracy, as per our expectation.
Figure 1 shows the absorption of a magnon side-
band of the (2A, 2T1) transition of CoF2 along with
the observed CD and MCD in a 2 T applied magnetic
field. The CD is also purely differential in shape and
is estimated to be an additive intrinsic splitting of the
order of �� = 0.015 cm–1. The CD in CoF2 becomes
unobservable in the vicinity of 10 K.
Figure 2 shows the absorption spectrum of the
so-called �1 and �2 sidebands accompanying the
ground state to 4T1g transition in MnF2. These transi-
tions are �-polarization active. A small CD signal is
observed that is clearly related to this transition. The
zero field splittings for the sidebands were determined
to be ��(�1) = 0.07 cm–1 and ��(�2) = –0.05 cm–1,
respectively.
The negative sign in ��(�2) signifies that the sense
of the splitting in this sideband is out of phase with
that of �1. Such a change in the sense of splitting is also
observed when uniaxial stress is applied to MnF2 along
the [110] direction in the basal plane. The behavior of
the CD was studied as a function of basal plane stress
allowing the intrinsic strain in the basal plane of or-
dered MnF2 to be estimated: � �� � � �d/d 12 10 5. .
This result highlights the advantages of chiral tech-
niques for exploring interactions in antiferromagnets.
FeF2 was studied in the 21500–28500 cm–1 spectral
region. The observed transitions without exception
exhibited MCD signals consistent with a simple
sublattice splitting and with a linear dependence on
magnetic field. No CD signals were encountered in any
of the �-active transitions investigated. The sensitivity
of our apparatus places an upper limit of 10–3 cm–1 on
any splitting that might occur in this compound at 4 K.
1034 Fizika Nizkikh Temperatur, 2005, v. 31, Nos. 8/9
K.R. Hoffman, D.J. Lockwood, and W.M. Yen
22800 22850 22900
–2
–1
0
1
2
�, cm–1
0,0
0,5
1,0
1,5
T=4K
Absorption
CD
H=0
CoF2
(
)
L(
)
R
(
)
(
V
)
Fig. 1. The CD spectrum of the one-magnon sideband near
22,800 cm–1 of CoF2 at 4 K and the corresponding absorp-
tion spectrum of the sideband.
18 470 18 480 18 49018 460
( )
�1
�2
0
C
ir
cu
la
r
d
ic
hr
oi
sm
+ (
)
(
)
�, cm–1
–
–
Fig. 2. The absorption and CD spectra of the so-called �1
and �2 sidebands of MnF2 at 2 K; these traces connect the
observed CD to the magnon sidebands.
4. Raman optical activity
Raman optical activity is observed experimentally
in the difference in a Raman line intensity when ex-
cited by right and left circularly polarized light, re-
spectively. The circular intensity difference (CID) is
normally used to quantify ROA [14]:
� �
� �
� �
�
I I
I I
R L
R L
, (1)
where I R
� and I L
� represent the scattered light inten-
sity with � polarization due to incident right (R) or
left (L) circularly polarized light. The normalization
term permits a simple comparison of data from differ-
ent spectroscopic systems. One major difficulty of
ROA measurements is the sensitivity of the technique
to intrinsic or extrinsic depolarization effects. Care
must be taken to account for any birefringent effects
in the steering and focusing optics. Additionally, the
crystal itself can contribute to significant depolariza-
tion effects that appear as CID in spectra that nor-
mally would not be expected to produce any differ-
ence. We will discuss this effect in more detail in the
next Section of the paper.
Fundamentally, the source of Raman scattering is
the modulation of the induced polarizability of a me-
dium by fundamental excitations in the medium. To
couple the incident light to the medium, its suscepti-
bility is expanded in terms of phonon or magnon ope-
rators. This results in a modulation of the polarizability
at the frequency of the fundamental excitation. In the
case of magnons, the inclusion of spin operators re-
sults in a change of the overall angular momentum
component along the z axis: �mj = ±1. Figure 3 illus-
trates the important mechanism for magnon light scat-
tering from antiferromagnetic fluorides. The energy
levels shown in the picture belong to the transition
metal ions on the two sublattices. These transitions
obey the same selection rules that contribute to CD or
MCD signals. The circularly polarized light couples to
different sublattices in the Raman process as well,
thus providing a way of distinguishing between the
two sublattices. The figure shows energy levels for
both sublattices in the medium and illustrates that left
and right circular polarizations create magnons on dif-
ferent sublattices.
Symmetry arguments can be applied to determine
selection rules for the various scattering processes and
are summarized in the form of Raman scattering ten-
sors. The scattering tensors for the tetragonal antifer-
romagnetic fluorides are presented in Table 1.
The tensors have been transformed so that they are
appropriate for [R, L, z] incident polarizations and
[x, y, z] scattered polarizations. The Raman tensors
describe the scattered electric field amplitudes and
thus the scattered light intensity in a specific polariza-
tion is calculated through multiplication of the tensor
by its conjugate value. To describe the expected CID
spectrum, we need to calculate the difference between
two intensity terms. For example, the A1g phonon
mode should not generate a CID signal for scattered
light polarized along the x axis:
� x g
x
R
x
L
x
R
x
L
A
I I
I I
a a
a a
( ) .1
2 2
2 2
0�
�
� (2)
Circular dichroism and Raman optical activity in antiferromagnetic transition metal fluorides
Fizika Nizkikh Temperatur, 2005, v. 31, Nos. 8/9 1035
J=S
J=S –1
S–1
S
J= –S
J=S+1
– S+1
–S
s_ z
s+ z
Fig. 3. The electronic levels of the magnetic ions in
antiferromagnetically ordered rutile materials responsible
for the observed ROA signal. The two sets of lines corre-
spond to the two antiferromagnetic sublattices. In both
cases a magnon is generated during the scattering process.
Table 1. The Raman tensors for circularly polarized incident light that is analyzed for linearly polarized light scattered at
90 degrees.
A g
a a
ia ia
b
1
1
2
0
0
0 0 2
� �
�
�
�
�
�
�
�
B g
c c
ic ic1
1
2
0
0
0 0 0
� �
�
�
�
�
�
�
�
B g
ie ie
e e2
1
2
0
0
0 0 0
�
��
�
�
�
�
�
�
3 4
1
1
1
2
0 0 2
0 0 2
0
� � � �
�
�
�
�
�
�
�
�
�
�
� �
�
�
�
�
� � � � �
i
( ) ( )
�
�
�
�
�
Eg d
id id
( )1
1
2
0 0 0
0 0 2
0
�
�
�
�
�
�
�
�
�
Eg
d
d d
( )2
1
2
0 0 2
0 0 0
0
�
�
�
�
�
�
�
�
In the case of the magnon scattering we can calcu-
late the CID for each sublattice separately. In the case
of the �3
+ band, we obtain the following CID expres-
sion:
� �z ( )
( * * * ) ( * * * )
( * *
3
2 2 2 2
2
�
� � � � � � � � � � � �
� � � � * ) ( * * * )
*
( * )� � � � � � � �
� �
� �
�
2 2 2
2 2
2
. (3)
Likewise for the �4
+ magnon band we get a similar
result:
� �z ( )
*
( * )4
2 2
2
�
� �
� � (4)
The negative sign reflects the coupling of left circu-
lar polarization to this particular sublattice. Notice
that if the energies of the two magnon bands are equal
and the line profiles equivalent, the total CID is the
sum of these terms and will be zero. The application of
a magnetic field lifts the degeneracy of the magnon
branches. As a result, the CID of the two branches
will no longer completely cancel. This result is analo-
gous to the differences observed between MCD and
CD spectra. The important distinction is that the
Raman spectra are determined only by the energies of
the magnon bands with no interference from the ex-
cited states of the ion.
5. Raman optical activity experiments
The experimental arrangement, shown in Fig. 4,
used the standard 90° Raman scattering geometry. The
incident beam was the 488-nm line of a continu-
ous-wave Spectra Physics 166 argon–ion laser. The
focused laser power density at the sample was
150 kW/cm2, well below the damage threshold of the
materials used. A Conoptics 370 electro-optic modula-
tor (EOM) was used to alternate between right and
left circular polarization. The scattered light was dis-
persed by a Spex 1401, 0.85-m double monochromator
and detected using a cooled RCA 31034-A02
photomultiplier. The signal was measured using a
Stanford SR-400 dual-gated photon counter and
stored on a computer. The signals counted on the two
gates, corresponding to the two incident polarizations,
were stored separately, permitting both IR – IL and
IR+ + IL to be easily calculated. Low temperatures
were achieved using a Cryosystems closed-cycle refrig-
erator. A permanent magnet assembly located inside
the refrigerator applied a constant magnetic field par-
allel to the crystal c axis. The magnets were drilled
through so that the incident laser beam could be fo-
cused in the crystal co-linear with the applied field
and allowed to pass on through the refrigerator.
The crystals used in this study were the rutile struc-
ture fluorides FeF2 and MgF2. These materials are
uniaxial, with FeF2 undergoing an antiferromagnetic
phase transition at TN = 78 K. The MgF2 crystal ori-
entation was determined from x-ray Laue-diffraction
measurements before the cuboid sample was cut (to
within a 1° accuracy) and polished. The c-axis orienta-
tion for the FeF2 sample was found using a
polariscope. In addition, a polished crown glass cube
was used as a reference material to check for instru-
mental artifacts.
Rayleigh and Raman CID in MgF2 and FeF2
Our measurements of the magnon and phonon CID
in FeF2 generated several signals that were difficult to
understand initially. Figure 5 shows the difference
spectrum IR – IL as well as the separate IR and IL
spectra from FeF2 with no applied magnetic field.
What is curious about these measurements is that the
magnon exhibits no apparent differential scattering
but the B1g phonon mode does. When the spectrum is
enlarged to encompass all of the phonon modes and
the Rayleigh line, we observe a similar CID spectrum
for the A1g and B2g modes and the Rayleigh line but
the sign of the difference signal is opposite to that of
the B1g mode. Referring back to the Raman tensors, no
CID is expected from any of the phonon modes nor
should Rayleigh scattering exhibit CID. MgF2 is also a
rutile structure fluoride, but it does not exhibit any
magnetic ordering. It showed a similar differential scat-
1036 Fizika Nizkikh Temperatur, 2005, v. 31, Nos. 8/9
K.R. Hoffman, D.J. Lockwood, and W.M. Yen
Fig. 4. The experimental setup for the ROA measure-
ments. The laser light was directed vertically through the
sample to align the beam path with the entrance slit of
the spectrometer.
tering pattern. An isotropic glass sample was used to
check for instrumental artifacts and none were found.
To explain the CID spectra of the phonons we
looked carefully at the effects of birefringence in the
crystal on the scattered intensity. Light propagating
along the c-axis will experience an isotropic index of
refraction so that no change in the polarization occurs
as it moves through the sample. A slight misalignment
results in distinct indices of refraction parallel and
perpendicular to the plane containing the c axis. As a
result, circularly polarized light becomes elliptically
polarized as it moves through the sample. The changes
in polarization affect the relative amplitude of the x
and y components of the incident light. The crucial
point is that changes of Ix and Iy are different for left
and right circular polarization so that any scattering
that depends on only one of these terms will exhibit a
CID spectrum. This result permits a very sensitive tool
for aligning the samples for ROA measurements so that
intrinsic birefringence effects can be eliminated.
Raman optical activity of Magnons in FeF2
Figure 6 presents the conventional Raman and ROA
spectrum of the one-magnon line of FeF2 in an applied
magnetic field. The modest applied field, B < 0.6 T, is
too small to observe the splitting of the magnon
branches within the magnon line width using conven-
tional (linearly polarized light) Raman techniques.
However, the magnon line exhibits a clear CID spec-
trum. To determine the energy difference between the
two branches we fit the data using offset Gaussian line
shapes to describe the one magnon scattering peak. In
this manner we can calculate the energy difference as a
function of temperature. Figure 7 shows the tempera-
ture dependence of the frequency splitting between
the two magnon branches in FeF2. As the temperature
increases, the energy splitting between the branches
decreases, while the linewidth increases resulting in a
larger uncertainty for the energy splitting between the
branches.
Circular dichroism and Raman optical activity in antiferromagnetic transition metal fluorides
Fizika Nizkikh Temperatur, 2005, v. 31, Nos. 8/9 1037
IR
IL
40 50 60 70 80
Energy, cm–1
3000
2000
1000
0
–1000
In
te
n
si
ty
(c
o
n
ts
)
IR IL–
FeF2
magnon
B1g phonon
Fig. 5. The ROA spectrum of FeF2 at 4 K. The presence of
a CID spectrum in the B1g phonon mode indicates a mis-
alignment of the incident laser and the crystalline c axis.
40 45 50 55 60 65
–400
0
400
800
1200
1600
2000
FeF2
28 K
B = 0.59 T
Frequency shift, cm –1
Conventional
Raman
CID
R
am
an
in
te
n
si
ty
,a
rb
. u
n
its
Fig. 6. The low temperature ROA spectrum of FeF2 in the
presence of an applied magnetic field. The upper trace
shows the conventional Raman spectrum. The lower trace
is the measured CID spectrum.
0 20 40 60 80
0.0
0.4
0.8
1.2
1.6
FeF 2
B = 0.59/0.55 T
Temperature, K
Theory
Expt.M
ag
n
o
n
sp
lit
tin
g
, c
m
–
1
Fig. 7. The temperature dependence of the magnon branch
splitting obtained from ROA measurements in FeF2. The
theoretical points are calculated using applied fields of
0.59 (filled diamonds) and 0.55 T (filled circles). The er-
ror bars on the theoretical points derive largely from the
uncertainty in the applied magnetic field strength.
Theoretically, [28] the energy splitting between
the two branches is given by
��
�( ) [ ]| |T g J HB� 2 (5)
where H is the applied magnetic field along the crys-
tal c-axis, �|| is the parallel susceptibility per spin
site, µB is the Bohr magneton, J is the dominant near-
est neighbor exchange, and g is the Lande g factor. At
low temperatures, �|| is nearly zero so:
��
( )T g HB� �0 2 (6)
whereas at temperatures near TN, �|| = g
B/2J so:
��
( )T T g HN B� � . (7)
This theory predicts that the splitting between the
branches should change by a factor of two when go-
ing from low temperature to the phase transition tem-
perature. Using parameters obtained from the experi-
ment or the literature the theoretical curve shown in
Fig. 7 was obtained. The theoretical curve accurately
describes the temperature variation of the energy
splitting between the magnon bands, but overall
there is a general offset between the measured data
and the theoretical curve. The primary uncertainty
for this measurement arises from determining the ac-
tual value of the magnetic field inside the refrigera-
tor. The magnetic field of 0.59 T for the permanent
magnet gap was measured at room temperature. The
magnet will be at a much lower temperature during
the Raman measurement, but the affects of the lower
temperature on the gap field strength is not known.
These measurements highlight the sensitivity of
this technique in measuring magnon branch splittings.
The line widths of the two peaks at 62 K were greater
than 10 cm–1, but the separation between the peaks
was only 0.4 cm–1.
For a simple antiferromagnetic, we would not ex-
pect to observe any CID spectrum in zero applied field
ROA measurements. However, careful measurements
revealed a small but repeatable CID spectrum in FeF2.
Figure 8 shows the CID spectrum and the separate I R
and I L Raman signals of FeF2. Care was taken to elim-
inate all of the depolarization effects by using the Ray-
leigh scattering spectrum to ensure alignment of the
crystal c-axis with the incident laser light. The solid
line fit determined the zero field splitting between the
two magnon branches to be �� = (0.09 ± 0.02) cm–1.
To our knowledge this result is the first measurement
of a zero field splitting of the magnon branches in
antiferromagnets using Raman scattering. Again the
high sensitivity and resolution of this technique is
highlighted in obtaining this result.
Finally, ROA measurements of the two-magnon
scattering peak in FeF2 showed a null result. The lack
of any CID signal confirms that the dominant mecha-
nism for two-magnon scattering involves exchange
coupled magnons on opposite sublattices. Other possi-
ble sources of two-magnon scattering are higher order
processes that would be expected to be much weaker
than the one-magnon scattering. Multiple spin-flip
transitions have been observed in dilute magnetic
semiconductors in the paramagnetic phase under reso-
nance excitation conditions. These transitions were
observed in a forward scattering geometry where the
scattered light was analyzed for the circular polariza-
tion as well. The intensities of the two spin flip signals
are much lower than the single spin flip signals consis-
tent with a higher order interaction.
6. Magnetic dipole–dipole interactions
In 1963 Loudon and Pincus [25] examined the ef-
fect of the classical dipole–dipole interaction between
magnetic moments on the spin wave spectrum of a sim-
ple uniaxial antiferromagnet. In the absence of an ap-
plied magnetic field, as discussed above, the spin wave
branches are degenerate in these systems. The magnon
dispersion relation is given by
� ���
H H H H b kA A E E
/2 2 2 2 1 22 2 ] (8)
where � = gµB is the gyromagnetic ratio, HA is the
uniaxial single-ion internal anisotropy field and HE is
the effective exchange field; k is the wave vector and
b = az–1/2 with a the nearest neighbor distance and z
the number of nearest neighbors. On inclusion of the
1038 Fizika Nizkikh Temperatur, 2005, v. 31, Nos. 8/9
K.R. Hoffman, D.J. Lockwood, and W.M. Yen
44 48 52 56 60
0
2
4
6
IR
IL
CID
Frequency shift, cm –1
FeF2
26 K
B = 0
R
am
an
in
te
n
si
ty
,a
rb
. u
n
its
Fig. 8. The zero-field ROA spectrum of the one magnon
line in FeF2. The solid line is a fit modeling the small
CID apparent in this spectrum.
dipolar fields, Eq. (8) splits into two modes with their
dispersions given by:
� �1
2 2 2 2 1 22 2�
[ ]H H H H b kA E A E
/ , (9)
� � � �2
2 2 2 2 1 22 4�
[ ( )( sin )]H H H b k H MA A E E s k
/
(10)
where Ms is the sublattice magnetization and �k is the
angle between the easy axis (z) of magnetization and
the direction of propagation of the spin wave, k. Only
the frequency of the second mode is affected by the
dipolar interaction. For spin waves with k = 0,
Eqs. (9) and (10) reduce to
�� �1
1 22�
[ )]H H HA A E
/ (11a)
� � � �2
2
0
1 22 8�
[ ( sin )]H H H MA A E s
/ . (11b)
In the case of longitudinal spin waves, �0 = 0, and
there is no dipolar effect. For the transverse spin
waves, however, �0 = �/2, and the dipolar effects are
at a maximum.
Subsequently, Harris [26] has shown that Eq. (10) is
an approximate result and has derived the full expres-
sion for the spin wave dispersion in the presence of a
dipolar field. However, the difference between Eq. (10)
and the complete result only becomes important when
4�Ms is comparable to HE, which is not the case here.
The Loudon and Pincus calculation is based on a
molecular field approach that neglects the affect of
the Lorentz field of one sublattice upon the other
sublattice. When this effect is included, White [27]
showed that the resonant frequencies for k = 0 are
given by:
�( ) ,k � �0 12
� � �
�
�
�
�
�
�
�
�
�
�
� � �H H M H H M NA A s A E s x
/
8
3
2 4
1 2
( ) ,
(12a)
�( ) ,k � �0 3 4
� � �
�
�
�
�
�
�
�
�
�
�
� � �H H M H H M NA A s A E s y
/
8
3
2 4
1 2
( )
(12b)
and for k ! 0 by
� � �( ) ( ),k H H H HA A E E k12
2 2 22 1� �
�
�
�
�
�
8
3
1
1 2
� �M H Hs A E k
/
[ ( )] , (13a)
� � �( ) ( ),k H H H HA A E E k3 4
2 2 22 1� �
�
�
�
�
�
�
�
�
�
�
�
8
1
3
12
1 2
� � �M H Hs k A E k
/
sin [ ( )] (13b)
where N is the demagnetization dyadic [27], and �k is
defined as (1/z) �� exp(ik•�), where the �’s are the
vectors to the z nearest neighbors.
Notice that for the uniform spin precession (k = 0)
in a sphere (Nx = Ny = Nz = 1/3), the Lorentz field
just cancels the surface demagnetizing field. There-
fore, there is no shift in the resonant frequency as com-
pared to non-dipolar cases. The symmetric occurrence
of Nx and Ny in Eq. (12) is due to the fact that the
normal modes labeled 1, 2 and 3, 4 are characterized
by net transverse magnetizations that are linearly po-
larized in the x and y directions, respectively. Notice
also that although the 1, 2 spin wave modes are inde-
pendent of � they are shifted by the Lorentz field.
Since the samples used in the optical studies had
rectangular cross sections in the basal plane, there
will be dipolar splittings of the magnons. Thus, from
Eq. (12),
�� � �
�
k
s A x y
A E A
M H N N
H H H
� � �
0 1 3
2
4
2
( )
. (14)
The demagnetizing factors are given by integrals
over the surface [27]. In order to obtain estimates of
these splittings we approximate Nx and Ny by 1/2
and 1/4, respectively, based on the sample dimen-
sions. The parameter values used in the calculations
and the resulting splittings are listed for MnF2, CoF2,
and FeF2 in Table 2.
7. Comparison of theory with experiment
Optical circular dichroism
The calculated splittings given in Table 2 are quite
small compared with the magnon energies in these
compounds. Given the nature of the natural line
widths, splittings of this magnitude are difficult to
measure and they have been resolved only recently
using Raman spectroscopy [11,15].
The calculated values for �� in MnF2 and CoF2
(see Table 2) are of similar magnitude to those found
experimentally for the magnon sidebands in the visi-
Circular dichroism and Raman optical activity in antiferromagnetic transition metal fluorides
Fizika Nizkikh Temperatur, 2005, v. 31, Nos. 8/9 1039
ble region: 0.02 cm–1 for CoF2 and 0.07 and 0.05 cm–1
for MnF2. Thus the origin of the observed splittings
reported in the CD measurements is well encompassed
by the effects of the magnetic dipole–dipole interac-
tion in these simple rutile antiferromagnets. The size
of the splittings detected by spontaneous magnetic
CD cannot be compared directly with those calculated
in Table 2, as the processes involved in the formation
of magnon sidebands are not as simple and direct as
those encountered in Raman scattering. For MnF2, for
example, the sideband involves a combination of an
exciton on one sublattice with essentially zero disper-
sion [25] with a magnon on the other with nearly 50
cm–1 dispersion; k conservation requires the creation
of exciton–magnon pairs with equal and opposite
wave vectors from throughout the Brillouin zone. Al-
though the splitting is mostly uniform across the
Brillouin zone, the CD is, in effect, sampling the de-
rivative of the joint density of states. It follows that
an accurate calculation of the CD requires the use of
Eq. (13), but the theory needs to be extended to in-
clude specific symmetry points in the Brillouin zone
and an appropriate summation over the entire zone.
Conceivably, another possible cause of CD in these
compounds could be a slight tilting of the spins away
from their ordered positions along the c-axis. How-
ever, at low temperatures and in zero applied magnetic
field, all experimental results to date point to collin-
ear spin alignment in all of these compounds [29].
Raman circular intensity difference
In the Raman equivalent of the CD measurements
discussed above, a �� of (0.09 ± 0.02) cm–1 was ob-
served in FeF2 at low temperatures using a 90° scatter-
ing geometry. From Eq. (11), the theoretical splitting
for magnons propagating at approximately 45° to the c
(z) axis is �� = 0.15 cm–1, which considering the ex-
pected large uncertainty in Ms is in good agreement
with the experimental results. Including the Lorenz
field correction, Eq. (14) gives �� = 0.065 cm–1 (see
Table 2), which is in even closer agreement with
experiment. The zero-field splitting of the magnon
branches observed in FeF2 is thus attributed to the
magnetic dipole–dipole interactions that can be im-
portant for long wavelength magnons [27].
Another way to quantitatively describe the interac-
tion is to define it in terms of an effective anisotropy
field, H", that is treated in the same manner as an exter-
nal magnetic field. In this approach the energy of the
magnons shows up in much the same form as presented
in Eq. (6) except the splitting is now due to H".
�� �( ) ' .T H� � � �0 2 0 09 1cm (15)
Table 2. Magnetic parameters at low temperature for vari-
ous transition metal fluorides possessing the rutile struc-
ture. The calculated dipole–dipole induced splittings,
��k=0 and ��k = �/2, are also given.
Parameter
MnF
2
[28–30]
FeF
2
[29,31]
CoF
2
[29,32]
Effective spin S 5/2 2 1/2
H
E
(T) 51.5 53.313 37.6
H
A
(T) 0.84 19.745 9.54
g 2.00 2.25 2.80
� (cm–1/T) 0.934 1.05 1.31
M
S
(T) 0.060 0.056 �0.06
��
k = 0
(cm–1) 0.016 0.065 0.083
��
k = �/2
(cm–1) 0.02 0.08 0.12
Using the parameter value � = 1.05 cm–1/T (Table 2)
we calculate the anisotropy field to be: H" = (0.04 ±
0.01) T.
By comparing Figs. 8 and 6 we see that the CID
spectra have opposite signs for the case of H = H" and
H = Happlied + H", respectively. Since the energy split-
ting of the two magnetic sublattices switches sign
with the external field, the direction of the applied
field must be opposite to the direction of the effective
anisotropy field. Going back to the temperature de-
pendence of the magnon branch splitting we can now
correct for the presence of the effective anisotropy
field in the material. Equation (5) becomes
��
�( ) [ ]( ')| |T g J H HB� �2 applied
� 2 0 55[ ]( . ).| |g JB
� T (16)
The second theoretical curve shown on Fig. 7 uses
this new value for the magnetic field and it gives
better agreement with the measured data.
8. Conclusions
The magnon sideband CD observed in rutile struc-
ture antiferromagnets can be quite readily explained
as being the result of magnetic dipole–dipole interac-
tions breaking the degeneracy of the two spin wave
branches. These optical measurements performed three
decades ago are seen now to represent the first obser-
vation of such splittings, as first predicted by Loudon
and Pincus in 1963. It would be informative to per-
form further experiments to confirm the sin2� depend-
ence of the splitting by varying the angle �0 from 0 to
�/2, i.e., the incident light direction is varied from
along to perpendicular to the easy axis, and also the
sample shape effect on the splitting.
1040 Fizika Nizkikh Temperatur, 2005, v. 31, Nos. 8/9
K.R. Hoffman, D.J. Lockwood, and W.M. Yen
The ability to measure an effective anisotropy field
in FeF2 highlights the sensitivity of ROA measure-
ments for determining small splittings between the
magnon branches in antiferromagnets. The measure-
ment of such a splitting due to magnetic dipole–dipole
interactions in FeF2 has permitted a final resolution of
questions concerning the origins of CD in optical mea-
surements completed over thirty years ago. The high
sensitivity of ROA also permits the study of anti-
ferromagnetic branch splittings for low applied mag-
netic fields.
The precise experimental work of Professor F.L.
Scarpace and Dr. Y.H. (Russ) Wong from thirty years
ago is noted with appreciation. We thank Professors
R. M. White and M. G. Cottam for useful discussions
and the National Science Foundation for continued
support spanning these three decades.
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