Magnetism in the high-Tc analogue Cs2AgF4 studied with muon-spin relaxation
T. Lancaster,1, ∗ S.J. Blundell,1 P.J. Baker,1 W. Hayes,1 S.R. Giblin,2 S.E.
McLain,2 F.L. Pratt,2 Z. Salman,1, 2 E.A. Jacobs,3 J.F.C. Turner,3 and T. Barnes3
Clarendon Laboratory, Oxford University Department of Physics, Parks Road, Oxford, OX1 3PU, UK
ISIS Facility, Rutherford Appleton Laboratory, Chilton, Oxfordshire OX11 0QX, UK
Department of Chemistry and Neutron Sciences Consortium,
University of Tennessee, Knoxville, Tennessee 37996, USA
(Dated: November 4, 2018)
We present the results of a muon-spin relaxation study of the high-Tc analogue material Cs2AgF4.
We find unambiguous evidence for magnetic order, intrinsic to the material, below TC = 13.95(3) K.
The ratio of inter- to intraplane coupling is estimated to be |J ′/J | = 1.9 × 10−2, while fits of the
temperature dependence of the order parameter reveal a critical exponent β = 0.292(3), implying an
intermediate character between pure two- and three- dimensional magnetism in the critical regime.
Above TC we observe a signal characteristic of dipolar interactions due to linear F–µ
+–F bonds,
allowing the muon stopping sites in this compound to be characterized.
PACS numbers: 74.25.Ha, 74.72.-h, 75.40.Cx, 76.75.+i
Twenty years after its discovery, high-Tc superconduc-
tivity remains one of the most pressing problems in con-
densed matter physics. High-TC cuprates share a lay-
ered structure of [CuO2] planes with strong antiferro-
magnetic (AFM) interactions between S = 1/2 3d9 Cu2+
ions [1, 2]. However, analogous materials based upon
3d transition metal systems such as manganites [3] and
nickelates [4] share neither the magnetic nor the super-
conducting properties of the high-TC cuprates, leading to
speculation that the spin- 1
character of Cu2+ is unique
in this context. A natural extension to this line of in-
quiry is to explore compounds based on the 4d analogue
of Cu2+, namely S = 1
4d9 Ag2+ [5]; this motivated the
synthesis of the layered fluoride Cs2AgF4 which contains
silver in the unusual divalent oxidation state [6, 7]. This
material possesses several structural similarities with the
superconducting parent compound La2CuO4; it is com-
prised of planes of [AgF2] instead of [CuO2] separated by
FIG. 1: (Color online.) Structure of Cs2AgF4 showing a pos-
sible magnetic structure. Candidate muon sites occur in both
the [CsF] and [AgF2] planes.
planes of [CsF] instead of [LaO] (Fig. 1).
Magnetic measurements [7] suggest that in contrast
to the antiferromagnetism of La2CuO4, Cs2AgF4 is well
modelled as a two-dimensional (2D) Heisenberg ferro-
magnet (described by the Hamiltonian H = J
〈ij〉 Si ·
Sj) with intralayer coupling J/kB = 44.0 K. The ob-
servation of a magnetic transition below TC ≈ 15 K
with no spontaneous magnetization in zero applied field
(ZF) and a small saturation magnetization (∼ 40 mT),
suggests the existence of a weak, AFM interlayer cou-
pling. This behavior is reminiscent of the 2D ferromag-
net K2CuF4 [8], where ferromagnetic (FM) exchange re-
sults from orbital ordering driven by a Jahn-Teller distor-
tion [9, 10]. On this basis, it has been suggested that in
Cs2AgF4 a staggered ordering of dz2−x2 and dz2−y2 hole-
containing orbitals on the Ag2+ ions gives rise to the
FM superexchange [7]. An alternative scenario has also
been advanced on the basis of density functional calcula-
tions in which a d3z2−r2 − p− dx2−y2 orbital interaction
through the Ag–F–Ag bridges causes spin polarization of
the dx2−y2 band [11].
Although inelastic neutron scattering measurements
have been carried out on this material [7], both Cs and
Ag strongly absorb neutrons, resulting in limited resolu-
tion and a poor signal-to-noise ratio. In contrast, spin-
polarized muons, which are very sensitive probes of local
magnetic fields, suffer no such impediments and, as we
shall see, are ideally suited to investigations of the mag-
netism in fluoride materials.
In this paper we present the results of a ZF muon-
spin relaxation (µ+SR) investigation of Cs2AgF4. We
are able to confirm that the material is uniformly ordered
throughout its bulk below TC and show that the critical
behavior associated with the magnetic phase transition
is intermediate in character between 2D and 3D. In ad-
dition, strong coupling between the muon and F− ions
allows us to characterise the muon stopping states in this
http://arxiv.org/abs/0704.0604v1
compound.
ZF µ+SR measurements were made on the MuSR in-
strument at the ISIS facility, using an Oxford Instru-
ments Variox 4He cryostat. In a µ+SR experiment 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 experiment
is the time evolution of the muon spin polarization, the
behavior of which depends on the local magnetic field B
at the muon site [12]. Polycrystalline Cs2AgF4 was syn-
thesised as previously reported [7]. Due to its chemical
reactivity, the sample was mounted under an Ar atmo-
sphere in a gold plated Ti sample holder with a cylindri-
cal sample space of diameter 2.5 cm and depth 2 mm.
A 25 µm thick window was screw-clamped onto a gold
o-ring on the main body of the sample holder resulting
in an airtight seal.
Example ZF µ+SR spectra measured on Cs2AgF4 are
shown in Fig. 2(a). Below TC (Fig. 2(c)) we observe
oscillations in the time dependence of the muon polar-
ization (the “asymmetry” A(t) [12]) which are character-
istic of a quasi-static local magnetic field at the muon
stopping site. This local field causes a coherent preces-
sion of the spins of those muons for which a component
of their spin polarization lies perpendicular to this local
field (expected to be 2/3 of the total spin polarization
for a powder sample). The frequency of the oscillations
is given by νi = γµ|Bi|/2π, where γµ is the muon gyro-
magnetic ratio (= 2π × 135.5 MHz T−1) and Bi is the
average magnitude of the local magnetic field at the ith
muon site. Any fluctuation in magnitude of these local
fields will result in a relaxation of the oscillating signal
[13], described by relaxation rates λi.
Maximum entropy analysis (inset, Fig. 2(c)) reveals
two separate frequencies in the spectra measured below
TC, corresponding to two magnetically inequivalent muon
stopping sites in the material. The precession frequen-
cies, which are proportional to the internal magnetic field
experienced by the muon, may be viewed as an effective
order parameter for these systems [12]. In order to ex-
tract the temperature dependence of the frequencies, the
low temperature data were fitted to the function
A(t) =
Ai exp(−λit) cos(2πνit) (1)
+A3 exp(−λ3t) +Abg,
where A1 and A2 are the amplitudes of the precession
signals and A3 accounts for the contribution from those
muons with a spin component parallel to the local mag-
netic field. The term Abg reflects the non-relaxing signal
from those muons which stop in the sample holder or
cryostat tail.
The ratio of the two precession frequencies was found
to be ν2/ν1 = 0.83 across the temperature range T < TC
and this ratio was fixed in the fitting procedure. The
FIG. 2: (Color online.) (a) Temperature evolution of ZF
µ+SR spectra measured on Cs2AgF4 between 1.3 K and
59.7 K. (b) Above TC low frequency oscillations are observed
due to the dipole-dipole coupling of F–µ+–F states. Inset:
The energy level structure allows three transitions, leading
to three observed frequencies. (c) Below TC higher frequency
oscillations are observed due to quasistatic magnetic fields
at the muon sites. Inset: Maximum entropy analysis reveal
two magnetic frequencies corresponding to two magnetically
inequivalent muon sites.
amplitudes Ai were found to be constant across the tem-
perature range and were fixed at values A1 = 1.66%,
A2 = 3.74% and A3 = 5.54%. This shows that the prob-
ability of a muon stopping in a site that gives rise to fre-
quency ν1 is approximately half that of a muon stopping
in a site that corresponds to ν2. We note also that A3 is in
excess of the expected ratio of A3/(A1 +A2) = 1/2. The
unambiguous assignment of amplitudes is made difficult
by the resolution limitations that a pulsed muon source
places on the measurement. The initial muon pulse at
ISIS has FWHM τmp ∼ 80 ns, limiting the response for
frequencies above ∼ τ−1mp [12]. We should expect, there-
fore, slightly reduced amplitudes or increased relaxation
(see below) for the oscillating components in our spectra
for which ν1,2 & 5 MHz. The amplitudes of the oscilla-
tions are large enough, however, for us to conclude that
the magnetic order in this material is an intrinsic prop-
erty of the bulk compound. Moreover, above TC there is
a complete recovery of the total expected muon asymme-
try. This observation, along with the constancy of A1,2,3
below TC, leads us to believe that Cs2AgF4 is completely
ordered throughout its bulk below TC.
Fig. 3(a) shows the evolution of the precession frequen-
11-T/TC
0 2 4 6 8 10 12 14
T (K)
FIG. 3: Results of fitting data measured below TC to Eq. (1).
(a) Evolution of the muon-spin precession frequencies ν1
(closed circles) and ν2 (open circles) with temperature. Solid
lines are fits to the function νi(T ) = νi(0)(1 − T/TC)
described in the text. Inset: Scaling plot of the precession
frequencies with parameters TC = 13.95(3) K and β = 0.292.
(b) Relaxation rates λ1 (closed circles), λ2 (open circles) and
λ3 (closed triangles), as a function of temperature showing a
rapid increase as TC is approached from below.
cies νi, allowing us to investigate the critical behavior as-
sociated with the phase transition. From fits of νi to the
form νi(T ) = νi(0)(1−T/TC)β for T > 10 K, we estimate
TC = 13.95(3) K and β = 0.292(3). In fact, good fits to
νi(T ) = νi(0)(1−T/13.95)0.292 are achieved over the en-
tire measured temperature range (that is, no spin wave
related contribution was evident at low temperatures),
yielding ν1(0) = 6.0(1) MHz and ν2(0) = 4.9(2) MHz
corresponding to local magnetic fields at the two muon
sites of B1 = 44(1) mT and B2 = 36(1) mT. A value
of β = 0.292(3) is less than expected for three dimen-
sional models (β = 0.367 for 3D Heisenberg), but larger
than expected for 2D models (β = 0.23 for 2D XY or
β = 0.125 for 2D Ising) [14, 15]. This suggests that in
the critical regime the behavior is intermediate in char-
acter between 2D and 3D; this contrasts with the mag-
netic properties of K2CuF4 where β = 0.33, typical of a
3D system, is observed in the reduced temperature re-
gion tr ≡ (TC − T )/TC > 7 × 10−2, with a crossover to
more 2D-like behavior at tr < 7 × 10−2, where β = 0.22
[16, 17, 18]. Our measurements probe the behavior of
Cs2AgF4 for tr ≥ 5.5 × 10−3, for which we do not ob-
serve any crossover.
A knowledge of TC and the intraplane coupling J , al-
lows us to estimate the interplane coupling J ′. Recent
studies of layered S = 1/2 Heisenberg ferromagnets us-
ing the spin-rotation invariant Green’s function method
[19], show that the interlayer coupling may be described
by an empirical formula
= exp
with a = 2.414 and b = 2.506. Substituting our value of
TC = 13.95 K and using |J |/kB = 44.0 K [7], we obtain
|J ′|/kB = 0.266 K and |J ′/J | = 1.9× 10−2. The applica-
tion of this procedure to K2CuF4 (for which TC = 6.25 K
and |J |/kB = 20.0 K [8]) results in |J ′|/kB = 0.078 K
and |J ′/J | = 3.9 × 10−3. This suggests that, although
highly anisotropic, the interlayer coupling is stronger
in Cs2AgF4 than in K2CuF4. This may account for
the lack of dimensional crossover in Cs2AgF4 down to
tr = 5.5× 10−3.
Both transverse depolarization rates λ1 and λ2 are seen
to decrease with increasing temperature (Fig. 3(b)) ex-
cept close to TC where they rapidly increase. The large
values of λ1,2 at low temperatures may reflect the re-
duced frequency response of the signal due to the muon
pulse width described above. The large upturn in the
depolarization rate close to TC, which is also seen in
the longitudinal relaxation rate λ3 (which is small and
nearly constant except on approach to TC), may be at-
tributed to the onset of critical fluctuations close to TC.
The component in the spectra with the larger precession
frequency ν1 has the smaller depolarization rate λ1 at all
temperatures. These features provide further evidence
for a magnetic phase transition at TC = 13.95 K.
Above TC the character of the measured spectra
changes considerably (Fig.2(a) and (b)) and we observe
lower frequency oscillations characteristic of the dipole-
dipole interaction of the muon and the 19F nucleus [20].
The Ag2+ electronic moments, which dominate the spec-
tra for T < TC, are no longer ordered in the param-
agnetic regime, and fluctuate very rapidly on the muon
time scale. They are therefore motionally narrowed from
the spectra, leaving the muon sensitive to the quasistatic
nuclear magnetic moments. This interpretation is sup-
ported by µ+SR measurements of K2CuF4 where simi-
lar behavior was observed [21]. In many materials con-
taining fluorine, the muon and two fluorine ions form
a strong hydrogen bond usually separated by approxi-
mately twice the F− ionic radius. The linear F–µ+–F
spin system consists of four distinct energy levels with
three allowed transitions between them (inset, Fig. 2(b))
giving rise to the distinctive three-frequency oscillations
observed. The signal is described by a polarization
function [20] D(ωdt) =
uj cos(ωjt)
, where
u1 = 1, u2 = (1 + 1/
3) and u3 = (1 − 1/
3). The
transition frequencies (shown in Fig. 2(b)) are given by
ωj = 3ujωd/2 where ωd = µ0γµγF/4πr
3, γF is the
nuclear gyromagnetic ratio and r is the µ+–19F separa-
tion. This function accounts for the observed frequencies
very well, leading us to conclude that the F–µ+–F bonds
are highly linear.
A successful fit of our data required the multiplication
of D(ωdt) by an exponential function with a small re-
laxation rate λ4, crudely modelling fluctuations close to
TC. The addition of a further exponential component
A5 exp(−λ5t) was also required in order to account for
those muon sites not strongly dipole coupled to fluorine
nuclei. The data were fitted with the resulting relaxation
function
A(t) = A4D(ωdt) exp(−λ4t) +A5 exp(−λ5t) +Abg, (3)
The frequency ωd was found to be constant at all
measured temperatures, taking the value ωd = 2π ×
0.211(1) MHz, which corresponds to a constant F–µ+
separation of 1.19(1) Å, typical of linear bonds [20]. The
relaxation rates only vary appreciably within 0.2 K of
the magnetic transition, increasing as TC is approached
from above, probably due to the onset of critical fluctua-
tions. This provides further evidence for our assignment
of TC = 13.95 K.
Our determination of νi(0) and observation of the lin-
ear F–µ+–F signal allow us to identify candidate muon
sites in Cs2AgF4. Although the magnetic structure of the
system is not known, magnetic measurements [7] suggest
the existence of loosely coupled FM Ag2+ layers arranged
antiferromagnetically along the c-direction. Dipole fields
were calculated for such a candidate magnetic structure
with Ag2+ moments in the ab planes oriented parallel
(antiparallel) to the a direction for z = 0 (z = 1/2). The
calculation was limited to a sphere containing ≈ 105 Ag
ions with localized moments of 0.8 µB [7]. The above
considerations suggest that the muon sites will be situ-
ated midway between two F− ions. Two sets of candidate
muon sites may be identified in the planes containing the
fluorine ions. Magnetic fields corresponding to ν2(0) are
found in the [CsF] planes (i.e. those with z = 0.145 and
z = 0.355) at the positions (1/4, 1/4, z), (1/4, 3/4, z),
(3/4, 1/4, z) and (3/4, 3/4, z). Sites corresponding to
the frequency ν1(0) are more difficult to assign, but good
candidates are found in the [AgF2] planes (at z = 0, 1/2)
at positions (1/4, 1/2, z), (3/4, 1/2, z), (1/4, 0, z) and
(3/4, 0, z). The candidate sites are shown in Fig. 1.
We note that there are twice as many [CsF] planes in a
unit cell than there are [AgF2] planes in agreement with
our observation that components with frequency ν2 oc-
cur with twice the amplitude of those with ν1. Such an
assignment then implies that the presence of the muon
distorts the surrounding F− ions such that their separa-
tion is ∼ 2.38 Å. This contrasts with the in-plane F–F
separation in the unperturbed material of 4.55 Å ([CsF]
planes) and∼ 3.2 Å ([AgF2] planes) [7]. Thus the two ad-
jacent F− ions in the magnetic [AgF2] planes each shift by
∼ 0.4 Å from their equilibrium positions towards the µ+,
demonstrating that the muon introduces a non-negligible
local distortion; however, the distortion in the Ag2+ ion
positions is expected to be much less significant.
In conclusion, we have shown unambiguous evi-
dence for magnetic order in Cs2AgF4 with an exchange
anisotropy of |J ′/J | ≈ 10−2 and critical behavior inter-
mediate in character between 2D and 3D. The presence
of coherent F–µ+–F states allows a determination of can-
didate muon sites and an estimate of the perturbation
of the system caused by the muon probe. This study
demonstrates that µ+SR is an effective and useful probe
of the Cs2AgF4 system. In order to further explore this
system as an analogue to the high-TC materials it is desir-
able to perform investigations of doped materials based
on the Cs2AgF4 parent compound.
Part of this work was carried out at the ISIS facility,
Rutherford Appleton Laboratory, UK. This work is sup-
ported by the EPSRC (UK). T.L. acknowledges support
from the Royal Commission for the Exhibition of 1851.
J.F.C.T and S.E.M acknowledge the U.S. National Sci-
ence Foundation under awards CAREER-CHE 039010
and OISE 0404938, respectively.
∗ Electronic address: t.lancaster1@physics.ox.ac.uk
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mailto:t.lancaster1@physics.ox.ac.uk

  We present the results of a muon-spin relaxation study of the high-Tc
analogue material Cs2AgF4. We find unambiguous evidence for magnetic order,
intrinsic to the material, below T_C=13.95(3) K. The ratio of inter- to
intraplane coupling is estimated to be |J'/J|=1.9 x 10^-2, while fits of the
temperature dependence of the order parameter reveal a critical exponent
beta=0.292(3), implying an intermediate character between pure two- and three-
dimensional magnetism in the critical regime. Above T_C we observe a signal
characteristic of dipolar interactions due to linear F-mu-F bonds, allowing the
muon stopping sites in this compound to be characterized.

Magnetism in the high-Tc analogue Cs2AgF4 studied with muon-spin relaxation
T. Lancaster,1, ∗ S.J. Blundell,1 P.J. Baker,1 W. Hayes,1 S.R. Giblin,2 S.E.
McLain,2 F.L. Pratt,2 Z. Salman,1, 2 E.A. Jacobs,3 J.F.C. Turner,3 and T. Barnes3
Clarendon Laboratory, Oxford University Department of Physics, Parks Road, Oxford, OX1 3PU, UK
ISIS Facility, Rutherford Appleton Laboratory, Chilton, Oxfordshire OX11 0QX, UK
Department of Chemistry and Neutron Sciences Consortium,
University of Tennessee, Knoxville, Tennessee 37996, USA
(Dated: November 4, 2018)
We present the results of a muon-spin relaxation study of the high-Tc analogue material Cs2AgF4.
We find unambiguous evidence for magnetic order, intrinsic to the material, below TC = 13.95(3) K.
The ratio of inter- to intraplane coupling is estimated to be |J ′/J | = 1.9 × 10−2, while fits of the
temperature dependence of the order parameter reveal a critical exponent β = 0.292(3), implying an
intermediate character between pure two- and three- dimensional magnetism in the critical regime.
Above TC we observe a signal characteristic of dipolar interactions due to linear F–µ
+–F bonds,
allowing the muon stopping sites in this compound to be characterized.
PACS numbers: 74.25.Ha, 74.72.-h, 75.40.Cx, 76.75.+i
Twenty years after its discovery, high-Tc superconduc-
tivity remains one of the most pressing problems in con-
densed matter physics. High-TC cuprates share a lay-
ered structure of [CuO2] planes with strong antiferro-
magnetic (AFM) interactions between S = 1/2 3d9 Cu2+
ions [1, 2]. However, analogous materials based upon
3d transition metal systems such as manganites [3] and
nickelates [4] share neither the magnetic nor the super-
conducting properties of the high-TC cuprates, leading to
speculation that the spin- 1
character of Cu2+ is unique
in this context. A natural extension to this line of in-
quiry is to explore compounds based on the 4d analogue
of Cu2+, namely S = 1
4d9 Ag2+ [5]; this motivated the
synthesis of the layered fluoride Cs2AgF4 which contains
silver in the unusual divalent oxidation state [6, 7]. This
material possesses several structural similarities with the
superconducting parent compound La2CuO4; it is com-
prised of planes of [AgF2] instead of [CuO2] separated by
FIG. 1: (Color online.) Structure of Cs2AgF4 showing a pos-
sible magnetic structure. Candidate muon sites occur in both
the [CsF] and [AgF2] planes.
planes of [CsF] instead of [LaO] (Fig. 1).
Magnetic measurements [7] suggest that in contrast
to the antiferromagnetism of La2CuO4, Cs2AgF4 is well
modelled as a two-dimensional (2D) Heisenberg ferro-
magnet (described by the Hamiltonian H = J
〈ij〉 Si ·
Sj) with intralayer coupling J/kB = 44.0 K. The ob-
servation of a magnetic transition below TC ≈ 15 K
with no spontaneous magnetization in zero applied field
(ZF) and a small saturation magnetization (∼ 40 mT),
suggests the existence of a weak, AFM interlayer cou-
pling. This behavior is reminiscent of the 2D ferromag-
net K2CuF4 [8], where ferromagnetic (FM) exchange re-
sults from orbital ordering driven by a Jahn-Teller distor-
tion [9, 10]. On this basis, it has been suggested that in
Cs2AgF4 a staggered ordering of dz2−x2 and dz2−y2 hole-
containing orbitals on the Ag2+ ions gives rise to the
FM superexchange [7]. An alternative scenario has also
been advanced on the basis of density functional calcula-
tions in which a d3z2−r2 − p− dx2−y2 orbital interaction
through the Ag–F–Ag bridges causes spin polarization of
the dx2−y2 band [11].
Although inelastic neutron scattering measurements
have been carried out on this material [7], both Cs and
Ag strongly absorb neutrons, resulting in limited resolu-
tion and a poor signal-to-noise ratio. In contrast, spin-
polarized muons, which are very sensitive probes of local
magnetic fields, suffer no such impediments and, as we
shall see, are ideally suited to investigations of the mag-
netism in fluoride materials.
In this paper we present the results of a ZF muon-
spin relaxation (µ+SR) investigation of Cs2AgF4. We
are able to confirm that the material is uniformly ordered
throughout its bulk below TC and show that the critical
behavior associated with the magnetic phase transition
is intermediate in character between 2D and 3D. In ad-
dition, strong coupling between the muon and F− ions
allows us to characterise the muon stopping states in this
http://arxiv.org/abs/0704.0604v1
compound.
ZF µ+SR measurements were made on the MuSR in-
strument at the ISIS facility, using an Oxford Instru-
ments Variox 4He cryostat. In a µ+SR experiment 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 experiment
is the time evolution of the muon spin polarization, the
behavior of which depends on the local magnetic field B
at the muon site [12]. Polycrystalline Cs2AgF4 was syn-
thesised as previously reported [7]. Due to its chemical
reactivity, the sample was mounted under an Ar atmo-
sphere in a gold plated Ti sample holder with a cylindri-
cal sample space of diameter 2.5 cm and depth 2 mm.
A 25 µm thick window was screw-clamped onto a gold
o-ring on the main body of the sample holder resulting
in an airtight seal.
Example ZF µ+SR spectra measured on Cs2AgF4 are
shown in Fig. 2(a). Below TC (Fig. 2(c)) we observe
oscillations in the time dependence of the muon polar-
ization (the “asymmetry” A(t) [12]) which are character-
istic of a quasi-static local magnetic field at the muon
stopping site. This local field causes a coherent preces-
sion of the spins of those muons for which a component
of their spin polarization lies perpendicular to this local
field (expected to be 2/3 of the total spin polarization
for a powder sample). The frequency of the oscillations
is given by νi = γµ|Bi|/2π, where γµ is the muon gyro-
magnetic ratio (= 2π × 135.5 MHz T−1) and Bi is the
average magnitude of the local magnetic field at the ith
muon site. Any fluctuation in magnitude of these local
fields will result in a relaxation of the oscillating signal
[13], described by relaxation rates λi.
Maximum entropy analysis (inset, Fig. 2(c)) reveals
two separate frequencies in the spectra measured below
TC, corresponding to two magnetically inequivalent muon
stopping sites in the material. The precession frequen-
cies, which are proportional to the internal magnetic field
experienced by the muon, may be viewed as an effective
order parameter for these systems [12]. In order to ex-
tract the temperature dependence of the frequencies, the
low temperature data were fitted to the function
A(t) =
Ai exp(−λit) cos(2πνit) (1)
+A3 exp(−λ3t) +Abg,
where A1 and A2 are the amplitudes of the precession
signals and A3 accounts for the contribution from those
muons with a spin component parallel to the local mag-
netic field. The term Abg reflects the non-relaxing signal
from those muons which stop in the sample holder or
cryostat tail.
The ratio of the two precession frequencies was found
to be ν2/ν1 = 0.83 across the temperature range T < TC
and this ratio was fixed in the fitting procedure. The
FIG. 2: (Color online.) (a) Temperature evolution of ZF
µ+SR spectra measured on Cs2AgF4 between 1.3 K and
59.7 K. (b) Above TC low frequency oscillations are observed
due to the dipole-dipole coupling of F–µ+–F states. Inset:
The energy level structure allows three transitions, leading
to three observed frequencies. (c) Below TC higher frequency
oscillations are observed due to quasistatic magnetic fields
at the muon sites. Inset: Maximum entropy analysis reveal
two magnetic frequencies corresponding to two magnetically
inequivalent muon sites.
amplitudes Ai were found to be constant across the tem-
perature range and were fixed at values A1 = 1.66%,
A2 = 3.74% and A3 = 5.54%. This shows that the prob-
ability of a muon stopping in a site that gives rise to fre-
quency ν1 is approximately half that of a muon stopping
in a site that corresponds to ν2. We note also that A3 is in
excess of the expected ratio of A3/(A1 +A2) = 1/2. The
unambiguous assignment of amplitudes is made difficult
by the resolution limitations that a pulsed muon source
places on the measurement. The initial muon pulse at
ISIS has FWHM τmp ∼ 80 ns, limiting the response for
frequencies above ∼ τ−1mp [12]. We should expect, there-
fore, slightly reduced amplitudes or increased relaxation
(see below) for the oscillating components in our spectra
for which ν1,2 & 5 MHz. The amplitudes of the oscilla-
tions are large enough, however, for us to conclude that
the magnetic order in this material is an intrinsic prop-
erty of the bulk compound. Moreover, above TC there is
a complete recovery of the total expected muon asymme-
try. This observation, along with the constancy of A1,2,3
below TC, leads us to believe that Cs2AgF4 is completely
ordered throughout its bulk below TC.
Fig. 3(a) shows the evolution of the precession frequen-
11-T/TC
0 2 4 6 8 10 12 14
T (K)
FIG. 3: Results of fitting data measured below TC to Eq. (1).
(a) Evolution of the muon-spin precession frequencies ν1
(closed circles) and ν2 (open circles) with temperature. Solid
lines are fits to the function νi(T ) = νi(0)(1 − T/TC)
described in the text. Inset: Scaling plot of the precession
frequencies with parameters TC = 13.95(3) K and β = 0.292.
(b) Relaxation rates λ1 (closed circles), λ2 (open circles) and
λ3 (closed triangles), as a function of temperature showing a
rapid increase as TC is approached from below.
cies νi, allowing us to investigate the critical behavior as-
sociated with the phase transition. From fits of νi to the
form νi(T ) = νi(0)(1−T/TC)β for T > 10 K, we estimate
TC = 13.95(3) K and β = 0.292(3). In fact, good fits to
νi(T ) = νi(0)(1−T/13.95)0.292 are achieved over the en-
tire measured temperature range (that is, no spin wave
related contribution was evident at low temperatures),
yielding ν1(0) = 6.0(1) MHz and ν2(0) = 4.9(2) MHz
corresponding to local magnetic fields at the two muon
sites of B1 = 44(1) mT and B2 = 36(1) mT. A value
of β = 0.292(3) is less than expected for three dimen-
sional models (β = 0.367 for 3D Heisenberg), but larger
than expected for 2D models (β = 0.23 for 2D XY or
β = 0.125 for 2D Ising) [14, 15]. This suggests that in
the critical regime the behavior is intermediate in char-
acter between 2D and 3D; this contrasts with the mag-
netic properties of K2CuF4 where β = 0.33, typical of a
3D system, is observed in the reduced temperature re-
gion tr ≡ (TC − T )/TC > 7 × 10−2, with a crossover to
more 2D-like behavior at tr < 7 × 10−2, where β = 0.22
[16, 17, 18]. Our measurements probe the behavior of
Cs2AgF4 for tr ≥ 5.5 × 10−3, for which we do not ob-
serve any crossover.
A knowledge of TC and the intraplane coupling J , al-
lows us to estimate the interplane coupling J ′. Recent
studies of layered S = 1/2 Heisenberg ferromagnets us-
ing the spin-rotation invariant Green’s function method
[19], show that the interlayer coupling may be described
by an empirical formula
= exp
with a = 2.414 and b = 2.506. Substituting our value of
TC = 13.95 K and using |J |/kB = 44.0 K [7], we obtain
|J ′|/kB = 0.266 K and |J ′/J | = 1.9× 10−2. The applica-
tion of this procedure to K2CuF4 (for which TC = 6.25 K
and |J |/kB = 20.0 K [8]) results in |J ′|/kB = 0.078 K
and |J ′/J | = 3.9 × 10−3. This suggests that, although
highly anisotropic, the interlayer coupling is stronger
in Cs2AgF4 than in K2CuF4. This may account for
the lack of dimensional crossover in Cs2AgF4 down to
tr = 5.5× 10−3.
Both transverse depolarization rates λ1 and λ2 are seen
to decrease with increasing temperature (Fig. 3(b)) ex-
cept close to TC where they rapidly increase. The large
values of λ1,2 at low temperatures may reflect the re-
duced frequency response of the signal due to the muon
pulse width described above. The large upturn in the
depolarization rate close to TC, which is also seen in
the longitudinal relaxation rate λ3 (which is small and
nearly constant except on approach to TC), may be at-
tributed to the onset of critical fluctuations close to TC.
The component in the spectra with the larger precession
frequency ν1 has the smaller depolarization rate λ1 at all
temperatures. These features provide further evidence
for a magnetic phase transition at TC = 13.95 K.
Above TC the character of the measured spectra
changes considerably (Fig.2(a) and (b)) and we observe
lower frequency oscillations characteristic of the dipole-
dipole interaction of the muon and the 19F nucleus [20].
The Ag2+ electronic moments, which dominate the spec-
tra for T < TC, are no longer ordered in the param-
agnetic regime, and fluctuate very rapidly on the muon
time scale. They are therefore motionally narrowed from
the spectra, leaving the muon sensitive to the quasistatic
nuclear magnetic moments. This interpretation is sup-
ported by µ+SR measurements of K2CuF4 where simi-
lar behavior was observed [21]. In many materials con-
taining fluorine, the muon and two fluorine ions form
a strong hydrogen bond usually separated by approxi-
mately twice the F− ionic radius. The linear F–µ+–F
spin system consists of four distinct energy levels with
three allowed transitions between them (inset, Fig. 2(b))
giving rise to the distinctive three-frequency oscillations
observed. The signal is described by a polarization
function [20] D(ωdt) =
uj cos(ωjt)
, where
u1 = 1, u2 = (1 + 1/
3) and u3 = (1 − 1/
3). The
transition frequencies (shown in Fig. 2(b)) are given by
ωj = 3ujωd/2 where ωd = µ0γµγF/4πr
3, γF is the
nuclear gyromagnetic ratio and r is the µ+–19F separa-
tion. This function accounts for the observed frequencies
very well, leading us to conclude that the F–µ+–F bonds
are highly linear.
A successful fit of our data required the multiplication
of D(ωdt) by an exponential function with a small re-
laxation rate λ4, crudely modelling fluctuations close to
TC. The addition of a further exponential component
A5 exp(−λ5t) was also required in order to account for
those muon sites not strongly dipole coupled to fluorine
nuclei. The data were fitted with the resulting relaxation
function
A(t) = A4D(ωdt) exp(−λ4t) +A5 exp(−λ5t) +Abg, (3)
The frequency ωd was found to be constant at all
measured temperatures, taking the value ωd = 2π ×
0.211(1) MHz, which corresponds to a constant F–µ+
separation of 1.19(1) Å, typical of linear bonds [20]. The
relaxation rates only vary appreciably within 0.2 K of
the magnetic transition, increasing as TC is approached
from above, probably due to the onset of critical fluctua-
tions. This provides further evidence for our assignment
of TC = 13.95 K.
Our determination of νi(0) and observation of the lin-
ear F–µ+–F signal allow us to identify candidate muon
sites in Cs2AgF4. Although the magnetic structure of the
system is not known, magnetic measurements [7] suggest
the existence of loosely coupled FM Ag2+ layers arranged
antiferromagnetically along the c-direction. Dipole fields
were calculated for such a candidate magnetic structure
with Ag2+ moments in the ab planes oriented parallel
(antiparallel) to the a direction for z = 0 (z = 1/2). The
calculation was limited to a sphere containing ≈ 105 Ag
ions with localized moments of 0.8 µB [7]. The above
considerations suggest that the muon sites will be situ-
ated midway between two F− ions. Two sets of candidate
muon sites may be identified in the planes containing the
fluorine ions. Magnetic fields corresponding to ν2(0) are
found in the [CsF] planes (i.e. those with z = 0.145 and
z = 0.355) at the positions (1/4, 1/4, z), (1/4, 3/4, z),
(3/4, 1/4, z) and (3/4, 3/4, z). Sites corresponding to
the frequency ν1(0) are more difficult to assign, but good
candidates are found in the [AgF2] planes (at z = 0, 1/2)
at positions (1/4, 1/2, z), (3/4, 1/2, z), (1/4, 0, z) and
(3/4, 0, z). The candidate sites are shown in Fig. 1.
We note that there are twice as many [CsF] planes in a
unit cell than there are [AgF2] planes in agreement with
our observation that components with frequency ν2 oc-
cur with twice the amplitude of those with ν1. Such an
assignment then implies that the presence of the muon
distorts the surrounding F− ions such that their separa-
tion is ∼ 2.38 Å. This contrasts with the in-plane F–F
separation in the unperturbed material of 4.55 Å ([CsF]
planes) and∼ 3.2 Å ([AgF2] planes) [7]. Thus the two ad-
jacent F− ions in the magnetic [AgF2] planes each shift by
∼ 0.4 Å from their equilibrium positions towards the µ+,
demonstrating that the muon introduces a non-negligible
local distortion; however, the distortion in the Ag2+ ion
positions is expected to be much less significant.
In conclusion, we have shown unambiguous evi-
dence for magnetic order in Cs2AgF4 with an exchange
anisotropy of |J ′/J | ≈ 10−2 and critical behavior inter-
mediate in character between 2D and 3D. The presence
of coherent F–µ+–F states allows a determination of can-
didate muon sites and an estimate of the perturbation
of the system caused by the muon probe. This study
demonstrates that µ+SR is an effective and useful probe
of the Cs2AgF4 system. In order to further explore this
system as an analogue to the high-TC materials it is desir-
able to perform investigations of doped materials based
on the Cs2AgF4 parent compound.
Part of this work was carried out at the ISIS facility,
Rutherford Appleton Laboratory, UK. This work is sup-
ported by the EPSRC (UK). T.L. acknowledges support
from the Royal Commission for the Exhibition of 1851.
J.F.C.T and S.E.M acknowledge the U.S. National Sci-
ence Foundation under awards CAREER-CHE 039010
and OISE 0404938, respectively.
∗ Electronic address: t.lancaster1@physics.ox.ac.uk
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mailto:t.lancaster1@physics.ox.ac.uk
