arXiv:0907.2152v1 [cond-mat.str-el] 13 Jul 2009
Spin freezing and dynamics in Ca3Co2−xMnxO6(x ≈ 0.95) investigated with implanted
muons: disorder in the anisotropic next-nearest neighbor Ising model
T. Lancaster,1, ∗S.J. Blundell,1P.J. Baker,1H.J. Lewtas,1W. Hayes,1F. L. Pratt,2H. T. Yi,3and S.-W. Cheong3
1Oxford University Department of Physics, Clarendon Laboratory, Parks Road, Oxford, OX1 3PU, UK
2ISIS Facility, Rutherford Appleton Laboratory, Chilton, Oxfordshire OX11 0QX, UK
3Rutgers Center for Emergent Materials and Department of Physics & Astronomy,
136 Frelinghuysen Road, Piscataway, New Jersey, 08854, USA
(Dated: July 13, 2009)
We present a muon-spin relaxation investigation of the Ising chain magnet Ca3Co2−xMnxO6
(x ≈ 0.95). We find dynamic spin fluctuations persisting down to the lowest measured temperature
of 1.6 K. The previously observed transition at around T ≈ 18 K is interpreted as a subtle change
in dynamics for a minority of the spins coupling to the muon that we interpret as spins locking
into clusters. The dynamics of this fraction of spins freeze below a temperature TSF ≈ 8 K, while a
majority of spins continue to fluctuate. An explanation of the low temperature behavior is suggested
in terms of the predictions of the anisotropic next-nearest-neighbor Ising model.
PACS numbers: 75.50.Ee, 75.50.Lk, 76.75.+i
The magnetic chain multiferroic Ca3Co2−xMnxO6
(x ≈ 1) has been the subject of considerable re-
cent investigation1,2,3,4. This material is based on
the Ising spin chain magnet Ca3Co2O6, with (close
to) half of the cobalt ions replaced with manganese5,7.
The observation of up-up-down-down (↑↑↓↓) order in
this system has led to the proposal that, at low
temperatures, Ca3Co2−xMnxO6 may be described by
the anisotropic next-nearest-neighbor Ising (ANNNI)
model1. This model8,9,10,11describes Ising spins on
a three-dimensional lattice in which, along one direc-
tion, there is nearest-neighbor ferromagnetic exchange
(JFM) and next-nearest-neighbor antiferromagnetic ex-
change (JAFM).For |JAFM/JFM| > 1/2 the ground
state magnetic order is of the ↑↑↓↓ type. As temper-
ature is increased from T = 0 the magnetic behav-
ior is determined by the existence of domain wall soli-
tons which separate regions with different commensurate
AFM spin arrangements9,10. Although a continuum de-
scription of the ANNNI model predicts an infinity of high
order commensurate AFM phases (known as the devil’s
staircase) a description in terms of a discrete Hamilto-
nian shows the possibility of metastable states of ran-
domly pinned solitons. In magnetic systems, these so-
called “chaotic states” are expected to lead to frozen-in
disorder or spin glass-like behavior8. Here we present
an investigation of the low temperature static and dy-
namic magnetism in Ca3Co2−xMnxO6 (x ≈ 1) that we
have observed at a local level using muon-spin relaxation
(µ+SR). We find that the low temperature magnetic state
of Ca3Co2−xMnxO6is reached through a complex freez-
ing out of dynamic processes and we conjecture that the
existence of chaotic states provides an explanation for the
disordered magnetism and persistent dynamics that we
observe at low temperature.
Ca3Co2−xMnxO6 is formed from chains of magnetic
ions arranged along the c-axis in alternating oxygen cages
of face-shared trigonal prisms and octahedra. Mn4+ions
preferentially occupy the octahedral sites while the trig-
onal prisms are occupied by Co2+ions. The magnetic
chains form a triangular lattice in the ab plane separated
by Ca2+ions. While it is agreed that 3d3Mn4+is in the
S = 3/2 high-spin configuration, the spin state of 3d7
Co2+has been questioned. Although fits to magnetic
neutron diffraction data1suggest a low spin S = 1/2
state, electronic structure calculations3,4and x-ray ab-
sorption spectroscopy3favor the high spin S = 3/2
state. The neutron diffraction and magnetic suscepti-
bility measurements1indicate that below TB ≈ 18 K
spins align along the c-axis, adopting the ↑↑↓↓ config-
uration with Mn and Co ordered moments of 0.66µBand
1.93µBrespectively. The considerable width of the mag-
netic Bragg peaks suggests that this is not a state of
true long-range order (LRO), but rather represents the
locking-in of spins into finite sized domains. Taken with
the Ising-like character of the magnetic ions, the obser-
vation of ↑↑↓↓ order is suggestive that this material can
be described as a realization of the ANNNI model, or
its extension12to the case of chains of unequal Ising
spins. It is also notable that the inversion symmetry
breaking of ↑↑↓↓ order, along with the alternating charge
order, results in magnetism driven ferroelectricity in this
µ+SR has proven useful in elucidating the static and
dynamic properties of Ising systems13including the par-
ent compound Ca3Co2O614,15. In a µ+SR experiment16,
spin-polarized positive muons are stopped in a target
sample, where the muon usually occupies an interstitial
position in the crystal. The observed property in the ex-
periment is the time evolution of the muon spin polariza-
tion, the behavior of which depends on the local magnetic
field at the muon site, and which is proportional to the
positron asymmetry function A(t). Measurements were
made at the Swiss Muon Source (SµS) using the GPS in-
strument. A polycrystalline sample of Ca3Co2−xMnxO6
with x = 0.95 (similar composition to Ref. 1), was packed
in a Ag foil packet and mounted on a Ag plate in a helium
0.00.10.2 0.30.4 0.5 0.6
t ( s)
0.00.1 0.20.3 0.4 0.5
t ( s)
0.00.1 0.20.30.4 0.5
t ( s)
110 K 14.6 K4.13 K
0 2 4 6 8 10 12 14 16 18 20
t ( s)
FIG. 1: ZF µ+SR spectra measured at (a) T = 110 K, (b) 14.6 K and (c) 4.13 K. Solid lines are fits to Eq. (1). Inset to (b): the
parameter y from Eq. (1) shows evidence of static freezing around T ≈ 8 K. Inset to (c): Spectra measured at 1.6 K, showing
a small, heavily damped oscillation. The solid line is a guide to the eye.
Spectra measured for Ca3Co1.05Mn0.95O6are shown in
Fig. 1. For temperatures T > 10 K [Fig. 1(a) and (b)]
we observe purely relaxing asymmetry spectra. We do
not observe oscillations in the asymmetry, which usually
signal the presence of long-range magnetic order (LRO).
For T < 8 K the form of the spectra alters, most no-
tably with an increase in the non-relaxing baseline of
the asymmetry. There is also a small undulation in the
asymmetry that emerges at early times in high statis-
tics spectra measured at the lowest temperatures [inset
Fig.1(c)]. This feature, which may represent a low am-
plitude, heavily damped oscillation displayed little tem-
perature dependence and was too indistinct to be fitted
systematically. Instead, the spectra are best modelled
over the entire temperature range by a sum of asymme-
try functions, one with a large relaxation rate λ1and the
other with a smaller rate λ2:
A(t) = Abg+ A0
+(1 − p)
where A0 represents the amplitude of the signal arising
from the sample and the term Abgrepresents a temper-
ature independent, nonrelaxing background from those
muons that stop in the sample holder or cryostat tails.
The parameter y = 0 above ≈ 10 K but takes nonzero
values at low temperatures (see below). The amplitude
p was found to be p = 0.75 across the entire measured
temperature regime. Exponential relaxation is often ex-
pected in cases where dynamic fluctuations in the local
magnetic field at the muon site represent the dominant
relaxation process17and it is likely that this mechanism
is at work in Ca3Co2−xMnxO6. Fast field fluctuations
lead to relaxation rates that vary as λi∝ γ2
γµis the muon gyromagnetic ratio, Biis the local mag-
netic field at the ith muon site and τ is a fluctuation
rate. In the present case, the interpretation of dynamic
fluctuations is supported by the observation that applied
longitudinal magnetic fields of up to 0.6 T do not decou-
ple the relaxation, as would be expected for relaxation
from static field inhomogeneities18.
The coexistence of two relaxation rates, λ1 and λ2,
implies the existence of two classes of spatially separate
muon sites in Ca3Co2−xMnxO6. In general, these sites
might differ in the width of the field distribution ?B2?, or
the fluctuation time at each site may be different. (It is
not possible to have a single class of site with two corre-
lation times giving rise to two different relaxation rates;
the shorter time will always dominate, giving the smaller
relaxation rate19.) The first class of muon site (with am-
plitude p and relaxation rate λ1) accounts for ≈ 75% of
the muon sites, with the second, with amplitude (1−p),
accounting for the remaining ≈ 25% of muon sites. As
there is no evidence (from our measurments or previous
work1) for phase separation in this system, it is proba-
ble that both classes arise from the intrinsic behavior of
the bulk of the material. Possibilities for this include the
coupling of each class of site preferentially to one or other
of the magnetic cations (i.e. one class sensitive to fields
arising from Co2+and the other sensitive to Mn4+), as is
the case in of X3V2O8(X=Ni,Co)20, or that the classes
of muon site are sensitive to different components of the
same spin distribution as in GeNi2O421. In the latter
case, a system with local site anisotropy might, for ex-
ample, give rise to relaxation timescales for longitudinal
and transverse fluctuations that could be quite different.
If different muons sites were selectively sensitive to lon-
gitudinal or transverse components, dynamics with two
timescales may arise from the same spin site.
The temperature dependence of λ1 and λ2 resulting
from fits of the data to Eq. (1) are shown in Fig. 2.
Both relaxation rates show the same trend of behavior
with their magnitude increasing as the temperature is re-
duced, followed by saturation of the relaxation rate at a
constant value below ≈ 30 K. This trend is sometimes
seen in the µ+SR of complex systems with low tem-
perature dynamics22,23and its origin is not completely
understood.It can arise due to the existence of two
relaxation channels, one of which is strongly T depen-
dent with a correlation time τs while the other shows
little variation with T and has a correlation time τw.
As noted above, in the presence of two competing relax-
ation processes, that with the shorter correlation time
wins out, giving the smaller relaxation rate.
temperature, therefore, we have τs(T) ≪ τw which re-
sults in a strongly T dependent relaxation which we can
fit phenomenologically by λi = Ciexp(Ui/T). At low
temperatures, where τw≪ τs(T), we have λi∼ λsat
sulting in a phenomenological fitting function 1/λi(T) =
+ 1/[Ciexp(Ui/T)]. Fitting this function to our
data allows us to parameterize the relaxation rates with
= 88(3) MHz, U1≈ 275(5) K for the fast relaxation
= 2.2(2) MHz, U2≈ 270(20) K for the slowly
relaxing component.Given the similarity between U1
and U2, it is likely that similar relaxation processes are
at work at both sites, sharing similar correlation times.
This would imply that the widths of the magnetic field
distributions differ for the two classes of site and are in
the ratio (?B2
further that, with the large activation energies involved,
it is unlikely that the temperature dependent behavior
is related to dynamics predicted by the ANNNI model.
This is not surprising, since models of this sort provide a
low energy description of real magnetic systems. Rather,
it is likely that the observed behavior arises due to the
single ion anisotropies of the magnetic ions.
Co2+, for example, the spin orbit coupling constant24
is −270 K, which resembles our energy scale U. This
is suggestive that the energy scale for the temperature
dependent contribution to the relaxation rates is set by
fluctuations between spin-orbit split components of the
low-lying electronic states of the magnetic Co2+ions. It
should be expected that Mn4+will affect the magnetic
field distribution at the muon sites making the true sit-
uation more complex.
i , re-
2)1/2≈ 6. We note
In addition to this general trend reflecting the dynam-
ics of the system, the smaller relaxing component, with
relaxation rate λ2, shows additional features at low tem-
perature (although it is not obvious why these features
are not seen in λ1). For the material in this regime,
where the ANNNI model provides a good effective de-
scription of the magnetic behavior, the spin fluctuations
will include excitations involving the domain wall soli-
tons described by the ANNNI model8, along with diffu-
sive modes resulting from coupling of the Ising spins to
non magnetic degrees of freedom. At around TB≈ 18 K
we see a small peak in the relaxation rate λ2which occurs
near the temperature below which (broadened) magnetic
Bragg peaks appear in neutron diffraction1. The absence
20 4060 80100 120
05 101520 25 303540
FIG. 2: (a) Evolution of the relaxation rates λ1 (open circles)
and λ2 (closed circles) as a function of temperature in ZF.
(b) Detail from (a) showing departure of λ2 from the trend
around the transition at TB and the spin freezing transition
TSF. Inset: Temperature evolution of the frequency of the
small oscillatory feature seen below 8 K at early times.
of a change at TBin the form of the signal or its ampli-
tudes indicates that this is not a transition to LRO or to
a static magnetic state. Rather, it is more likely to repre-
sent the freezing out of one or more relaxation processes,
likely to involve the free motion of domain walls, leading
to the formation of poorly correlated clusters of ordered
spins. This is consistent with the observation of broad-
ened peaks in the neutron diffraction1signalling that the
order that gives rise to the magnetic Bragg peaks is not
truly long range.
The more significant change in the slowly relaxing com-
ponent of the muon asymmetry occurs below 8 K. Here
the parameter y in Eq. (1) increases on cooling from
y = 0 to y = 1 at the lowest measured temperatures,
as shown in Fig. 1, indicating a transition to a static
local field distribution at low T. Such a distribution,
whether ordered or disordered, will only dephase those
muon-spin components that lie perpendicular to the ini-
tial muon-spin polarization direction, expected to be 2/3
of the total polarization for a powder sample, leaving the
other 1/3 polarized. Although the magnetic field dis-
tribution experienced by these muons below T ≈ 8 K
is static on the muon timescale, it is unlikely that the
system locks into true LRO. Instead of the oscillations
that would usually be observed in the presence of LRO,
we see only a very small minimum in the asymmetry at
early times [inset Fig. 1(c)] in the minority component of
our spectra measured below 8 K. This may originate from
Kubo-Toyabe-like relaxation17typical of a static ensem-
ble of disordered moments or may be a highly damped
oscillation due to static order occurring over only a short
length scale. Assuming the latter interpretation, the in-
set to Fig. 2(b) shows the frequency ν(= γµBi/2π) of this
feature26. The static disordered magnetism observed in
our experiments, taken together with the peak measured
in the χ′′susceptibility and broad maximum in the heat
capacity observed around 8 K1, points to a freezing of
domain walls to form spin clusters, resulting in a static
disordered or glassy magnetic system25. There are also,
of course, still significant slow dynamic fluctuations in
the system, measured by the majority asymmetry com-
ponent with amplitude p.
The observed disorder may be explained in terms of the
predictions of the ANNNI model8. Although the devil’s
staircase of commensurate spin structures is predicted
from a continuum treatment of the model, a treatment
based on a discrete Hamiltonian predicts the existence of
an array of “chaotic” states at a higher energy than the
devil’s staircase, comprising a random array of domain
wall solitons and antisolitons. Such states are metastable,
but are separated from the true Devil’s staircase of stable
states by relatively high energy barriers, making it impos-
sible for a system to relax into the devil’s staircase states
in a finite amount of time8. Instead of LRO, the chaotic
state possesses an intrinsic randomness which means it
may be described as spin glass-like. This picture there-
fore provides an explanation within the framework of the
ANNNI model for the disordered magnetic state that we
observe in CaCo1.05Mn0.95O6. We note, however, that
additionals factors not considered in this simple model
may contribute to the glassy character, including the pos-
sibility of clustering of the Mn4+and Co2+ions and the
effect of complex interchain exchange interactions27.
Finally we note that it is possible that the behavior ob-
served in CaCo1.05Mn0.95O6reflects the existence of both
series spin relaxation processes (where the freezing of one
relaxation process allows another to freeze at a lower
temperature) and also coexistent parallel relaxation pro-
cesses which persist independently. Both of these types of
process have been advanced to explain dynamics in glassy
systems28and it is likely that they are important in a
complex dynamic system such as Ca3Co1.05Mn0.95O6.
Part of this work was carried out at SµS, Paul Scherrer
Institut, Villigen, CH. We are grateful to H. Luetkens
and A. Amato for experimental assistance and to J.M.
Yeomans for useful discussions. This work is supported
by the EPSRC (UK). Work at Rutgers was supported by
the DOE under Grant No. DE-FG02-07ER46382.
∗Electronic address: email@example.com
1Y.J. Choi et al., Phys. Rev. Lett. 100, 047601 (2008).
2Y.J. Jo et al., Phys. Rev. B 79, 012407 (2009).
3H. Wu et al., Phys. Rev. Lett. 102, 026404 (2009).
4Y. Zhang, H.J. Xiang and M.-H. Whangbo, Phys. Rev. B
79, 054432 (2009)
5S. Rayaprol, K. Sengupta and E.V. Sampathkumaran,
Solid State Commun. 128, 79 (2003).
6C.H. Hervoches et al., J. Solid State Chem. 182, 331
7V.G. Zubkov et al., J. Solid State Chem. 160, 293 (2001).
8P. Bak, Rep. Prog. Phys. 45, 587 (1982) and references
9P. Bak and J. von Boehm, Phys. Rev. B 21, 5297 (1980).
10M. E. Fisher and W. Selke, Phys. Rev. Lett. 44, 1502
11J.M. Yeomans in Solid State Physics 41, 151 (Academic
Press, Orlando, 1988).
12J.-J. Kim, S. Mori and I. Harada, J. Phys. Soc. Japan, 65,
13L.P. Le et al., Phys. Rev. B 65, 024432 (2001).
14S. Takeshita et al., J. Phys. Soc. Japan 75, 034712 (2006).
15J. Sugiyama et al, Phys. Rev. B 72, 064418 (2005).
16S.J. Blundell, Contemp. Phys. 40, 175 (1999).
17R.S. Hayano et al., Phys. Rev. B 20, 850 (1979).
18Decoupling implies limt→∞A(t) = A(0). The maximum
available field using the GPS spectrometer is 0.6 T.
19R.H. Heffner et al., Phys. Rev. Lett. 85, 3285 (2000).
20T. Lancaster et al., Phys. Rev B. 75 064427 (2007).
21T. Lancaster et al., Phys. Rev. B 73, 184436 (2006).
22T. Lancaster et al., J. Phys. Condens. Matter 16, S4563
23J.S. Gardner et al., Phys. Rev. Lett. 82, 1012 (1999).
24W. Low Paramagnetic Resonance in Solids (Academic
Press, New York, 1960) and references therein.
25K. Binder and A.P. Young, Rev. Mod. Phys. 58 801 (1986).
26A full analysis of this feature is difficult and requires
the replacement of e−λ2tin Eq.(1) with (1 − a)e−λ2t+
ae−λ3tcos(2πνt), where a(≪ 1) is an amplitude and λ3 a
large relaxation rate. Since this approach does not alter
our interpretation we do not persue it here.
27V. Kiryukhin et al., Phys. Rev. Lett. 102, 187202 (2009).
28R.G. Palmer et al., Phys. Rev. Lett. 53, 958 (1984) and