Table Of ContentChiral fluctuations in MnSi above the Curie temperature
B. Roessli1, P. B¨oni2, W. E. Fischer1 and Y. Endoh3
1Laboratory for Neutron Scattering, ETH Zurich & Paul Scherrer Institute, CH-5232 Villigen PSI
2 Physik-Department E21, Technische Universit¨at Mu¨nchen, D-85747 Garching, Germany
3Physics Department, Tohoku University, Sendai 980, Japan
(February 1, 2008)
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0
0
Polarized neutrons are used to determine the antisymmetric part of the magnetic susceptibility in
2
non-centrosymmetric MnSi. The paramagnetic fluctuations are found to be incommensurate with
n thechemical latticeand tohaveachiral character. Weargue thatantisymmetricinteractions must
a betakenintoaccounttoproperlydescribethecritical dynamicsinMnSiaboveTC. Thepossibility
J
of directly measuring the polarization dependent part of the dynamical susceptibility in a large
8 class of compounds through polarised inelastic neutron-scattering is outlined as it can yield direct
1
evidence for antisymmetric interactions like spin-orbit coupling in metals as well as in insulators.
]
PACS numbers: 75.25+z, 71.70.Ej, 71.20.lp
l
e
-
r
t Ordered states with helical arrangement of the mag- acter of the spin fluctuations due to spin-orbit coupling
s
. netic moments aredescribedby a chiralorderparameter anddiscuss the experimental results in the frameworkof
at C~ = S~ ×S~ , which yields the left- or right-handed ro- self-consistentrenormalisationtheoryofspin-fluctuations
1 2
m tation of neighboring spins along the pitch of the helix. in itinerant magnets [7].
- Examplesforcompoundsofthatsortarerare-earthmet- Beingaprototypeofaweakitinerantferromagnet,the
d
alslikeHo[1]. Spins onafrustratedlatticeformanother magnetic fluctuations in MnSi have been investigated in
n
class of systems, where simultaneous ordering of chiral the past in detail by means of unpolarizedand polarized
o
c and spin parameters can be found. For example, in the neutron scattering. The results demonstrate the itiner-
[ triangular lattice with antiferromagnetic nearest neigh- ant nature of the spin fluctuations [8–10] as well as the
bor interaction, the classical ground-state is given by a occurrence of spiral correlations [11] and strong longitu-
1
v non-collinear arrangement with the spin vectors form- dinal fluctuations [13].
7 ing a 120◦ structure. In this case, the ground state is MnSi has a cubic space groupP2 3 with a lattice con-
1
2 highly degenerate as a continuous rotation of the spins stanta=4.558˚Athatlacksacenterofsymmetryleading
3
in the hexagonal plane leaves the energy of the system toaferromagneticspiralalongthe[111]directionwitha
1
unchanged. Inaddition,itispossibletoobtaintwoequiv- period of approximately180 ˚A [14]. The Curie tempera-
0
2 alent ground states which differ only by the sense of ro- ture is TC =29.5 K.The spontaneous magnetic moment
0 tation (left or right) of the magnetic moments from sub- of Mn µ ≃ 0.4µ is strongly reduced from its free ion
s B
/ lattice to sub-lattice,hence yielding anexampleofchiral value µ = 2.5µ . As shown in the inset of Fig. 1 the
t f B
a
degeneracy. fourMnandSiatomsareplacedatthepositions(x,x,x),
m
As a consequence of the chiral symmetry of the or- (1+x,1−x,−x),(1−x,−x,1+x),and(−x,1+x,1−x)
2 2 2 2 2 2
- der parameter, a new universality class results that is with x =0.138 and x =0.845, resepctively.
d Mn Si
n characterized by novel critical exponents as calculated We investigated the paramagnetic fluctuations in a
o by Monte-Carlo simulations [2] and measured by neu- large single crystal of MnSi (mosaic η = 1.50) of about
c tronscattering[3]intheXY-antiferromagnetCsMnBr . 10cm3 onthe triple-axisspectrometerTASPattheneu-
3
:
v Aninterestingbutstillunresolvedproblemisthecharac- tron spallation source SINQ using a polarized neutron
i terization of chiral spin fluctuations that have been sug- beam. The single crystal was mounted in a 4He refriger-
X
gestedtoplayanimportantrolee.g. inthedopedhigh-T ator of ILL-type and aligned with the [0 0 1] and [1 1 0]
c
r
a superconductors [4]. The measurement of chiral fluctua- crystallographic directions in the scattering plane. Most
tions is,however,a difficult taskandcanusually onlybe constantenergy-scanswereperformedaroundthe(011)
performed by projecting the magnetic fluctuations on a Bragg peak and in the paramagnetic phase in order to
field-induced magnetization [5,6]. relax the problem of depolarizationof the neutron beam
In this Letter, we show that chiral fluctuations can be in the ordered phase. The spectrometer was operated
directly observed in non-centrosymmetric crystals with- in the constant final energy mode with a neutron wave
out disturbing the sample by a magnetic field. We vector~k =1.97˚A−1. Inorderto suppresscontamination
f
present results of polarized inelastic neutron scattering by higher order neutrons a pyrolytic graphite filter was
experiments performedinthe paramagneticphase ofthe installed in the scattered beam. The incident neutrons
itinerantferromagnetMnSi that confirmthe chiralchar- werepolarizedbymeansofaremanent[15]FeCoV/TiN-
1
type bender that was inserted after the monochromator (Qˆ~ ·P~ )(Qˆ~ ·B~) (2)
i
[16]. Thepolarizationoftheneutronbeamatthesample
positionwas maintainedby a guide fieldB =10G that andvanishesforcentro-symmetricsystemsorwhenthere
g
defines also the polarization of the neutrons P~ with re- isnolong-rangeorder. Intheabsenceofsymmetrybreak-
i
spect to the scattering vector Q~ =~k −~k at the sample ingfieldslikeexternalmagneticfields,pressureetc.,sim-
i f
position. ilar scans with polarized neutrons would yield a peak
In contrast to previous experiments, where the polar- of diffuse scattering at the zone center and no scatter-
izationP~ ofthescatteredneutronswasalsomeasuredin ing that depends on the polarization of the neutrons.
f
order to distinguish between longitudinal and transverse However,anintrinsicanisotropyofthe spinHamiltonian
fluctuations [13], we did not analyze P~ , as our goalwas in a system that lacks lattice inversion symmetry may
f
todetectapolarizationdependentscatteringthatispro- provide anaxialinteractionleading to a polarizationde-
portional to σ ∝(Qˆ~ ·P~ ) as discussed below. pendent cross section. The polarization dependent scat-
p i
tering obtained in the present experiments is therefore
A typical constant-energy scan with h¯ω = 0.5 meV
an indication of fluctuations in the chiral order param-
measured in the paramagnetic phase at T = 31 K is
eter and points towards the existence of an axial vector
shown in Fig. 1 for the polarization of the incident neu-
tronsP~ parallelandanti-paralleltothescatteringvector B~ that is not necessarilycommensurate with the lattice.
i
Q~. ItisclearlyseenthatthepeakpositionsdependonP~ Hence,accordingtoEq.2theneutronscatteringfunction
i
and appear at the incommensurate positions Q~ = ~τ ±~δ in MnSi contains a non-vanishing antisymmetric part.
Because the crystal structure of MnSi is non-
with respect to the reciprocal lattice vector ~τ of the
011
centrosymmetric and the magnetic ground-state forms
nuclear unit cell. Obviously, this shift of the peaks with
a helix with spins perpendicular to the [1 1 1] crys-
respect to (0 1 1) would be hardly visible with unpolar-
tallographic direction, it is reasonable to interpret the
izedneutronsandcouldnotobservedinpreviousinelastic
polarization-dependent transverse part of the dynami-
neutron works.
cal susceptibility in terms of the Dzyaloshinskii-Moriya
In order to discuss our results we start with the gen-
(DM)interaction[20,21]similarlyasitwasdoneinother
eral expression for the cross-section of magnetic scatter-
non-centrosymmetricsystemsthatshowincommensurate
ing with polarized neutrons [12]
ordering [18,19].
d2σ Usually the DM-interaction is written as the cross
dΩdω ∼ (δα,β −QˆαQˆβ)Aαβ(Q~,ω) product of interacting spins HDM = l,mD~l,m ·(~sl ×
Xα,β ~s ), where the direction of the DMP-vector D~ is de-
m
+ (Qˆ~ ·P~ ) ǫ QˆγBαβ(Q~,ω) (1) termined by bond symmetry and its scalar by the
i α,β,γ
Xα,β Xγ strength of the spin-orbit coupling [21]. Although the
DM-interactionwasoriginallyintroducedonmicroscopic
where (Q~,ω) are the momentum and energy-transfers grounds for ionic crystals, it was shown that antisym-
from the neutron to the sample, Qˆ~ = Q~/|Qˆ|, and metric spin interactions are also present in metals with
non-centrosymmetriccrystalsymmetry [22]. Ina similar
α,β,γ indicate Cartesian coordinates. The first term
way as for insulators with localized spin densities, the
in Eq. 1 is independent of the polarization of the inci-
antisymmetric interaction originates from the spin-orbit
dent neutrons, while the second is polarization depen-
dent through the factor (Qˆ~ ·P~ ). P~ denotes the direc- couplingintheabsenceofaninversioncenterandafinite
i i contribution to the the antisymmetric part of the wave-
tion of the neutron polarization and its scalar is equal
vector dependent dynamical susceptibility is obtained.
to 1 when the beam is fully polarized. Aαβ and Bαβ
For the caseof a uniformDM-interaction, the neutron
are the symmetric and antisymmetric parts of the scat-
cross-section depends on the polarization of the neutron
tering function Sαβ, that is Aαβ = 1(Sαβ +Sβα) and
2 beam [23] as follows
Bαβ = 1(Sαβ −Sβα). Sαβ are the Fourier transforms
2
of the spin correlation function < sαsβ >, Sαβ(Q~,ω) = d2σ
2π1N −∞∞dte−iωt ll′eiQ~(X~l−X~l′) <slαl sl′βl′(t)>. The vec- (cid:18)dΩdω(cid:19)np ∼ℑ(χ⊥(~q−~δ,ω)+χ⊥(~q+~δ,ω)),
torsXR~ldesignatePthepositionsofthescatteringcentersin d2σ ∼(D~ˆ ·Qˆ~)(Qˆ~ ·P~ )
the lattice. The correlationfunction is relatedto the dy- (cid:18)dΩdω(cid:19) i
p
namicalsusceptibilitythroughthefluctuation-dissipation
theorem S(Q~,ω)=2h¯/(1−exp(−¯hω/kT))ℑχ(Q~,ω). ×ℑ(χ⊥(~q−~δ,ω)−χ⊥(~q+~δ,ω)). (3)
FollowingRef.[17]we define nowanaxialvectorB~ by
Here, ~q designates the reduced momentum transfer with
αβǫαβγBαβ = Bγ(Q~,ω), that represents the antisym- respecttothenearestmagneticBraggpeakat~τ±~δ. The
Pmetric part of the susceptibility which, hence, depends first line of Eq. 3 describes inelastic scattering with a
on the neutron polarization as follows
non-polarized neutron beam. The second part describes
2
inelastic scattering that depends on P~ as well as on D~. In conclusion, we have shown that chiral fluctuations
i
Eq. 3 shows that the cross section for P~ ⊥ Q~ is in- can be measured by means of polarized inelastic neu-
i
deed independent of P as observed in Fig. 2. By sub- tron scattering in zero field, when the antisymmetric
i
tracting the inelastic spectra taken with P~ parallel and part of the dynamical susceptibility has a finite value.
i
anti-paralleltoQ~,thepolarizationdependentpartofthe We have shown that this is the case in metallic MnSi
cross-section can be isolated, as demonstrated in Fig. 3 that has a non-centrosymmetric crystal symmetry. For
for two temperatures T =31 K and T =40 K. this compound the axial interaction leading to the po-
Closeto T , the intensityis ratherhighandthe cross- larizedpart of the neutron cross-sectionhas been identi-
C
ingatQ=(011)issharp. At40Ktheintensitybecomes fied as originating from the DM-interaction. Similar in-
smallandthetransitionat(011)israthersmooth,which vestigations can be performed in a large class of other
mirrors the decreases of the correlation length with in- physical systems. They will yield direct evidence for
creasingtemperature. Wehavemeasured(d2σ/(dΩdω)) thepresenceofantisymmetricinteractionsinformingthe
p
in the vicinity of the (0 1 1) Bragg peak at T = 35 K. magnetic ground-state in magnetic insulators with DM-
The result shown as a contour plot in Fig. 4 indicates interactions, high-T superconductors (e.g. La CuO
c 2 4
that the DM-interaction vector in MnSi has a compo- [24]),nickelates[25],quasi-onedimensionalantiferromag-
nent along the [0 1 1] crystallographic direction which nets [26] or metallic compounds like FeGe [27].
induces paramagnetic fluctuations centered at positions
incommensurate with the chemical lattice.
In order to proceed further with the analysis we as-
sume for the transverse susceptibilities in Eq. 3 the ex-
pressionfor itinerant magnets as givenby self-consistent
re-normalizationtheory (SCR) [7]
[1] V.P. Plakhty et al., Phys.Rev.B 64, 100402(R), 2001.
χ⊥(~q±~δ,ω)=χ⊥(~q±~δ)/(1−iω/Γ ). (4) [2] H. Kawamura, Phys.Rev.B 38 4916 (1988).
q~±~δ [3] T.E. Mason et al., Phys.Rev.B 39 586 (1989).
~δ is the ordering wave-vector,χ⊥(~q±~δ)=χ⊥(∓~δ)/(1+ [4] P.E. Sulewski et al., Phys. Rev.Lett. 67, 3864 (1991).
q2/κ2) the static susceptibility, and κ the inversecorre- [5] S. V. Maleyev, Phys.Rev. Lett. 75, 4682 (1995).
δ δ [6] V. P. Plakhty et al., Europhys. Lett. 48, 215 (1999).
lationlength. For itinerantferromagnetsthe dampingof
[7] T. Moriya, in Spin Fluctuations in Itinerant Electron
the spinfluctuationsis givenby Γ =uq(q2+κ2)with
q~±~δ δ Magnetism 56, Springer-Verlag, Berlin Heidelberg New-
u=u(~δ) reflecting the damping of the spin fluctuations. York Tokyo, 1985.
Experimentally, it has been found from previous inelas- [8] Y. Ishikawa et al., Phys. Rev.B 16, 4956 (1977).
tic neutron scattering measurements that the damping [9] Y. Ishikawa et al., Phys. Rev.B 25, 254 (1982).
of the low-energy fluctuations in MnSi is adequately de- [10] Y. Ishikawa et al., Phys. Rev.B 31, 5884 (1985).
scribed using the results of the SCR-theory rather than [11] G. Shiraneet al., Phys.Rev.B 28, 6251 (1983).
the qz (z = 2.5) wave-vector dependence expected for a [12] e.g. Yu A.Izyumov,Sov.Phys. Usp. 27, 845 (1984).
[13] S. Tixier et al., Physica B 241-243, 613, (1998).
Heisenberg magnet [9].
[14] Y.Ishikawaetal.,Solid.State.Commun.19,525(1976).
The solid lines of Figs. 1 to 3 show fits of
[15] No spin flipping devices are necessary due to the rema-
(d2σ/(dΩdω)) to the polarized beam data. It is seen
p nent magnetization of the supermirror coatings of the
that the cross section for itinerant magnets reproduces benders. For details see: P. B¨oni et al., Physica B 267-
thedatawellifthe incommensurabilityisproperlytaken 268, (1999) 320.
into account. Using Eqs. 3 and 4 and taking into ac- [16] F.Semadeni,B.Roessli,andP.B¨oni,PhysicaB297,152
countthe resolutionfunctionofthe spectrometer,weex- (2001).
tract values κ = 0.12 ˚A−1 and u = 27 meV˚A3 in rea- [17] S.W. Lovesey and E. Balcar, Physica B 267-268, 221
0
sonable agreement with the analysis given in Ref. [10]. (1999).
The smaller value for u when compared with u = 50 [18] A. Zheludev et al., Phys. Rev.Lett. 78, (1997) 4857.
meV˚A3 from Ref. [9] indicates that the incommensura- [19] B. Roessli et al., Phys. Rev.Lett. 86 (2001) 1885.
[20] L. Dzyaloshinskii, J. Phys. Chem. Solids 4, 241 (1958).
bility ~δ = (0.02,0.02,0.02)was neglected in the analysis [21] T. Moriya, Phys.Rev. 120, 91 (1960).
of the non-polarized neutron data. At T = 40 K, the [22] M. Kataoka et al., J. Phys.Soc. Japan 53, 3624 (1984).
chiralfluctuations are broad(Fig. 3) due to the increase [23] D.N. Aristov and S.V. Maleyev Phys. Rev. B 62 (2000)
ofκδ withincreasingT,i.e. κδ(T)=κ0(1−TC/T)ν. We R751.
note thatthe mean-field-likevalueν =0.5obtainedhere [24] J.BergerandA.Aharony,Phys.Rev.B46,6477(1992).
isclosetotheexpectedexponentν =0.53forchiralsym- [25] W. Koshibae, Y. Ohta and S. Maekawa, Phys. Rev. B
metry [2]. This suggeststhata chiral-orderingtransition 50, 3767 (1994).
also occurs in MnSi in a similar way to the rare-earth [26] I. Tsukada et al., Phys. Rev.Lett.87, 127203 (2001).
[27] B. Lebech, J. Bernhard, and T. Freltoft, J. Phys.: Con-
compoundHo,pointingtowardtheexistenceofauniver-
dens. Matter 1, 6105 (1989).
sality class in the magnetic ordering of helimagnets [1].
3
400 250
350 200
T=40K
7 min. 235000 PPoollaarr.. aalloonngg −QQ MT=n3S1iK 2*7 min. 11505000 T=31K MnSi
Neutron Counts/ 112050000 Neutron Counts/ --11-5500000
50 -200
-250
0 0.7 0.8 0.9 1 1.1 1.2 1.3
0.7 0.8 0.9 1 1.1 1.2 1.3
(0,q,q) (r.l.u.)
(0,q,q) (rlu)
FIG.3. DifferenceneutroncountsforpolarizationP~iofthe
incidentneutronbeamparallelandanti-paralleltoQ~ inMnSi
FIG. 1. Inelastic spectra in MnSi (¯hω = 0.5 meV) at T =31 K and 40 K,respectively. Thesolid linesare fit to
at T = 31 K for the neutron polarization parallel and thedatausingtheSRC-resultforthedynamicalsusceptibility
anti-paralleltothescatteringvectorQ~,respectively. Thesolid with theparameters given in thetext.
lines are fits to the data. The inset shows the Mn atoms in
the crystal structure of MnSi. Note that MnSi is not cen-
tro-symmetric. FIG.4. Contour-map of the polarization dependent scat-
teringforanenergytransfer¯hω=0.5meVasmeasurednear
the(0 1 1) reciprocal lattice point at T=35K.
200
150 Polar. perp. Q MnSi
Polar. perp. −Q T=35K
u.)
a.
nts ( 100
u
o
C
on 50
utr
e
N
0
Difference Counts
-50
0.7 0.8 0.9 1 1.1 1.2 1.3
(1-q,q,q) (rlu)
FIG. 2. Neutron spectra in MnSi for an energy-transfer
¯hω=0.5meVasmeasured atT =35KforP~i perpendicular
to Q~ and −Q~, respectively. The solid line shows a fit to the
data and the small symbols represent the difference signal
that is independentof P~i. See text for details.
4
