Hybrid excitations due to crystal-field, spin-orbit coupling and spin-waves in LiFePO4
Yuen Yiua , Manh Duc Leb , Rasmus Toft-Petersonc , Georg Ehlersd , Robert McQueeneya , and David Vaknina
a
DMSE, Ames Laboratory and Department of Physics and Astronomy,
Iowa State University, USA
b
ISIS Neutron and Muon Source,
Rutherford Appleton Laboratory, UK
c
Helmholtz-Zentrum Berlin f¨
ur Materialen und Energie, Germany
d
Quantum Condensed Matter Division,
Oak Ridge National Laboratory, USA
(Dated: August 4, 2016)
We report on the spin waves and crystal field excitations in single crystal LiFePO4 by inelastic
neutron scattering over a wide range of temperatures, below and above the antiferromagnetic transition of this system. In particular, we find extra excitations below TN = 50 K that are nearly
dispersionless and are most intense around magnetic zone centers. We show that these excitations
correspond to transitions between thermally occupied excited states of Fe2+ due to splitting of the
S = 2 levels that arise from crystal field and spin-orbit interaction. These excitations are further
amplified by the highly distorted nature of the oxygen octahedron surrounding the iron atoms.
Above TN , magnetic fluctuations are observed up to at least 720 K, with an additional excitation
broad in energy is observed around ± 4 meV, likely caused by single-ion splittings through spin-orbit
and crystal field interactions. The latter weakens slightly at 720 K compared to 100 K, which is
consistent with calculated cross-sections using a single-ion model. Our theoretical analysis, using
the MF-RPA model, provides both detailed spectra of the Fe d− shell and estimates of the average ordered magnetic moment and TN . By applying the MF-RPA model to a number of existing
spin-wave results from other LiM PO4 (M = Mn, Co, and Ni), we are able to obtain reasonable
predictions for the moment sizes and transition temperatures.
I.
INTRODUCTION
Various members of the lithium-orthophosphates have
gained renewed attention in the last decade as candidates
of electrodes in lithium based rechargeable batteries1–4
by virtue of their high Li-ion conductivity through channels that are present in their olivine crystal structure1,5,6 .
Thus, although most research efforts have recently been
focused on their electro-chemical properties, they have
long been known to exhibit intriguing magnetic properties. In particular, the transition-metal based lithiumorthophosphates display strong magnetoelectric (ME) effect, where an applied magnetic field induces an electric
polarization and vice-versa, applied electric field induces
magnetization. Naturally, the ME effect is invoked by the
coupling among orbital, magnetic, and electrostatic degrees of freedom, that have been at the forefront of recent
research in condensed matter physics. Prominent examples of such coupling have been found in the iron- and
copper-based unconventional superconductors7,8 , the giant magnetoresistance in Mn-based oxides9,10 , or the
magnetoelectric effect in transition metal oxides11,12 .
Often, interrelated magnetic, electric and/or structural
transformations raise the question of their origin, but it
is generally accepted that the spin-orbit coupling (SOC)
drives a secondary transition, namely magnetic or structural that follows structural or magnetic respectively.
The SOC can also lead to higher order asymmetric exchange terms such as the Dzyaloshinskii-Moriya interaction and can lift the degeneracy of crystal field levels. In the case of the lithium-orthophosphates, it has
been observed that the relative strength of the ME effect correlates with the effective total orbital moment
with LiCoPO4 (L = 3) and LiMnPO4 (L = 0) displaying
the largest and smallest ME coefficients13–16 . Here, we
report on the magnetic excitations of LIFePO4 and expand on recent elastic and inelastic neutron studies of the
lithium-orthophosphates17–19 . Similar to other lithiumorthophosphates, LiFePO4 possesses an intricate magnetic structure, where the Fe2+ moments (S = 2; L = 2)
order antiferromagnetically at TN = 50 K with moments
pointing mainly along the b-direction20,21 . However, subsequent studies have revealed zero field spin canting along
a and c, which are both forbidden by Pnma symmetry,
hinting that the crystal structure symmetry might be
lower than Pnma below TN 22 . The observed spin-canting
implies the presence of Dzyaloshinsky-Moriya (DM) interactions, which can be linked to the ME response. The
local symmetry of the magnetically ordered Fe sublattice
may also be reflected by the crystal field level structure
of Fe2+23 .
Several inelastic neutron scattering (INS) efforts have
measured the spin wave spectrum in LiFePO4 in restricted regions of the Brillouin zone22,24 . The most
comprehensive model for LiFePO4 deduced from these
measurements includes 5 exchange interaction terms
(1 in-plane nearest neighbour, 2 in-plane next-nearestneighbours, and 2 out-of-plane interactions), and 2
single-ion anisotropy terms (along a and c). In this study
we complete the INS picture for spin waves measured
along all directions a, b, and c and compare our results
with the existing Hamiltonian. We also report on new

2
low-energy excitations found below the spin waves excitations, which persist from below TN up to 720 K. We
argue that such excitations are due to single-ion splitting
of the S = 2 manifold from the crystal field, spin orbit
and ordered moment exchange field. Modifications of the
existing spin Hamiltonian model will include these hybrid
excitations that account for those new excitations.
II.
EXPERIMENTAL DETAILS
For the inelastic neutron scattering experiment, a single crystalline sample of LiFePO4 was used. The crystal
was selected from a batch of single crystals synthesized
using the standard flux growth method, using the same
recipe prescribed by Ref. 22. The high quality single crystal weighs approximately 200mg, and its structure and
stoichiometry were confirmed by laboratory X-ray and
by single crystal neutron diffraction22 . Inelastic neutron
scattering (INS) data were collected at the Cold Neutron
Chopper Spectrometer (CNCS)25 of the Spallation Neutron Source, Oak Ridge National Laboratory. The sample was aligned with the bc-plane horizontal with some
detector coverage along the a (vertical) direction. The
incident neutron energy was set at Ei = 12 meV for the
optimal resolution and flux for the (Q, ω) region of relevant interest. INS data was collected at three separate
temperatures T = 35, 100, and 720 K.
III.
RESULTS AND DISCUSSION
Figure. 1(a-c) shows contour plots of the INS data collected at T = 35 K along the (H10), (0K0), and (00L)
reciprocal space directions, with an incident neutron energy of Ei = 12 meV and integrated over a range of
|Q⊥ | of ±0.25 reciprocal lattice units. Along (0K0) two
spin-wave branches are clearly visible, as expected by the
known antiferromagnetic structure, which contains spins
precessing perpendicular to the moment direction b22 :
One branch where the two oppositely aligned spins precess in-phase, and one branch where the two precess outof-phase. Also shown in Fig. 1 are dotted lines obtained
from linear spin wave calculations using the existing spin
Hamiltonian described below22 .
1X
H=
Jij Si · Sj + HSIA
(1)
2 i,j
The Hamiltonian considers 5 exchange interaction terms,
among them are one nearest neighbor Jbc , two next nearest neighbors Jb and Jc , and two out-of-plane interactions
Jab and Jac . Previous reports based on single crystal
data have found magnetic anisotropy along a and c20,24 ,
and as a result the Hamiltonian requires the inclusion
of single-ion anisotropy (SIA) terms Da and Dc , allowing for an anisotropic hard plane while the easy axis b
direction is set by Db = 0. For the linear spin wave calculation shown here, the J and D values from previously
published inelastic single crystal data of the same crystal
are used22 . These values are listed in Table I.
TABLE I. Exchange interactions Jij and single ion anisotropy
Da and Dc (in meV) used in calculating the spin-waves shown
in Fig. 1. Hmf is calculated from the Jij ’s in the mean-field
approximation
Fig.1(a-c) Fig. 1(d-f)
Ref. 22 this study
Jbc 0.77 (2)
0.46 (2)
Jb 0.30 (2)
0.09 (1)
Jc 0.14 (2)
0.01 (1)
Jab 0.14 (2)
0.09 (1)
Jac 0.05 (1)
0.01 (1)
Da 0.62 (1)
0.86 (2)
Dc 1.56 (4)
2.23 (2)
Hmf
4.84
3.69
A.
Zeeman splitting
As illustrated in Fig.1(a-c), the linear spin wave calculations (white dotted lines) track the measured magnon
dispersion closely, however it does not account for the extra excitations visible near 4.5 meV indicated by the red
arrows. These extra excitations are nearly dispersionless
in energy and centered around the magnetic zone centers. We argue that these extra excitations arise from
transitions from a thermally populated excited state of
the spin S = 2 multiplet that are not considered by
spin wave theory, which is a T = 0 theory and maps
the spin ground state to a vacuum state and transitions
from (to) this state to (from) the nth excited state to
creation (annihilation) of n magnons. The 4.5 meV excitation corresponds to a transition between two excited
states, n = 1 and n = 2, as illustrated schematically in
Fig. 2. We thus use the mean-field random-phase approximation (MF-RPA)26 to calculate the higher temperature
magnetic spectrum.
Fig.2 shows schematically how the different singleion interactions splits the d-electron energy levels. The
largest energy splitting, of the order of an electron-volt, is
from the crystal field, which only acts on the orbital angular momentum, leaving the spin states degenerate. In
the lithium orthophosphates, the Fe2+ ions are located
at a low (monoclinic Cs ) symmetry site, so the crystal
field splitting lifts all orbital degeneracy resulting in five
orbital singlets, which each have a five-fold spin degeneracy. The on-site spin-orbit interaction is much weaker
than the crystal field interaction in transition metals and
only splits the spin states by a few meV . Finally, in the
ordered phase, the ordered moments generate a internal
magnetic field which further Zeeman splits the spin energy levels.
The effects of the spin-orbit and crystal field interactions on the S = 2 levels can be parameterised by

3
FIG. 1. (color online) (a-c): Contour plots of energy transfer from the inelastic neutron scattering (INS) data collected at
T = 35 K along (H10), (0K0), and (00L). The dashed lines are calculated linear spin wave dispersions based on the model
described in the text. The red arrows indicate extra excitations visible around 4.5 meV that corresponds to Zeeman splitting
levels by internal mean field induced by the ordered moments. (d-f): Virtual INS data calculated based on the same model
using mean-field random-phase approximation. As can be seen the simulation correctly predicts the hybrid extra excitations
near 4.5 meV in addition to an extra less intense excitation near 3 meV, both originating from internal-Zeeman splitting.
the single-ion anisotropy parameters Da and Dc , whilst
the Heisenberg term leads to a Zeeman-like interaction,
in the mean-field (MF) approximation. Thus we can
rewrite equation 1 in the MF-approximation as an effective single-ion Hamiltonian:
(1)
Hmf = HSIA + HZeeman
(2)
where, after choosing the moment direction as the quantisation axis, that is z||b,
HSIA = Da Sˆx2 + Dc Sˆy2
HZeeman = −Hmf Sˆz
(3)
(4)
where Da , Dc and Hmf (which depends on the Jij parameters) are given in Table I. Diagonalising this Hamiltonian in the |Sz i basis with the Zeeman term results
in the level scheme shown in Table II for the ordered
(AFM) phase, whereas setting Hmf = 0 gives the level
scheme shown for the paramagnetic phase. Furthermore,
inspection of the results of diagonalising the Hamiltonian
with different values of Da , Dc and Hmf shows that when
Hmf > (Da + Dc ), the difference in energy between the
n = 1 and n = 2 levels is ∆12 = Hmf + (Da + Dc )/2.
The parameters given in Table I are determined by fitting the measured data at 35 K using the MF-RPA as im-
TABLE II. Calculated level scheme due to the crystal field
and spin-orbit interactions, parameterized by the single-ion
parameters Da = 0.86, Dc = 2.23 meV, in the paramagnetic
phase and in the magnetically ordered phase with an additional Zeeman splitting term parameterised by Hmf = 3.69
meV.
Energy
(meV)
| Sz i components
Paramagnetic Phase
n=0
0
0.67
√ (| − 2i + | + 2i) + 0.32 |0i
1 0.81
1/√2 (| − 2i − | + 2i)
2 3.39
1/√2 (| − 1i + | + 1i)
3 7.49
1/ 2 (| − 1i − | + 1i)
4 7.79 −0.23 (| − 2i + | + 2i) + 0.95 |0i
Ordered Phase
n=0
0
0.99 | + 2i + 0.12 | 0i
1 8.00 0.97 | + 1i + 0.25 | − 1i
2 12.73 0.80 | 0i + 0.60 | − 2i
3 16.21 0.80 | − 2i + 0.60 | 0i
4 16.45 0.97 | − 1i + 0.25 | + 1i
plemented in the program M cP hase27–29 within the data
analysis environment provided by the program Horace30 .
In order to find starting parameters for the fit, we
generate 105 random sets of exchange Jij and single-ion
anisotropy Dα parameters and then use the previously

4
FIG. 2. (color online) Multi-electron states arising from the
crystal field splitting, each of these orbital singlets has a fivefold spin degeneracy. The ground state multiplet is further
split as indicated by the blue lines, either by spin-orbit coupling or internal magnetic field when the system is magnetically ordered.
reported 2 K data and linear spin-wave model22 to filter out parameter sets which do not reproduce the low
temperature dispersion. This procedure produces several
sets of parameters which differ from the published set by
having smaller Jij but larger Dα parameters, which although having slightly higher χ2 values still fit the 2 K
data well. This is because the calculated spin-wave bandwidth and energy gap are governed by both the Jij and
Dα parameters in a complex fashion, so that the same
bandwidth and gap can result from smaller Jij and larger
Dα parameters. However, as Hmf is determined by the
exchange parameters Jij , and has a larger effect on the
energy ∆12 of the excited state transition, weaker Jij parameters fit the 35 K measurements better, as the Hmf
from the parameters of Ref. 22 overestimates the energy
of the excited state mode.
Furthermore, the MF-RPA predicts the physical properties better, since the calculated transition temperature
in the mean-field approximation is TNmf = 73 K for the
parameters of Ref. 22 compared to TNmf = 62 K for the
set obtained from the fit of the neutron spectrum at 35 K,
as presented in Fig.1(d-f). The calculated spectrum gives
four excitation branches at 35 K rather than the three observed. The fourth mode, calculated to have an energy
of 3.7 meV, corresponds to an excited state transition
from the n = 2 to the n = 4 level, and is expected to
have one fifth the intensity of the 4.5 meV excitation
(from n = 1 to n = 2), with a calculated cross-section
of 7 mb/sr/Fe2+ compared to 36 mb/sr/Fe2+ . Thus it is
probably too weak to be observed experimentally.
Inspection of the fitted exchange parameters show that
the nearest-neighbour superexchange interaction Jbc be-
tween Fe2+ ions via a single oxygen ligand is more dominant than previously thought, with Jbc /Jb ≈5 (this
study) rather than Jbc /Jb ≈2.5 (previous study22 ). Furthermore, Jb and Jab have similar magnitudes and are
both larger than Jc and Jac which are extremely weak
but non-zero. These next nearest neighbour interactions
are mediated by superexchange via two oxygen ligands,
but whilst Jc and Jac couple Fe2+ ions lying in octahedra
which are tilted in opposite directions giving Fe-O-O-Fe
bond angles closer to 90◦ , the FeO6 octahedra linked by
Jb and Jab are tilted in the same direction, resulting in
more obtuse bond angles and hence greater overlap for
the oxygen p-orbitals.
However, a potential shortfall of modeling the spinorbit and crystal field interactions using the effective SIA
parameters Da and Dc is that it ignores higher order and
odd-component crystal field terms which are permitted
here because of the low symmetry of the crystallographic
site occupied by the Fe2+ ions. Using the full crystal field
Hamiltonian,
i
HCF
= λLi · Si +
k
X X
Bkq Okq ,
(5)
k=2,4 q=−k
in place of HSIA requires many more parameters, including on-site spin-orbit coupling, ζ, and the crystal field
parameters Bkq . We choose to restrict λ=12.75 meV to
the free-ion value determined by optical spectroscopy and
atomic calculations31,32 , and use the point-charge model
to determine the Bkq parameters, including charges within
a cutoff range of 3.3 ˚
A from the Fe2+ ions. The point
charge model has known shortcomings, such as not accounting for charge transfer or bonding effects, but allows
us to reduce the number of parameters to three: the effective charges on each ligand atomic species. Starting
parameters are obtained by requiring the point charges
to approximately reproduce the energy level scheme in
Table II. Fitting the 35 K inelastic dataset with this new
point charge model, we obtained qO =-1.87(3) |e| for the
oxygen ligands, qP =0.58(3) |e| for the phosphorous and
qLi =0.04(2) |e| for the lithium. Thus, it seems that the
flowing Li ions contribute very little to the anisotropy
of the iron spins, as may be expected, whilst the largest
effect is due to the distorted oxygen octahedron. The
calculations does give the b direction as the easy direction, but overestimates the ordered moment µcalc =4.67
µB compared to a measured value of 4.09(4) µB 22 .
More importantly, the calculated inelastic neutron
spectrum is virtually identical to that shown in Fig. 1(df) using the SIA parameters Da and Dc . Thus, the higher
order crystal field terms appear to have little effect on
the dispersion or intensities of the spin waves or excited
state transition. This can be understood by reference
to Table.II, where we see that the ground state (n = 0)
and the first excited state (n = 1) are both nearly pure
states, that is the n = 0 state is 99% |Sz = +2i and
the n = 1 state is 98% |Sz = +1i, so the wavefunctions
and hence the transition matrix elements which deter-

5
FIG. 3. Energy transfer at (010) and (001) for T = 35 and
100 K. The integrated Q range is ±0.25 along each direction.
The red arrows indicates the extra excitation observed near
4 meV for 35 K along both directions. The thick red bar
represents approximately the instrumental resolution width.
mine dispersion and scattering intensity are dominated
by the ordered phase Zeeman field.
TABLE III. Calculated and reported moment directions and
sizes for LiM PO4 . Calculations for Co33 , Ni17 , and Mn19 are
based on reported values of exchange parameters. Note that
for small amount of canting is present in LiFePO4 but has
been omitted here22 .
M Calc. moment Exp. moment
(a, b, c) (in µB ) (a, b, c) (in µB )
Fe
(0, 4.67, 0) (0, 4.09, 0)22,34
Co
(0, 3.87, 0)
(0, 3.35, 0)35
Ni (0.41, 0, 2.80) (0.3, 0, 2.2)17
Mn (5.00, 0, 0)
(4.29,0,0)19,36
Nonetheless, the point charge model does show us that
the b easy-axis direction and anisotropic hard plane can
be explained by the distorted geometry of the oxygen octahedron surrounding the Fe2+ spins. Indeed, applying
the same point charge model as fitted to the inelastic
neutron scattering data from LiFePO4 to other lithiumorthophosphates satisfactorily reproduces the measured
ordered moment directions in LiM PO4 , although the
magnitude of the ordered moments are overestimated
by MF-RPA. Table III provide a list of magnetic moment sizes and orientations calculated using the point
charge model, with comparison to the reported values
for LiM PO4 (M =Fe, Co, Ni, and Mn).
B.
Spin-orbit splitting
Fig.3 shows the energy responses of INS intensity
around (010) and (001) at T = 35 and 100 K. In order to
correctly analyze the energy response, the negative energy transfer portion (i.e. neutron energy gain) of the
FIG. 4. (a): Measured energy transfer at (010) for T = 100,
296 and 720 K. The dashed lines show the fitted Lorentzian
centered at E = 0 and a Gaussian centered near 4 meV. The
solid colored lines show the sum of the fitted Lorentzian and
Gaussian curves. (b): Calculated energy transfer at (010) for
the same temperatures.
data have been processed using the principle of detailed
balance37 . The spectrum at T = 35 K, which is below
TN and in the spin ordered phase, shows well defined
peaks that correspond to the spin wave branches. The
red arrows indicate the extra excitations observed near
4.5 meV as shown above. Above TN = 50K the system
becomes paramagnetic and both the magnon dispersions
and the hybrid 4.5 meV excitation disappear.
Fig.4(a) shows a closer look of the INS energy responses at higher temperatures 100, 296, and 720 K. At
these temperatures the spectra are relatively broad in
energy, which includes a Lorentzian component centering at 0 meV and a Gaussian component centering near
4 meV, all represented as dashed lines in Fig.4(a). The
Lorentzian response is expected of magnetic fluctuations
that are quasi-elastic in nature and short-lived in time,
however, there is little change in intensity and width over
a temperature range many times TN .
Broad excitations are detected around ±3.4(4) meV
at 100 K and ±4.2(7) meV at 720 K, determined by the
fitted Gaussian curves as represented by the two dashed
Gaussian lines in Fig.4(a). Since the internal fields generated by the ordered moments should be absent at these
temperatures, these excitations cannot be the same excited state excitation discussed above, seen at 35 K. However, as shown in Table II, in the paramagnetic phase
the spin-orbit and crystal field interactions combined still
splits the S = 2 levels, for instance the energy for a n =
2 to 0 transition is 3.39 meV, which may be what is observed at 100 K. At 720 K, we should expect that all the

6
excited states have become populated, allowing higher
transitions to have greater spectral weight, and shifting
the observed peak to a higher energy. However, the data
is much broader than the instrumental resolution, and
MF-RPA theory we have used to analyse the low temperature data does not account for thermal broadening
of the excitations. Instead we have convoluted the calculation with Gaussian curves with a full width at half
maximum of 1.2 meV to obtain the curves in figure 4,
whereas instrumental resolution at these energy transfers
is expected to be less than 0.5 meV. It can be seen that
this still does not fully account for the measurements.
In addition, the calculations predict that the intensity
should fall off much faster with increasing temperature
than is observed: the calculate intensity ratio of the peak
area at 720 K vs 100 K is 0.35, which is half that observed (I720K /I100K ≈ 0.7). Thus, whilst the energies of
the peaks observed at high temperatures may be satisfactorily explained by the MF-RPA theory, their intensity
and linewidths needs a more sophisticated approach.
Fe2+ due to splitting of the S = 2 levels that arise from
the effects of the crystal field and spin-orbit interactions
amplified by the highly distorted nature of the oxygen octahedron surrounding the iron spins. Above TN the magnetic fluctuations are observed to relatively high temperatures, with little temperature dependence between 100
and 720 K. Additional excitations, broad in energy, are
observed around ± 4 meV that are due to the single-ion
splittings caused by the spin-orbit and crystal field interactions. These excitations weaken slightly at 720 K
compared to 100 K, consistent with the calculated crosssections from our single-ion model. Our theoretical analysis using the MF-RPA model provides detailed spectra
of the d−shell in LiFePO4 and also enables estimates of
the average ordered magnetic moment and TN . Applying
it to spin-wave results of other members of the LiM PO4
(M = Mn, Co, and Ni) compounds provides reasonable
ordered moments and transition temperatures showing
the approach is robust.
V.
IV.
SUMMARY
In summary, we present a inelastic neutron scattering study in LiFePO4 that complements a previous spin
wave excitations study. In particular, we find an extra
excitation at 4.5 meV at T = 35 K < TN that is nearly
dispersionless and is most intense around magnetic zone
centers. We show that these excitations correspond to
transitions between thermally occupied excited states of
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