Papers in Physics, vol. 7, art. 070009 (2015)

Received: 17 May 2015, Accepted: 12 June 2015
Edited by: A. Vindigni
Reviewed by: M. Perfetti, Dipartimento di Chimica, Universitá di Firenze, Italy
Licence: Creative Commons Attribution 3.0
DOI: http://dx.doi.org/10.4279/PIP.070009

www.papersinphysics.org

ISSN 1852-4249

An intermediate state between the kagome-ice and the fully polarized
state in Dy2Ti2O7

S. A. Grigera,1, 2∗ R. A. Borzi,3 D. G. Slobinsky,1† A. S. Gibbs,1

R. Higashinaka,4 Y. Maeno,5 T. S. Grigera3

Dy2Ti2O7 is at present the cleanest example of a spin-ice material. Previous theoretical
and experimental work on the first-order transition between the kagome-ice and the fully
polarized state has been taken as a validation for the dipolar spin-ice model. Here we inves-
tigate in further depth this phase transition using ac-susceptibility and dc-magnetization,
and compare this results with Monte-Carlo simulations and previous magnetization and
specific heat measurements. We find signatures of an intermediate state between the
kagome-ice and full polarization. This signatures are absent in current theoretical models
used to describe spin-ice materials.

I. Introduction

Spin-ice materials are deceptively simple in their
constitution: classical Ising spins with nearest-
neighbour ferromagnetic interactions forming a py-
rochlore lattice. This crystal structure can be
thought as an alternating stack of kagome and tri-

∗E-mail: sag2@st-and.ac.uk
†Now at: Departamento de Ingenieŕıa Mecánica, Facul-

tad Regional La Plata, Universidad Tecnológica Nacional,
1900 La Plata, Argentina.

1 School of Physics and Astronomy, University of St An-
drews, North Haugh, St Andrews KY16 9SS, UK

2 Instituto de F́ısica de Ĺıquidos y Sistemas Biológicos,
UNLP-CONICET, 1900 La Plata, Argentina

3 Instituto de Investigaciones Fisicoqúımicas Teóricas
y Aplicadas UNLP-CONICET and Departamento de
F́ısica, Facultad de Ciencias Exactas, Universidad Na-
cional de La Plata, 1900 La Plata, Argentina

4 Graduate School of Science, Tokyo Metropolitan Univer-
sity, Hachioji, Tokyo 192-0397, Japan.

5 Department of Physics, Kyoto University, Kyoto 606-
8502, Japan.

angular lattices along the [111] direction. The spins
sit at the vertices of tetrahedra and can point ei-
ther to their center or towards the outside. The
magnetic frustration can be seen at the level of a
single tetrahedron: the energy is minimized by hav-
ing two spins pointing inwards and two outwards.
This is the ice rule, which corresponds exactly to
the Pauling rules for protons in water ice; like the
latter, it also leads to zero-point entropy, a charac-
teristic signature of spin-ice systems [1].

We have chosen to work on Dy2Ti2O7 as the
cleanest example of a spin-ice material. Its ground
state properties can be well described by a model
with only an effective nearest neighbour exchange
interaction Jsi of ≈ 1.1 K [2]. Within this frame-
work, when one applies an external magnetic field
H in [111] below 1 K, the polarization of the sys-
tem will happen in two steps. First, the spins in the
triangular lattice that lie parallel to [111] will ori-
ent along the magnetic field, removing part of the
residual entropy but with no change in the configu-
rational energy [3, 4]. When the magnetic moment
of this sublattice has saturated, the magnetization
M cannot be further increased without breaking

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Papers in Physics, vol. 7, art. 070009 (2015) / S. A. Grigera et al.

the spin-ice rule. This leads to a plateau as a func-
tion of field at M = 3.33 µB/Dy-ion, characteris-
tic of the kagome ice state. At higher fields, the
spins in the kagome lattice are finally fully polar-
ized, leading to a sudden but continuous increase
in M towards its saturation. This behavior was
predicted theoretically and found in Monte Carlo
simulations [5, 6]. In spite of this, something differ-
ent happens in real spin-ice materials.

Since the magnetic moment of the magnetic ions
in spin-ice materials is quite large – the one asso-
ciated with Dy3+ ions in Dy2Ti2O7 is near 10µB–
, long range dipolar interactions have to be con-
sidered [7]. These interactions do not alter the
zero field ground state [8], but have a big effect
on its excitations. In relation to this, the transi-
tion to the fully polarized state —which is the main
concern of this paper— experiments a qualitative
change. T. Sakakibara and collaborators [9] studied
experimentally the magnetization with H// [111]
to temperatures much smaller than Jsi. After a
well defined plateau at ≈ 3.33 µB/Dy-ion, they ob-
served a very sharp increase in the magnetization.
The presence of hysteresis was a convincing argu-
ment that the real system reaches the fully polar-
ized state through a metamagnetic first order phase
change at the lowest temperatures. The change
in M becomes continuous at the critical end-point
Tc = 360 ± 20mK and µ0Hc ≈ 0.93 T [9].

The change in character of this transition —
from a crossover to a discontinuity when dipolar
interactions are included— was later understood in
terms of the defects associated with the breaking of
the ice rules, or monopoles. The nearest neighbors
model corresponds to the case of free non-conserved
monopoles sitting in a diamond lattice. Including
dipolar interactions implies turning on a Coulomb
interaction between these charges, allowing them
to condense through a real first order transition
[10]. Numerical simulations (including Ewald sum-
mations to take into account long range interac-
tions) proved this picture right, and provided an
additional validation to the dipolar model [10]. The
Mvs. H curves obtained in these simulations are
quite symmetrical around Hc. The jump in the
magnetization ∆M(T) when crossing the first order
transition line grows very abruptly with decreasing
temperature: for T only ≈ 10% below Tc, ∆M(T)
amounts to ≈ 90% of the total change in magne-
tization from the kagome ice to full saturation. In

other words, almost full order is achieved in the sys-
tem for temperatures just below Tc and a magnetic
field of 1 T.

Specific heat Cp measurements confirmed the ex-
istence of a critical end-point —a sharp peak is
clearly seen very near the precise spot in field and
temperature specified by Sakakibara et al. [11].
However, the identification of a single first-order
line below Tc is less clear. The Cp(T)vs.H curves
show peaks at the fields Hc(T) identified in [9] as
the first-order line, albeit of much smaller ampli-
tude than that at Tc. Additionally, below 300 mK,
a second peak at higher fields is discernible [11].
Even at the lowest temperatures (100 mK), mag-
netic fields above 2 T are needed to coerce the spe-
cific heat down to 0. This suggests that, in spite
of the absence of thermal excitations, the system
does not reach full polarization immediately after
the first order transition from the kagome ice, and
an intermediate state establishes between these two
well-known phases. This specific heat features were
confirmed by ac-susceptibility measurements on the
same samples [12]. In all cases, the sample sat at a
fixed platform with respect to the magnetic field
and therefore the alignment with respect to the
[111] direction was within a few degrees. An an-
gular dependent study of the magnetization with
Sato and coworkers [19] showed these asymmetries,
and additional features in the polarization tran-
sition were seen at small angles away from [111].
The implications of these results in the current un-
derstanding and modeling of the spin-ice materials
have not been considered.

In this paper, we study in detail this additional
intermediate state, and show that it cannot be ex-
plained by any of the models currently used to
study spin-ice materials. Working at small angles
away from [111], we looked for a magnetic signa-
ture by repeating the static magnetization mea-
surements in several samples. In addition, improv-
ing the sensitivity by three orders of magnitude, we
measured ac-susceptibility at different frequencies,
which also allowed us to do a characterisation of
the dynamics of the observed transitions. In order
to gain further insight into this possible interme-
diate state, we performed Monte Carlo simulations
of the experimental situation using the currently
accepted models including Ewald summations and
exchange interactions up to the third nearest neigh-
bor [13, 14].

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Papers in Physics, vol. 7, art. 070009 (2015) / S. A. Grigera et al.

Figure 1: Real (left) and Imaginary (right) parts of the ac-susceptibility as a function of temperature and
magnetic field, for temperatures between 50 mK and 6000 mK and magnetic fields between -4 and 4 T.
The oscillatory field was of amplitude 0.05 Oe and at a frequency of 87Hz. The zero-field Schottky-type
anomaly corresponding to the onset of spin-ice correlations, and the peaks corresponding to the critical
point at ±1 T and ≈400 mK are clearly seen.

II. Methods

For our work, we measured several Dy2Ti2O7 single
crystals grown in Kyoto and in St Andrews with the
floating-zone method. Samples were oriented us-
ing Laue diffraction and cut into 3 mm long prisms
of square or octagonal section of approximately
1 mm2, with the [111] direction along the long axis
to reduce demagnetising effects with the field in
the vicinity of [111] (≈ 5o). The experiments were
performed in a dilution refrigerator in St Andrews.
Samples were thermally grounded to the mixing
chamber through gold wires attached into them
with silver paint. For susceptibility, we used a
drive field of 3.3·10−5 T r.m.s., and counter-wound
pickup coils each consisting of approximately 1000
turns of 12 µm diameter copper wire. The filling
factor of the sample in the pick up coil was of ap-
proximately 90%. We measured using drive fields of
frequencies varying from approx. 10 Hz to 1.0 kHz.
Low temperature transformers mounted on the 1 K
pot of the dilution refrigerator were used through-
out to provide an initial signal boost of approx-
imately a factor of 100. The magnetization was
measured using a home-built capacitance Faraday
magnetometer [15].

III. Results and Discussion

Figure 1 shows the real (∆χ′, left) and imaginary
(∆χ′′, right) parts of the ac-susceptibility χ as a
function of temperature and magnetic field in the

whole area of interest. The excitation frequency
in this case is ω = 87 Hz; the main features we
describe in the following are qualitatively indepen-
dent of ω. At zero field, there is a very noticeable
peak in both ∆χ′ and ∆χ′′ for T ≈ 2 K. This cor-
responds to the Schottky-type anomaly associated
with the onset of spin-ice correlations of the system.
The magnetic field axis spans from -4 to 4 T, and
we can clearly see in the real part two peaks (at
positive and negative fields) corresponding to the
critical point at ≈ ±1 T and ≈ 400 mK. For tem-
peratures below 400 mK, we see a much smaller fea-
ture in ∆χ′′, which has a correspondence in ∆χ′′:
a ridge with an amplitude that decreases as a func-
tion of temperature. The magnitude of the latter is
comparatively very small. At low temperatures and
for fields 0.3 T< |µ0H| < 0.9 T, and 2 T < |µ0H|,
the susceptibility is very low, in accordance to the
kagome ice plateau and the saturation in the mag-
netization, respectively.

We now concentrate on the real part of the sus-
ceptibility at temperatures below Jsi. In Fig. 2,
we can see a series of curves at fixed temperatures
(from 50 to 500 mK) and fields between −3.5 and
3.5 T. The excitation field used was 0.05 Oe, and
the frequency 87 Hz. The curves have been off-
set by 30% for clarity. The field was swept from
negative to positive values. Before the kagome
ice state is established, the low field susceptibility
(|µ0H| < 0.3 T) at temperatures below 600 mK is
strongly dependent on the magnetic field sweep rate
and direction (increasing or decreasing), both signs

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Papers in Physics, vol. 7, art. 070009 (2015) / S. A. Grigera et al.

Figure 2: Low temperature real part of the ac-
susceptibility as a function of field at fixed temper-
atures as indicated in the plot. The excitation field
was 0.05 Oe at a frequency of 87Hz. The curves are
offset by 30% for clarity. As temperature is lowered
from 400 mK, the peak at approx. 1 T at 400mK
rapidly decreases in amplitude, and splits into two
peaks at lower temperatures.

of out-of-equilibrium behavior. At higher magnetic
fields, we only observe a small difference in the
height of the peaks at around ±1 T, depending on
whether the transitions are swept upwards or down-
wards in field. The position changes very little,
and the shape of the features is unaltered. As we
lower the temperature, the peak at ≈ 1 T decreases
markedly in amplitude, but without a correspond-
ing change in its high field side shoulder. Below
400 mK, it eventually splits into two distinct fea-
tures. Their separation in the field axis (≈ 0.1 T
at 300 mK) is consistent with previous measure-
ments for a similar sample orientation with respect
to [111] [19]. While the first set of peaks has a
correlate in the imaginary part of ∆χ (not shown
here), no feature is discernible in ∆χ′′ for the peaks
at higher fields.

In Fig. 3, we have plotted the position of these
peaks as a function of field and temperature (white
circles), and the position of the critical point (black
circle). We have taken the specific heat data from
reference [11] and determined the position of the
peaks in C vs. H for different temperatures. These
are plotted in this same graphic as red symbols.

Figure 3: Phase diagram with field slightly tilted
from [111] (θ ≤ 5o). An intermediate phase is seen
between the kagome-ice and fully polarized regions.
The black circle is the critical point as identified
from a peak in the real part of the ac-susceptibility,
χ′. The dotted white circles correspond to a small
doubled peaks seen in χ′ with a corresponding fea-
ture in the imaginary part χ′′, while the white cir-
cles denote small peak in χ′ with no signature in χ′′.
The red circles are taken from peaks in the specific
heat (C) measurements of reference [11]. The main
divergence of C seen in reference [11] and identified
as a critical point coincides with the critical point
(black circle).

The coincidence between these two experiments of
different quantities, on different samples, laborato-
ries and experimental setup is remarkable.

As mentioned before, this secondary peak at
higher fields is absent in the dM/dH data presented
on Ref. [9]. We measured the magnetization using
a Faraday balance on the same samples and un-
der similar temperature and field conditions than
before [16]. The main body of Fig. [17] shows our
dM/dH as a function of field, compared with curves
of ∆χ at T = 100 mK and frequencies spanning two
orders of magnitude (from ω ≈ 10 to 1000 Hz). For
clarity, we have multiplied ∆χ by a factor of twenty.
The peak in dM/dH is markedly asymmetric, with
an extended tail in the high field side but no addi-
tional feature is seen at high fields, in coincidence
with Sakakibara’s observations. On the other hand,
the second peak is clearly seen for low tempera-
ture (T < 400 mK) at all measured frequencies in

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Papers in Physics, vol. 7, art. 070009 (2015) / S. A. Grigera et al.

0.5 1.0 1.5
0

10

20

 

 [
B
/ T

 D
y]

0.5 1.0 1.5 2.0
0

5

10

15

20

25
 

 [
B
/ T

 D
y]

Field (tesla)

20X

 dM/dH

Figure 4: dM/dH (dotted line) and real part of ac-
susceptibility measured at different excitation fre-
quencies, from top to bottom: 19, 37, 77, 136, 277,
561, and 1117 Hz, and for T = 100mK. For ease
of comparison, the latter have been multiplied by a
factor 20, and normalized to the amplitude of sec-
ond peak (no imaginary part has been measured
for this feature). The inset shows both sets of data
in the same scale.

the ac-susceptibility. While these two experiments
seem to be in mutual contradiction, the issue can be
easily explained in terms of the resolutions of both
techniques. The inset of Fig. 4 shows both sets of
data on the same scale; we can see that the high
field shoulder on the dM/dH peak directly corre-
sponds (in the limit of long measurement times or
low frequencies) to the second peak detected with
ac-susceptibility.

Through this analysis, we can see that the exper-
imental volume of data concerning this transition
seems to be compatible. Between ≈ 300mK and
the lowest temperatures (50 mK in Ref. [9]), only
≈ 60% of the total change in magnetization occurs
when traversing the first order transition line. The
remaining 40% is delivered gradually when the field
is further increased to values well above 1.5Hc, in
a fashion that does not seem to depend much on
temperature (see Fig. 3 on Ref. [9]). This gradual
(as opposed to discontinuous) change is behind the
asymmetric shape of the magnetization curves, and
the second set of peaks in Cp and ∆χ.

The theoretical prediction for the transition be-
tween kagome-ice to fully polarized state with field

in [111] was of a single transition—the “dimer to
monomer” transition of refs. [6, 18]. A small ad-
ditional perpendicular field – present in the exper-
iments at small angles away from [111]– induces
order in the dimers in the kagome-ice state, but
does not change the prediction of a single transi-
tion into the fully polarised “monomer” state [18].
This might hold true when further interactions are
added, such as dipolar or further neighbor exchange
interactions. In order to investigate this, we per-
formed a numerical check. We did extensive Monte
Carlo simulations of the dipolar model including
Ewald summations to account for the dipolar long
range interactions. We also added exchange inter-
actions up to third nearest neighbors (taking the
exchange constants and other parameters within
the constraints given by refs. [13, 14]). We ex-
plored a wide range of field angles around [111], but
were unable to detect a double feature in CV at low
temperatures compatible with the experimental ob-
servations. It is then worth stressing that the very
observation of a second feature –even when taking
into account a possible sample misalignment– asks
for new ingredients in the Hamiltonians that are
regularly used to describe spin-ice materials.

Given these considerations, it is difficult to dis-
cuss on the nature of this intermediate state. It is
tempting to think of some sort of “charge” ordering
in the diamond lattice (2-in 2-out tetrahedra within
a majority of 3-1 and 1-3), previous to the final
Zn-blende arrangement, where only ≈ 40 − 50% of
the sites are occupied by single monopoles. Note
that this does not rule out the possibility of still
storing some residual entropy, since there are dif-
ferent spin configurations that generate the same
charge within a given tetrahedron. We have not
found previous data of the evolution of the entropy
as a function of field at temperatures well below
Tc. However, the very asymmetric shape of the en-
tropy at 350 mK obtained using the magnetocaloric
effect shows that at this temperature the system is
already experiencing a strong first order metamag-
netic transition, as mentioned in Ref. [20]. This
work shows that a big fraction of the residual en-
tropy stored in the kagome planes remains in the
system well above Hc [20], suggesting that the in-
termediate state is indeed a partially disordered
one.

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Papers in Physics, vol. 7, art. 070009 (2015) / S. A. Grigera et al.

IV. Conclusions

In conclusion, we observe an intermediate state be-
tween the kagome-ice and the fully polarized state
when the field is slightly tilted from the [111] di-
rection. The signature of a double step we find
in ac-susceptibility and magnetization measure-
ments is also present in earlier calorimetric mea-
surements, and suggested by magnetocaloric effect
experiments. This feature cannot be captured by
the models regularly used to describe spin-ice sys-
tems, fact that asks for further model refinements.
At present, this data stands as a challenge for the
development of a realistic theoretical model of spin-
ice materials.

Acknowledgements - We thank Joseph Betouras,
Andrew Green and Chris Hooley for useful discus-
sions. SAG would like to acknowledge financial
support from the Royal Society (UK), RAB and
TSG from CONICET, UNLP and ANPCYT (Ar-
gentina).

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Papers in Physics, vol. 7, art. 070009 (2015) / S. A. Grigera et al.

ac-susceptibility and the magnetization measure-
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