Table Of ContentAnharmonic Phonons and Magnons in BiFeO
3
O. Delaire1, M.B. Stone1, J. Ma1, A. Huq1, D. Gout1, C. Brown2, K.F. Wang3, and Z.F. Ren3
1. Oak Ridge National Laboratory; Oak Ridge TN 37831 USA
2. NIST Center for Neutron Research; Gaithersburg MD 20899 USA
3. Department of Physics; Boston College; Boston MA 02467 USA
(Dated: January 12, 2012)
Thephonondensityofstates(DOS)andmagneticexcitationspectrumofpolycrystallineBiFeO
3
weremeasuredfortemperatures200≤T ≤750K,usinginelasticneutronscattering(INS).Ourre-
sultsindicatethatthemagneticspectrumofBiFeO closelyresemblesthatofsimilarFeperovskites,
3
such as LaFeO , despite the cycloid modulation in BiFeO . We do not find any evidence for a spin
3 3
2
gap. A strong T-dependence of the phonon DOS was found, with a marked broadening of the
1
whole spectrum, providing evidence of strong anharmonicity. This anharmonicity is corroborated
0
bylarge-amplitudemotionsofBiandOionsobservedwithneutrondiffraction. Aclearanomalyis
2
seenintheT dependenceofBi-dominatedmodesacrosstheN´eeltransition. Theseresultshighlight
n the importance of spin-phonon coupling in this material.
a
J PACSnumbers: 63.20.kk,75.30.Ds,75.85.+t,78.70.Nx
1
1
I. INTRODUCTION II. NEUTRON DIFFRACTION
]
i
c
A stoichiometric mixture of Bi O (99.99%, Aldrich)
s Multiferroic materials exhibiting a strong magneto- 2 3
- andFe O (99.99%,Aldrich)wasball-milledfor10hours
rl electric coupling are of great interest for potential appli- [22]. T2he3resulting powder was hot-pressed at 900◦C for
cations in spintronic devices and actuator systems [1, 2].
t
m BiFeO (BFO) is one of the few known systems exhibit- 5min in a half-inch diameter graphite die, with a 2ton
3 force applied, and a heating rate of 300◦C/min. The
. ing simultaneous magnetic and ferroelectric ordering at
t pressedpelletswereannealedinvacuumat750Kfor24h.
a T > 300K, and as such is a strong candidate for ap-
m plications [1–3]. BFO crystallizes in a rhombohedrally- The total sample weight was about 15g.
- distorted perovskite structure (space group R3c) [4–7]. Neutron diffraction measurements were performed us-
d The Bi lone-pair is thought to be responsible for the off- ing the POWGEN time-of-flight diffractometer at the
n centering of Bi atoms, which induces the ferroelectricity, Spallation Neutron Source (SNS), at Oak Ridge Na-
o
with a high Curie temperature T (cid:39)1100K [2]. The Fe tional Laboratory. The sample was placed inside a
c C
[ ions, inside oxygen octahedra, carry large magnetic mo- thin-wall vanadium container, and loaded in a radia-
ments (cid:39) 4µB [2], and order antiferromagnetically (AF) tive vacuum furnace. Data were collected at T =
3 below the N´eel temperature, T (cid:39) 640K, with some 300,473,573,623,723,773K. The time-of-flight diffrac-
v N
canting of the spins, and a long-period cycloid modu- tometer uses a broad incident spectrum of neutrons, and
3 lation [2, 8, 9]. Unravelling the behavior of phonons was configured with a center-wavelength λ = 1.066˚A.
7
8 and magnons, and their interactions, is crucial to un- The diffraction data were refined with GSAS [23], us-
3 derstanding and controlling multiferroic properties [1]. ing the R3c space-group. The fits indicated good crys-
0. Phonons couple to the ferroelectric order, and magnons tallinityandgoodsamplepuritywith∼98%BiFeO3 and
1 to the magnetic order, and it is expected that phonons ∼2%ofasecondaryphase, indexedasFe2O3, atalltem-
1 and magnons strongly interact in a system exhibiting peratures. For T < 640K, the data were refined with a
1 simultaneous ferroelectric and antiferromagnetic order, G-typeantiferromagneticorder,withoutcycloidmodula-
: such as BFO [1]. This interaction also gives rise to mix- tion. Since the inclusion of the G-type AF order did not
v
i ing of the excitations, resulting in electromagnons, for change the refinements significantly, it is likely that the
X example [1, 2]. To our knowledge, there are currently furtherincorporationofthecycloidstructurewouldonly
r no reported experimental data of the full magnon and have a minimal effect on our results. Results are sum-
a
phonon spectra in BFO, however. Multiple Raman mea- marizedinTableI.ThediffractiondataandRietveldfits
surements have been performed, but these only probe for T = 300K and 773K are shown in Fig. 1 (for one of
modes at small wavevectors q → 0 [10–20]. Here, we three detector banks). We observe a constant intensity
report the first INS measurement of the phonon DOS ratio for (104) and (110) reflections at all temperatures
in polycrystalline BiFeO , as a function of T, as well measured, and thus our data do not support the R3c–
3
as more detailed measurements of the magnon spectrum R3m transition reported in Ref. [25].
than previously reported [21]. From these data, we ex- The refined lattice parameters and atomic positions
tract the exchange coupling constant of BFO, and we areingoodagreementwithpriorreports[5,6]. Thetem-
identify a strong anharmonicity of the phonons, provid- perature dependences of the lattice constants and mean
ing evidence for strong spin-phonon coupling. square thermal displacements (squares of quantities in
2
TABLE I: Results of Rietveld refinements of time-of-flight neutron diffraction data for BiFeO (R3c).
3
T χ2 x a,b c rmsU rmsU rmsU rmsU rmsU rmsU rmsU
Fe2O3 Bi⊥c Bi(cid:107)c Fe⊥c Fe(cid:107)c O,long O,mid O,short
(K) (%) (˚A) (˚A) (˚A) (˚A) (˚A) (˚A) (˚A) (˚A) (˚A)
300 1.285 2.0 5.5752 13.8654 0.0977 0.0819 0.0694 0.0722 0.1038 0.0899 0.0698
473 2.306 2.0 5.5862 13.9013 0.1291 0.1035 0.0914 0.0919 0.1303 0.1148 0.0956
573 2.151 1.9 5.5930 13.9222 0.1455 0.1163 0.1008 0.1014 0.1458 0.1261 0.1058
623 2.099 2.0 5.5966 13.9332 0.1527 0.1189 0.1046 0.1088 0.1531 0.1327 0.1094
723 1.714 1.7 5.6033 13.9513 0.1645 0.1301 0.1153 0.1170 0.1658 0.1437 0.1189
773 1.749 1.7 5.6065 13.9594 0.1718 0.1353 0.1190 0.1225 0.1704 0.1493 0.1264
Table I) are shown in Fig. 2. The mean-square displace- 3.55meV, in up-scattering mode (excitation annihila-
ments were refined with an anisotropic harmonic model tion) [27]. In these conditions, the energy resolution was
for all atoms, which assumes Gaussian probability dis- ∼0.12meV at the elastic line, increasing to ∼1.2meV at
tributions for atom positions. The atomic mean-square 25meV neutron energy gain. For DCS measurements,
displacements are largest for Bi, followed by O. Our re- the sample was encased in the same Al can as in the
sults for thermal displacements are also in good agree- ARCS measurements. The empty Al sample container
ment with Ref. [6], although we find displacements that wasmeasuredinidenticalconditionstothesampleatall
are larger for Bi than for Fe, both along the (trigonal) c- temperatures, and subtracted from the data. DCS mea-
axisandinthe(basal)a,bplane. ForBiandFevibration surements at T =200,300,470,570K were performed in
modes, the behavior of (cid:104)u2(cid:105) is linear in T, as expected a high-temperature He refrigerator, and measurements
in the high-T regime of an harmonic oscillator (from the at T =570,670,720K were performed in a radiative fur-
theoretical partial phonon DOS reported in [36], the av- nace.
eragephononenergiesforBiandFevibrationsare12and
28meV, respectively equivalent to 140K and 320K). In
thisregime, theamplitudeofvibrationsdoesnotdepend
A. Magnetic Spectrum
on the mass, but only on an effective force-constant, K,
(cid:104)u2(cid:105)∝T/K [24]. Linear fits to the Bi and Fe data indi-
Figure 3 shows the orientation-averaged scattering
catethattheeffectiveK forBimotionsinthe(a,b)plane
function, S(Q,E), obtained with ARCS, as a function
isabouthalfoftheforce-constantforBimotionsalongc,
of temperature (Q and E are the momentum and energy
which itself is comparable to those for either types of Fe
transfer to the sample, respectively). At T =300K, the
motions. We note that our fits did not use anharmonic
data for Q < 4˚A−1 clearly show steep spin-waves, em-
displacement models, which could slightly affect the re-
anating from strong magnetic Bragg peaks at Q = 1.37
sults. The results for (cid:104)u2(cid:105) of oxygen atoms in Fig. 2-c
and 2.62˚A−1, and extending to E ∼70meV. This range
shows a departure from linearity around 600K for the
of energies overlaps with much of the phonon spectrum
short and long axes of the thermal ellipsoids. This effect
(see below). The intensity of the magnetic Bragg peaks
mayberelatedtothemagnetictransitionatT =640K,
N decreases with increasing T, vanishing for T ≥ 670K,
although there could also be effects of phonon thermal
in good agreement with the reported T = 640K. The
occupations, since the average energy of O vibrations is N
magnetic scattering nearly vanishes for Q > 6˚A−1, ow-
45meV, corresponding to 520K [36].
ing to the magnetic form factor of Fe3+ ions. The high-
E, optical part of the spin-waves is strongly damped for
T = 470,570K, well below T . The low-E part of the
N
III. INELASTIC NEUTRON SCATTERING spin-waves is also clearly seen in the DCS data in Fig. 4-
a,b, corresponding to sharp vertical streaks emanating
INS spectra were measured using the ARCS direct- fromthemagneticBraggpeaks. Theacousticpartofthe
geometrytime-of-flightchopperspectrometerattheSNS spin-waves shows some Q-broadening with increasing T
[26]. In the ARCS measurements, the sample was en- below T . Magnetic correlations remain for T > T ,
N N
cased in a 12mm diameter, thin-walled Al can, and but are much broader (Figs. 3/4-c,d). We note that the
mounted inside a low-background furnace for measure- orientation-averaged spin-waves show a strong similarity
mentsatT =300,470,570,690,750K.Allmeasurements between BiFeO and other AFeO perovskites, such as
3 3
wereperformedunderhighvacuum. Anincidentneutron ErFeO , LaFeO and YFeO [28–30].
3 3 3
energyE =110meVwasusedandtheenergyresolution An important question is whether low-energy spin-
i
(fullwidthathalfmax.) was∼3meVat80meVneutron waves are present in BFO, and whether an energy gap
energy loss, increasing to ∼7meV at the elastic line. exists, potentially associated with an anisotropy in the
AdditionalINSmeasurementswereperformedwiththe exchange coupling, or single-ion anisotropy. Figure 5-b
Disk Chopper Spectrometer (DCS) at the NIST Cen- shows the magnetic scattering intensity, S (E), mea-
mag
ter for Neutron Research, with incident energy E = suredwithDCS,integratedoverthespin-wavedispersing
i
3
(a) 13.96
300K (cid:129)) 13.92
c ( 13.88
5.60
(cid:129))
b ( 5.58 (a)
a,
Bi a,b plane
0.025
2(cid:129)) BFei ca-,abx ipslane
e ( 0.020 Fe c-axis
F
Bi, 0.015
or
d-spacing (Å) 2u> f 0.010 (b)
<
0.005
(b)
773K
O, long
0.025
O, mid
2(cid:129)) 0.020 O, short
O (
or 0.015
> f
2u 0.010
< (c)
0.005
300 400 500 600 700
T (K)
FIG. 2: Results of Rietveld refinements (space-group R3c)
for lattice parameters and anisotropic atomic mean-square
displacements, from POWGEN neutron diffraction data.
d-spacing (Å)
Straight lines in (b) are fits to the data, while straight lines
in (c) are guides to the eye.
FIG. 1: Neutron dffraction patterns (markers) and Rietveld
refinements (red line) for BiFeO at 300K (a) and 773K
3
(b). Blues curves are the bottom of each panel are difference
curves. Upper and lower tick marks are the reflection posi- 100 105 T=300K 105 T=570K
tions for the BiFeO3 and Fe2O3 phases, respectively. Peak 104 (a) 104 (b)
labels are for the BiFeO phase. The refinements were done
3 80
with the R3c space-group. The data for T < 640K were 103 103
refined with a G-type antiferromagnetic structure (without V)60
e
m
cycloid modulation). E (
40
20
fgrroomunQd o=n1ei.t3h7e˚Ar−s1id,ea.nTdhciosmfipguarreedclteoartlhyeshpohwonsotnhabtatchke- 1000 2105 4 6 8 1T0=6(9c10)2K 2105 4 6 8 1T0=7(5d10)2K
104 104
S (E) intensity from the spin-wave persists down to
mag 80
|E| (cid:39) 0.3meV, where it merges with the elastic scatter- 103 103
ingsignal. Thus,wecanestimateanupper-boundforany eV)60
m
magnetic anisotropy gap, Eg <0.3meV, if any such gap E (40
exists. Thisisatoddswithreportsofagap,E =6meV,
g
derived from modeling the low-T specific heat C [31]. 20
P
Although the powder average of excitations does not
0
allow us to determine if multiple spin-wave branches ex- 2 4 6 8 10 12 2 4 6 8 10 12
|Q| (Å-1) |Q| (Å-1)
ist within the magnon DOS, we can compare our results
withrecentRamanscatteringstudiesthatreportedmul-
tiple electromagnon modes in BFO between 1.5 and 7.5 FIG. 3: S(Q,E) for BiFeO3 at different temperatures, mea-
sured using ARCS (logarithmic intensity).
meV [12, 13]. We do not see these excitations in the
powder spectrum at either 200 or 300K (see Fig. 5-a),
4
0 although high-resolution INS measurements on single-
T=300K T=570K
crystals may be needed to observe such effects. Our re-
-5 (a) (b)
sultsforthemagneticspectrumshowasinglemaximum,
-10 around 65meV at 300K, see Fig. 5-b. This is at odds
V)
me-15 with the magnetic spectrum reported in Ref. [21], which
E ( showed additional maxima around 30 and 55meV. How-
-20 1 1
ever,theextrapeaksin[21]arelikelyduetopeaksinthe
0.1 0.1
-25 phonon DOS at these energies (see below). We use the
0.01 0.01
magnetic DOS to estimate the exchange coupling, with
-30
0 a simple spin-wave model for a collinear Heisenberg G-
T=670K T=720K
-5 (c) (d) type antiferromagnet in an undistorted perovskite struc-
ture. A similar model successfully described the spin
-10
waves in the orthoferrite compounds ErFeO , TmFeO
V) 3 3
E (me-15 [c2lo8s],eamsawgenlletaics LstarFuecOtu3reasn.dTYhFiesOd3oe[2s9n,o3t0]c,awphtuicrhehpaovse-
-20 1 1
sible effects from the spiral spin structure in BFO, but
-25 0.1 0.1 these are expected to be limited, owing to the long pe-
0.01 0.01
riodofthemodulation. Withinthismodel,twoexchange
-30
1 2 3 4 5 1 2 3 4 5 constants, J andJ(cid:48), describeagaplessspin-wavedisper-
|Q| (Å-1) |Q| (Å-1)
sion, with acoustic and optic modes that meet at the
magnetic zone boundaries. This behavior can be seen
FIG. 4: S(Q,E) for BiFeO at different temperatures, mea-
3 in the large patch of scattering intensity in the ARCS
sured using DCS (logarithmic intensity). 300K data near 65 meV (Fig. 3-a). The J[J(cid:48)] exchange
constantcorrespondstocoupledmomentswithspin-spin
distances of 3.968[5.613±0.025] ˚A. Assuming S = 5/2
Fe3+ moments, and comparing the maximum energy of
(a) the measured and model spin-wave spectra, we are able
0.6 T=300K (DCS)
to place limits on the values of J and J(cid:48). We find that
0.5 1.15 < Q < 1.30 (cid:129)-1 J = 1.6+0.4 and J(cid:48) = −0.25+0.07 where there is a lin-
b. units) 00..43 11..3405 << QQ << 11..4650 (cid:129)(cid:129)--11 eJa(cid:48)r=de0p.9e−n0(d02.e2)n−ceJo0n.4th0(e1se).pTarhaem−ce0a.t1le7cruslwatietdhinmtahgenseetibcoDunOdSs
sity (ar 0.2 fpohroJno=n s1u.b6tarancdteJd(cid:48)T==−300.205K3 mdaetVaashgroewens winelFliwg.it5h-bt.he
n
nte 0.1 There is a clear softening and broadening of the spin-
I
0.0 wave spectrum with increasing temperature from 300K
-8 -6 -4 -2 0 to 570K. Magnon-magnon and magnon-phonon interac-
Energy (meV)
(b) tionsarelikelybothresponsibleforthissoftening[13,32].
6x10-3 (ARCS) 750K ItispossiblethatthestrongsofteningofSmag(E)inthis
690K range is related to the an anomalous magnetization be-
units) 54 557700KK cool lowTN [31]. AboveTN, weobserveLorentzianscattering
b. 470K intensitycenteredatE =0meV,typicalofparamagnetic
sity (ar 32 3c0a0lcK. bphehoanvoinorsu(bthtreadctipionb)e.low 20meV is a result of imperfect
n
e
nt 1
c i
eti 0
n
g B. Phonons
a 0 20 40 60 80 100
M
Energy (meV)
InbothFigs.3and4,thehorizontalbandsofintensity
increasingasQ2 areorientation-averagedphonondisper-
FIG. 5: (a) Integrated INS intensity (DCS, 300K) from
sions. In the DCS data, three main horizontal bands are
spin-wave, 1.3 (cid:54) Q (cid:54) 1.45˚A−1, compared with background
clearlyobservedat|E|(cid:39)6.5,8,11meV,correspondingto
phononsignal(1.15(cid:54)Q(cid:54)1.3˚A−1 and1.45(cid:54)Q(cid:54)1.6˚A−1).
the top of acoustic phonon branches and low-E optical
(b) Magnetic excitation spectra obtained from INS data
(ARCS),integratedover2(cid:54)Q(cid:54)6˚A−1,correctedforphonon branches,whichmainlyinvolveBivibrations(seebelow).
scattering (6 (cid:54) Q (cid:54) 10˚A−1, scaled). The thick line is the Theacousticbranchesareseendispersingoutofanuclear
magnon spectrum calculation (see text). Error bars indicate Bragg peak at Q (cid:39) 2.25˚A−1. The phonon cutoff corre-
one standard deviation. sponding to the top of oxygen-dominated optic branches
is seen at E (cid:39) 85meV in Fig. 3-a. A strong broadening
ofthephononmodeswithincreasingtemperaturecanbe
5
(a) 0.04
750K ARCS 6† Q †10(cid:129)-1 DFT (Wang et al.)
/meV) 0.04 655977000KKK cooling /meV) 0.03 DACRCS,S ,3 30000KK
OS (1 0.03 437000KK S (1
D O
n D
no 0.02 n 0.02
o o
alized Ph 0.01 d Phon
Gener 0.00 alize 0.01
0 20 40 60 80 er
(b) Energy (meV) n
e
G 0.00
720K DCS 0 20 40 60 80
V) 0.02 670K
/me 557700KK E (meV)
1 470K
S ( 300K
n DO 200K FIG. 7: Generalized phonon DOS of BiFeO3 measured
no 0.01 with INS (ARCS and DCS) at 300K, compared with first-
o
h
P principles calculation of Wang et al. [36].
d
e
z
ali
er
n
Ge 0.00
0 5 10 15 20
Energy (meV) σ/M for (Bi, Fe, O) are (0.044, 0.208, 0.265), in units of
barns/amu, respectively. Thus, the modes involving pri-
marily Bi motions are under-emphasized in S(Q,E) and
FIG. 6: (a) Generalized phonon DOS obtained from ARCS
g(E) (but there is relatively little weighting of Fe modes
data. (b) Low-E part of generalized DOS from DCS data,
showing strong broadening of Bi-dominated modes. Curves compared to O modes).
for different temperatures are vertically offset for clarity. While the energy-range of the spectrum is comparable
with other Fe perovskites [30], what is striking is the se-
vere broadening of the spectrum with increasing T. The
gDOS measured at 300K is in good agreement overall
withthefirst-principlescalculationofWangetal. within
seen in both Figs. 3 and 4. This broadening indicates a
spin-polarizedDFT+U[36],ascanbeseeninFig.7. This
strong damping of phonons, over the full spectrum. The
agreementallowsforaclearidentificationofthemainfea-
T rangeoverwhichthisoccursisinagreementwithprior
tures in the DOS. The three peaks at E (cid:39)6.5,8,11meV
Raman measurements [14], and points to a spin-phonon
(Fig. 6(b)) arise from the top of acoustic branches and
coupling effect.
thelowest-E opticmodes, andtheyaredominatedbyBi
The S(Q,E) data were analyzed to extract the gen-
motions. The lower two Bi peaks (6.5meV and 8meV)
eralized phonon DOS, g(E), in the incoherent scattering
are softer than predicted by DFT by about 10% [36].
approximation. The data from ARCS were integrated These Bi modes are responsible for the peak in C /T3
over 6 (cid:54) Q (cid:54) 10˚A−1, which minimizes any contribu- P
around 25K reported in Ref. [31]. The phonon spec-
tion from magnetic scattering, and the elastic peak was
trum for E > 40meV is mainly comprised of oxygen
subtracted, and extrapolated with a quadratic E depen-
vibrations, and is stiffer in the measured DOS than in
denceforE <5meV.Acorrectionformultiphononscat-
the DFT calculation [36]. While the agreement is gen-
tering was performed at all T [33, 34]. The gDOS from
erally good between the DFT calculation of Wang et al.
the DCS data was obtained by integrating over the full
and the phonon DOS measured at low temperature, the
range of Q (no multiphonon correction) [35]. The mea-
T dependence of the DOS is strongly affected by anhar-
sured signal from the empty container was much weaker
monicity and spin-phonon coupling, as we discuss in the
than from the sample, and was easily subtracted. The
next section.
results are shown in Fig. 6(a) for the full gDOS (ARCS)
andpanel(b)forE <20meV(DCS).AlthoughtheDCS
data include a magnetic contribution, this effect is lim-
ited,andtheDOSsformARCSandDCSareinexcellent IV. DISCUSSION
agreement, considering the difference in instrument res-
olution (see Fig. 7). The coarser energy resolution in As already pointed out above, the phonon DOS ex-
ARCS data washes out the three Bi-dominated peaks hibits a severe broadening with increasing T, which af-
at E (cid:54) 15meV, but the two DOS curves are other- fects the whole spectrum. In particular, the broaden-
wise very similar. In BFO, the different elements have ing of Bi modes at low-E is obvious in the DCS data,
different ratios of cross-section over mass, σ/M, result- shown in Fig. 6-b. It is particularly strong between
ing in a weighted phonon DOS (gDOS). The values of 470K and 570K, with little additional broadening ob-
6
(a)
11.3 tive change in volume to be about 1.4% between 200K
11.2
V) 11.1 and 570K from our diffraction measurements (this was
me 11.0 linearly extrapolated over the range 200−570K, since
E (3 1100..98 our diffraction data were limited to T (cid:62)300K). The av-
erage relative decrease of E , E , and E is −4.2±0.7%
1 2 3
8.5 over the same T range, yielding a Gru¨neisen parameter
V) 8.4 γ = 3.0±0.5. Such a large value of γ is a further cor-
me 8.3
(2 8.2 roboration of the anharmonicity of these Bi-dominated
E modes. The lack of softening above 570K indicates that
6.7 anincreaseininteratomicforce-constantsassociatedwith
V) 6.6 the loss of magnetic order compensates for the effect of
me 6.5 thermal expansion.
(1 We note that Bi and O atoms undergo large ampli-
E 6.4
(b) 200 300 400 500 600 700 tude vibrations, according to both our measurements
1.00 T (K) and reports of others [5, 6]. Since the Bi and O modes
0.99 are sharp at T (cid:54) 300K, the large displacements ob-
served in diffraction data are dynamic in nature, rather
K 0.98
00 than associated with static disorder. These large ampli-
2
E/E 0.97 E1 tudes of vibration for Bi and O are related to the anhar-
E
0.96 2 monic scattering of phonons, which leads to the broad-
E
3 eningandsofteningoffeaturesintheDOS.Wehavealso
0.95
200 300 400 500 600 700 performed measurements of the Fe-partial phonon DOS
T (K) withnuclear-resonantinelasticx-rayscattering(NRIXS)
on 57Fe-enriched samples, and observed a more limited
broadening of Fe modes, in agreement with the smaller
FIG. 8: (a) Centroids of the three low-energy peaks in DOS
thermal displacements of these atoms [38].
around 6.5, 8.5, 11meV (resp. E , E , E ), as a function
1 2 3
We suggest that the large thermal displacements and
of temperature. (b) Relative change in these energies with
respect to their value at 200K. The vertical dashed line at anharmonicity of Bi and O modes lead to structural
T =640K denotes the N´eel temperature. fluctuations, such as variations in Fe-O-Fe bond lengths
and bond angles (tuning the superexchange interaction)
throughtiltsandrotationsofFeO octahedra,thatcould
6
lead to fluctuations in magnetic coupling. The magni-
served further above T . This indicates a coupling be- tude of O thermal motions perpendicular to Fe-O bonds
N
tweenanharmonicityandthelossoftheAForder. These actually leads to fluctuations in the Fe-O-Fe bond angle
results are consistent with the reported observations of that are larger ((cid:39)6−10◦) than the variation of average
strong broadening of Raman phonon modes [14]. The angle with T. Reciprocally, the loss of magnetic order
oxygenmodes,dominatingtheDOSbetween40meVand induces a change in interatomic force-constants, stiffen-
85meV, are also strongly broadened in this T-range. ing the Bi vibration modes at low E. The motion of Bi
atoms is also directly related to the ferroelectricity. The
Wenowanalyzeinmoredetailthetemperaturedepen-
large-amplitude structural fluctuations could thus lead
dence of the three peaks at 6.5, 8.5, 11meV (resp. E ,
1
to steric effects between Bi motions and the rotations of
E , E ). The peaks were fitted with Gaussians, and the
2 3
oxygen octahedra, yielding a complex coupling between
resulting peak centers are plotted as a function of tem-
ferroelectric and AF magnetic orders.
perature in Fig. 8. As may be seen on this figure, all
three peaks undergo a pronounced but gradual softening
between200Kand570K,besidesthestrongbroadening.
However, this softening stops around T . This is com- V. SUMMARY
N
patiblewiththerecentlyreportedbehaviorofA Raman
1
modes [14]. E3 actually appears to stiffen above TN, but We have systematically investigated the temperature
this may partly be due to the influence of broadening dependence of the magnetic excitation spectrum and
andsofteningofmodesatE (cid:62)13meV,causingaskewed phonon density of states of BiFeO over the range 200(cid:54)
3
contribution to the E3 peak. T (cid:54) 750K, using inelastic neutron scattering. In ad-
The change in phonon softening behavior around T dition, we performed neutron diffraction measurements
N
is further evidence of the influence of spin-phonon cou- and refined the lattice parameters and thermal displace-
pling. The softening of phonon modes with increasing ment parameters over 300 (cid:54) T (cid:54) 770K. We separated
T, as observed here for T (cid:54) 570K, can generally be re- the magnon and phonon contributions in the S(Q,E)
latedtothethermalexpansionofthesystemthroughthe data, and observed a strong resemblance of the magnon
quasiharmonic approximation dlnE = −γdlnV, with γ spectrum with that of related compounds LaFeO and
3
the Gru¨neisen parameter [37]. We determined the rela- YFeO . Themagnonspectrumwasfitsatisfactorilywith
3
7
asimplecollinearHeisenbergmodelforaG-typeantifer- constants from the loss of magnetic order were pointed
romagnet, indicating the limited role of the cycloid on out.
the spin dynamics, as expected from the long period of
thecycloidmodulation. ThephononDOSobtainedfrom
thehigh-QpartoftheS(Q,E)dataisingoodagreement
at low temperatures with the first-principles calculation
VI. ACKNOWLEDGEMENTS
of Wang et al. [36]. However, the phonon DOS shows
a strong temperature dependence, with in particular a
pronounced broadening. Also, both the softening and The Research at Oak Ridge National Laboratory’s
broadeningoffeaturesintheDOScorrelatewiththeloss Spallation Neutron Source was sponsored by the Scien-
of antiferromagnetic order around T = 640K, indicat- tific User Facilities Division, Office of Basic Energy Sci-
N
ing the presence of significant spin-phonon coupling, in ences, US DOE. This work utilized facilities supported
agreement with recently reported Raman measurements inpartbytheNationalScienceFoundationunderAgree-
[14]. The potential influence of large atomic displace- mentNo. DMR-0944772. TheworkperformedatBoston
ments on the modulation of the superexchange interac- CollegeisfundedbytheUSDepartmentofEnergyunder
tion, and the concomitant effect of the change in force- contract number DOE DE-FG02-00ER45805 (ZFR).
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