Thermal Relaxation and Heat Transport in the Spin Ice Material Dy2Ti2O7
ABSTRACT The thermal properties of single crystalline Dy2Ti2O7 have been studied in a temperature range from 0.3K to 30K and magnetic fields applied along  direction up to 1.5T.
Based on a thermodynamic field theory various heat relaxation and thermal transport measurements were analysed. So we were
able to present not only the heat capacity of Dy2Ti2O7 in the whole temperature and magnetic field range, but also the different contributions of the magnetic excitations and their
corresponding relaxation times in the spin ice phase. In addition, the thermal conductivity and the shortest relaxation time
were determined by thermodynamic analysis of steady state heat transport measurements. Finally, we were able to reproduce
the temperature profiles recorded in heat pulse experiments on the basis of the thermodynamic field theory using the previously
determined heat capacity and thermal conductivity data without additional parameters. Thus, the thermodynamic field theory
has been proved to be thermodynamically consistent in describing three thermal transport experiments on different time scales.
The observed temperature and field dependencies of heat capacity contributions and relaxation times indicate the magnetic
excitations in the spin ice material Dy2Ti2O7 as thermally activated monopole-antimonopole defects.
KeywordsFrustrated magnetism–Magnetic monopoles–Specific heat–Thermal conductivity
JLTP manuscript No.
(will be inserted by the editor)
B. Klemke · M. Meissner · P. Strehlow ·
K. Kiefer · S. A. Grigera · D. A. Tennant
Thermal Relaxation and Heat Transport
in the Spin Ice Material Dy2Ti2O7
Received: date / Accepted: date
Abstract The thermal properties of single crystalline Dy2Ti2O7have been studied in a temperature range
from 0.3 K to 30 K and magnetic fields applied along  direction up to 1.5 T. Based on a thermodynamic
field theory various heat relaxation and thermal transport measurements were analysed. So we were able
to present not only the heat capacity of Dy2Ti2O7in the whole temperature and magnetic field range, but
also the different contributions of the magnetic excitations and their corresponding relaxation times in the
spin ice phase. In addition, the thermal conductivity and the shortest relaxation time were determined by
thermodynamic analysis of steady state heat transport measurements. Finally, we were able to reproduce
the temperature profiles recorded in heat pulse experiments on the basis of the thermodynamic field theory
using the previously determined heat capacity and thermal conductivity data without additional parameters.
Thus, the thermodynamic field theory has been proved to be thermodynamically consistent in describing three
thermal transport experiments on different time scales. The observed temperature and field dependencies of
heat capacity contributions and relaxation times indicate the magnetic excitations in the spin ice material
Dy2Ti2O7as thermally activated monopole-antimonopole defects.
Keywords frustrated magnetism · magnetic monopoles · specific heat · thermal conductivity
PACS 75.50.Lk · 75.40.-s · 66.70.-f
For more than a decade, the dipolar spin ice compound Dy2Ti2O7has attracted much interest as a model
material for geometrically frustrated spin systems. Forming a pyrochlore lattice of cubic symmetry, the strong
magnetic moments of the rare-earth ion Dy3+occupy a 3-dimensional network of corner sharing tetrahedra.
Among the four Dy-spins in a tetrahedron the ferromagnetic nearest neighbour interactions force geometrical
frustration: each spin is aligned parallel to the local  axes, however, two spins point inward and two
spins outward on each tetrahedron. This is called the “2-in & 2-out” rule or – in analogy to the disordered
B. Klemke · D. A. Tennant
Helmholtz-Zentrum Berlin f¨ ur Materialien und Energie GmbH, Institut komplexe magnetische Materialien, Berlin, Germany
K. Kiefer · M. Meissner
Helmholtz-Zentrum Berlin f¨ ur Materialien und Energie GmbH, Abteilung Probenumgebung, Berlin, Germany
Physikalisch-Technische Bundesanstalt, Institut Berlin, Germany
Technische Universit¨ at Berlin, Institut f¨ ur Thermodynamik, Germany
B. Klemke · M. Meissner · D. A. Tennant
Technische Universit¨ at Berlin, Institut f¨ ur Festk¨ orperphysik, Germany
S. A. Grigera
School of Physics and Astronomy, St. Andrews, UK
Instituto de F´ ısica de L´ ıquidos y Sistemas Biol´ ogicos, CONICET, UNLP, La Plata, Argentina
protons in hexagonal water ice – the “spin ice” state . Under the ice rule, the ground state spin structure
has a six-fold degeneracy and there remains a residual entropy almost the same as the Pauling value for
water ice . In the absence of any structural disorder in the host lattice, a non-zero entropy indicates that
spin ice represents a new state of magnetism . In 2008, Castelnovo et al.  proposed the existence of
magnetic quasi-particles in spin ice materials like Dy2Ti2O7. In the constraint of a “2-in & 2-out” situation,
the flipping of one spin breaks the ice rule and leads to a “1 in & 3 out” and “3 in & 1 out” configuration,which
constitute a pair of topological defects. Continuation of the spin flips over consecutive tetrahedra separates
the defects and creates a string of dipoles with a monopole and an anti-monopole at its ends. In a series of
neutron scattering and muon spectroscopy experiments on Dy2Ti2O7and Ho2Ti2O7, respectively, signatures
of magnetic monopoles have been evidenced: the density and orientation of Dirac strings by diffuse neutron
scattering , the existence of a magnetic Coulomb phase by polarized neutron scattering  and by muon
spin rotation . From these experimental results it must be concluded that magnetic charges exist in a spin
ice material and that they interact by Coulombs law.
Since the early studies on the spin ice problem, bulk measurements such as the heat capacity of Dy2Ti2O7
have revealed the various spin states in this frustrated material with respect to temperature and magnetic
field parallel to the main crystallographic directions [5, 8–10]. However, the published data exhibit obvious
differences, especially in the temperature range below 1 K. Whereas for temperatures above 1 K all data
fairly agree by ±5 %, the differences increase considerably with decreasing temperature below 1 K. These
discrepancies in the magnitude of heat capacity forced us to re-evaluate our measurement technique and above
all the thermodynamic analysis of the measurement.
At low temperatures, heat capacity measurements are mostly based on the relaxation type method. Here,
the sample is thermally linked to the heat sink at constant reference temperature TR. After applying a con-
stant heating power ˙QS (at t = 0) the temperature increase? T(t) = T(t) − TR is recorded. Based on the
to? T =˙QS/KR(1−exp(−t/τSB)), where KRis the conductance of the thermal link at TR. The sample heat ca-
In Fig. 1 the temperature increase? T(t) is shown as recorded in a relaxation type calorimeter for a single
first law of thermodynamics the temperature increase can be calculated in linear approximation according
pacity CRat the reference temperature TRthen follows from the sample-to-bath time constant τSB=CR/KR.
crystalline Dy2Ti2O7sample at different reference temperatures.
time t (s)
0 20 40 60 80
Fig. 1 Temperature increase? T(t) as recorded in a relaxation type calorimeter for a single crystalline Dy2Ti2O7sample at
in time according to? T(t) =? Tmax·(1−exp(−t/τSB)) as shown by the black dashed lines. The relative temperature deviation
(see inset). For TR≤ 530 mK, deviations from the exponential time dependence become obvious. The description of the non-
exponential temperature profiles requires a suitable thermodynamic field theory. The full line graphs represent solutions of
thermodynamic field equations for the proper initial values of thermal relaxation experiment (see section 3.1).
different reference temperatures TR. For a thermodynamically simple material the temperature increase is expected to vary
1−? T(t)/? Tmaxfrom? Tmaxin the semi-logarithmic plot would then follow a straight line; this is only valid for TR= 1029 mK
In the case that the relative temperature deviations 1−? T(t)/? Tmaxfrom the maximal temperature increase
seen in the inset of Fig. 1, this is really true for a reference temperature of 1.029 K. Below 1 K, however,
non-exponential temperature profiles become obvious.
Certainly, several corrections are required to evaluate accurately the heat capacity from the recorded tem-
perature profiles obtained in a relaxation type calorimeter [11, 12]. But the observed non-exponential tem-
perature profiles rather indicate that the spin ice material Dy2Ti2O7does not behave as a thermodynamically
simple system, and additional internal variables have to be introduced to describe its thermodynamic state. So,
in order to analyse the non-exponential temperature profiles, we need a suitable thermodynamic field theory.
Based on the kinetic theory of phonons and the assumption of localized magnetic excitations, thermody-
namic field equations were derived [13, 14]. The solution of these field equations for initial and boundary
conditions, which can be controlled in the experiment, was used to analyse various heat relaxation and ther-
mal transport measurements. So we were able to present not only the heat capacity C(TR,B) of the spin ice
material Dy2Ti2O7in the temperature range from 0.3 K to 30 K and in magnetic fields up to B = 1.5 T, but
also the different contributions of the magnetic excitations and their corresponding relaxation times in the
spin ice phase. By thermodynamic analysis of steady state heat transport, the thermal conductivity and the
shortest relaxation time were determined. In addition, we were able to reproduce the temperature profiles
? T(x,t) recorded in heat pulse experiments on the basis of the previous determined heat capacity and ther-
thermodynamically consistent in describing three thermal transport experiments on different time scales.
The observed temperature and field dependencies of heat capacity contributions and relaxation times in-
dicate the magnetic excitations in the spin ice as thermally activated monopole-antimonopole defects.
? Tmax=˙QS/KRis plotted on a logarithmic scale as function of time, one expects a straight line. As can be
mal conductivity data without additional parameters. Thus, the thermodynamic field theory was proved to be
2 Thermodynamic Field Theory
This section contains an outline of the fundamental thermodynamic concept underlying an interpretation of
the experimental temperature responses obtained in the various heat relaxation and transport measurements.
The main objective of this concept is the determination of macroscopic field variables such as temperature and
heat flux in all positions of the sample and for all times. In order to achieve this objective one needs field equa-
tions. For an insulating crystalline sample the suitable field equations can be derived from the kinetic theory
of a phonon gas. In general, an infinite hierarchy of macroscopic equations based on the Boltzmann-Peierls
equation is required to describe the entire spectrum of the time- and space dependent behaviour of a phonon
gas. The first set of equations in this hierarchy is the nine-moment system which involves the phonon energy
density, the heat flux vector and the deviatoric part of the pressure tensor. For this system, field equations can
be obtained using Callaway’s relaxation time approximation . It is expected that this nine-moment sys-
tem covers all of the thermal transport phenomena to be observed in crystalline systems at low temperatures;
namely ballistic propagation of phonons, second sound and thermal diffusion.
In most crystals, however, the thermodynamic variable space defined by the (nine) phonon state variables
has to be extended by additional macroscopic field variables such as the energy densities of magnetic excita-
tions. This is especially true for magnetic defects in geometrically frustrated magnets treated as noninteracting
quasiparticles. In the following it is assumed that the magnetic excitations are localized, so that heat is trans-
ported by phonons, only.
In the kinetic theory, the state of a phonon gas is described by the phase density f(k,x,t) which represents
the number density of phonons of wave vector k at a position x and timet. For small values of k, the dispersion
relation for all types of phonons has the form ω = cDk, where ω is the angular frequency and cDis the Debye
velocity of phonons. The phase density obeys the Boltzmann-Peierls equation
whereξ isthephasedensityofproduction.Multiplicationofequation(1)withthephononenergy ¯ hω = ¯ hcDk,
the phonon momentum ¯ hkiand the deviatoric part of the phonon momentum flux ¯ hcDk<ikj>/k, respectively,
and integration over all momentum space provide the macroscopic moment equations
¯ hcDkξ d3k
¯ hkiξ d3k(2)
The first equations in the hierarchy of momentum equations (2) contain the phonon energy density ephand
the heat flux qi= cD2pi, given by
¯ hcDk f d3k and(3)
¯ hkif d3k.(4)
The momentum flux Nij= eph/3δij+N<ij>includes the phonon energy density eph, so that the components
of the deviatoric part
f d3k (5)
extend the thermodynamic variable space (eph, qi, N<ij>) of a pure phonon gas in a nine-field theory.
In order to close the system of momentum equations (2), we expand the phase density f around a non-
equilibrium state which is defined through the nine momentum (eph, qi, N<ij>). For small deviation from
equilibrium, the phase density assumes the form
f = fE−
where the phase density in equilibrium is given by
Making use of the phase density (6), the highest momentum M<ijk>appearing in the momentum equations
vanishes as expected in a linear theory.
In Callaway’s relaxation time approximation the collision term ξ is divided into two parts,
ξ = −1
τR(f − fE)−1
τN(f − fN),(8)
where τRis the relaxation time of resistive scattering processes which do not conserve phonon momentum
and which tend to push the phonon gas back to the equilibrium fE. The relaxation time τNis the mean time
of free flight of phonons in normal scattering processes. These momentum conservation processes attempt
to force the phonon gas to a non-equilibrium state, in which the phase density fNtakes the form (6) with
By using both the relaxation time approximation (8) and the linear phase density (6), the field equations
for the phonon gas variables (eph, qi, N<ij>) can be calculated from
The first equation of (9) for the phononenergy density is a conservation law in a pure phonongas. This follows
from the conservation of the energy in normal and resistive scattering processes. As the non-equilibrium part
of the phase density (6) does not contribute to the moment (3), the phonon energy density is obtained directly
from the phase density in equilibrium (7). Thus one finds the well known formula
Consequently, the phonon energy density ephis expressed in terms of the measurable quantity temperature.
To take into account n additional localized excitations in a crystal, the thermodynamic variable space of
a phonon gas (T, qi, N<ij>) has to be extended to the energy densities eν(ν = 1,2,...,n). In the relaxation
time approximation the energy balance equations for these localized excitations reads
The relaxation time τR
phonon gas. For small deviation? T = T −TRfrom a reference temperature TR, the energy density of excitation
νcharacterizes the time needed for a system of excitation ν to reach equilibrium with the
in equilibrium eE
νcan be expanded,
ν(T) = eE
where the specific heat of excitation ν, cR
temperature, ? eν= eν−eE
ν, is defined as the temperature derivative of the energy density
νfrom its equilibrium value at reference
ν(TR), it follows from (11)
νin equilibrium at TR. For the deviation of the energy density eE
The linear field equations for the 9 plus n variables (? T, qi, N<ij>, ? eν) describing the thermal transport in a
The system of field equations (14) provides the basis for the interpretation of our thermal relaxation and
heat transport measurements on Dy2Ti2O7at low temperatures.
dielectric crystal with localized (magnetic) excitations at low temperature are given by
3 Experimental Methods
With our present work we have studied heat relaxation and transport by measurements of the heat capacity, the
thermal conductivity and heat pulse propagation on an isotopic high purity and crystallographic high-quality
single crystal of Dy2Ti2O7. This crystal was grown with enriched162Dy (95 % – 98 %) to a rod shaped single
crystal of a length of L = 50 mm and a cross section of A = 7.55 mm2. The measurements have been
performed in a temperature range from 0.28 K to 30 K and magnetic fields up to 1.5 T at the Laboratory
for Magnetic Measurements (LaMMB) within the HZB neutron scattering facility. The heat capacity of a
small cut-off sliced sample with mass m = 4.24 mg and orientation  parallel to magnetic field was
measured with an in-house developed apparatus designed for the relaxation method operating with heat flow
and heat pulse technique as well . For the thermal transport measurements on the 5 cm long slab with
orientation  parallel to magnetic field, short-time thermometry and heater technique has been developed
using electronic data acquisition up to 100 µs time resolution . For both types of measurements, two
liquid helium based cryomagnets have been used equipped with sorption pumped3He-cryostat inserts .
3.1 Thermal Relaxation Calorimetry
The heat capacity C of a solid body in a magnetic field is defined as the derivative of its internal energy
U(T,F,B) with respect to the temperature T at constant values of the deformation gradient F (or volume
V ∝ detF) and the magnetic flux density B. In measurements using the relaxation method, the heat capacity is
derived from thermodynamic modelling of the whole thermal response to an applied heating power˙Qs(t). In
the case that the measurement time is long compared to the internal thermal diffusion time of the sample, the
thermal relaxation can be described by uniform fields of the thermodynamic variables. This requirement can
be satisfied by using thin samples and a proper conductance KRof the thermal link between sample and bath
at the constant reference temperature TR. Neglecting spatial variations of the thermodynamic variables, the
field equations (14) decouple into a set of partial energy equations (for? T and ? eν) and rate equations for both
then describes the homogeneous thermodynamic process of thermal relaxation in insulating crystals including
localized (magnetic) excitations.
It must be pointed out, however, that in thermal relaxation experiments the temperature response? Tch=
˙Qs(t) are located on the sapphire calorimeter chip. In Fig. 2 the relaxation type calorimeter is sketched to-
gether with the thermal schematic model underlying the description of the heat flow in the calorimeter.
the heat flux q and the deviatoric part of the momentum flux N. The solution of the partial energy equations
Tch−TRis recorded by a chip thermometer. The chip thermometer and the heater generating the heat supply
Fig. 2 Schematic view onto the relaxation calorimeter (on right) used for the heat capacity measurements and the subsequent
thermal model (on left).
A single crystalline sample of isotopically pure Dy2Ti2O7of slab size geometry (0.35 mm thickness,
2.55 mm length, 0.85 mm width) was attached onto the sapphire calorimeter chip using a small amount of
Apiezon N grease of a mass less than 10 µg. The crystal’s long slab axis (which is parallel to ) was
aligned to the magnetic field axis by better than 1◦. The calorimeter temperature Tchis measured with a bare
chip Cernox thermometer type BC 1030  which is glued onto the sapphire calorimeter chip. The calorime-
the thermal boundary resistance between thermometer and sapphire chip and heater and sapphire chip, respec-
tively, has not been taken into account for the thermal model of the calorimeter. Thermometer and heater are
electrically connected with four phosphor bronze wires  leading to four soldering pads thermally anchored
at the copper mount. Four additional nylon strings give mechanical stability to the sapphire chip. The ther-
mal model conductance KR(TR), i. e. the thermal link between sapphire chip and copper base, results from
the metallic conductance of the four phosphor bronze wires only. As the sample is attached to the sapphire
calorimeter chip, the thermal resistance provided by the grease layer and by the acoustic mismatch, as well,
has been taken into account by the thermal model conductance KS(TR).
The temperature responses? Tch(t), measured with the Cernox resistance thermometer, have been recorded
300 mK). The time response of the Cernox chip recorded with the electronic equipment was experimentally
tested to be below 1 ms (for a temperature below 1 K) using pulsed laser light and glass fibre technique .
With respect to the thermal model of the calorimeter as shown in Fig. 2, the linearized field equations,
describing thermal relaxation of the uniform fields (? T(t),? Tch(t),? eν(t)) in a relaxation calorimeter experiment,
by a lock-in electronic technique providing a minimum sampling time of 0.1 s and a rms-noise of 3 µK (at
?? T −? Tch
+KR? T −˙QS= 0
?? Tch−? T
Here,Caddenotes the heat capacity of the calorimeter chip including the thin film heater, the thermometer and
the amount of Apiezon N grease fixing the sample. The set of coupled ordinary differential equations (15) for
the variables? Tch(t),? T(t) and ? eν(t) can be solved numerically with initial values? Tch(0) =? T(0) = ? eν(0) = 0.
function. The quasi-adiabatic heat pulse method was applied for temperatures above 1.5 K. In this calorimet-
ric technique˙Qs(t) is given by a rectangular pulse with a typical width of several seconds.
For temperatures below 1.5 K the heat flow method was used for which the heating power˙Qs(t) is a step
In Fig. 3 we present the measured temperature response? Tch(t) at TR=530 mK where the non-exponential
nential saturation at Tmax
up to 300 s. Just for reference, the green line provides a comparison according
to a single exponential temperature rise, as expected for a simple material. The red line is a best fit to data
calculated from Eqns. (15) and taking into account only one magnetic excitation (n = 1). It is obvious that
additional magnetic excitations with particularly shorter relaxation times (< 5 s) are required to describe the
data at the outset of the temperature response. Satisfactory agreement between theory and experiment was
obtained by including a number of three (ν = α,β,γ) magnetic excitations with contributions cR
to the magnetic heat capacity Cmag= V(cR
The quality of the modelling according to the thermodynamic field theory is demonstrated in the upper part
of Fig. 3, where the deviation of fit to data is plotted accordingly. For the temperature profile recorded in the
time domain from 1 s to 300 s the agreement of theory and experiment is better than ±2 %.
Three magnetic excitations have been also identified at all temperature from 0.3 K to about 1 K and for
magnetic fields up to 1.5 T. Thereby, all fitting procedures have been performed with an unbiased Levenberg-
Marquardt algorithm. We would like to point out that the three identified magnetic excitations exhibit a char-
acteristic temperature and field dependence, and thus making them also interpretable. This is the reason to
limit the number of magnetic excitations to n = 3. For temperatures above 1 K, the deviations of? Tch(t) from
behaviour becomes obvious. Plotted on a log-time scale, the recorded data (open circles) exhibit no expo-
γ) and relaxation times τR
γ(see black line).
Fig. 3 Measured temperature profile at TR=
530 mK (open circles, as show in Fig. 1). The
green line denotes a fit to the data according to a
thermodynamically simple material (with an ex-
ponential temperature profile). The red line ex-
hibits a fit to data according to the thermody-
namic field theory assuming only one magnetic
excitation (n = 1). The black line was calculated
according to the thermodynamic field theory as-
suming three magnetic excitation (n = 3), see
text. On top, the deviation of the fits from the
data are plotted accordingly.
time t (s)
TR= 530 mK
a single exponential behaviour is decreased due to the considerable shortening of relaxation times as the tem-
perature is increased. In this case the Dy2Ti2O7crystal behaves as a thermally homogeneous material without
any internal energy relaxation, or more precisely, the thermal equilibration between magnetic excitations and
the phonon gas occurs on a time scale much shorter than the time resolution of measurement.
Consequently, we are restricted in presenting the results for the individual specific heat contributions cR
1 K. For temperatures above 1 K, we can not distinguish between the different contributions, and thus, we
present the total specific heat only. Due to their characteristic temperature and magnetic field dependencies,
however, the individual contributions of the magnetic specific heat can be identified in the overall temperature
range 0.3 K ≤ TR≤ 30 K. This is shown and discussed in section 4.
Furthermore, the time resolution of the thermal relaxation experiments does not permit us to identify the
exact value of the shortest relaxation time τR
γ. But then the shortest relaxation time corresponding to the
largest relaxation rate affects primarily the thermal conductivity measured in the steady state heat transport
experiments. So we are also able to determine the shortest magnetic relaxation time and its temperature
γof magnetic excitations and their relaxation times τR
γto the temperature range below
3.2 Steady State Heat Transport
The thermal conductivity was measured using the set-up for a “one heater & two thermometer” method .
Enforcing one-dimensional heat flow, a long rod geometry is appropriate where a constant heater power˙Q is
applied to the top face of the crystal (at x = 0). The lower end of the crystal (at x = L = 50 mm) is thermally
fixed to the thermal bath at TR. With two thermometers T1(at position x1= 15 mm) and T2(at position
x2= 28 mm) the temperature gradient ΔT/Δx = (T1−T2)/(x1−x2) has to be measured accurately. The
thermal conductivity is then determined by κ = −(˙Q/A)·Δx/ΔT.
copper base platform
Fig. 4 Schematic view of the set-up for the thermal conductivity measurements.
In Fig. 4 our set-up for the measurements of the thermal conductivity κ(T,B) is sketched. The sample rod
was mounted to the copper base of a3He sorption cryostat inserted in a 8 T cryomagnet . The crystal’s
long  axis was aligned to the magnetic field axis to better than 1◦. The lower end of the crystal was
glued into a matching hole of the copper base platform using GE-7031 varnish. The temperatures T1and T2
resulting from the stationary heat flow by the heater were measured with bare chip Cernox type BC 1030 
thermometers, again superglued to the outer wall side of the crystal at defined positions x1and x2. Typical
temperature gradients were about 5 % of the reference temperature TR. The thermometers and the heater were
electrically wired with NbTi twisted pair wires to the copper base platform, each wire having a length of about
100 mm to ensure low thermal conductance between crystal and thermal bath.
Due to the finite size of the Cernox thermometers (1 mm width) an error for distance Δx of about 4 % had
to be accepted. Also, the variance in the cross section A along the length of the crystal produced an error of
about 6 %. In total, the error to the absolute values of our thermal conductivity measurement adds up to 10 %.
A third Cernox thermometer T3was mounted to the crystal wall near to the thermal contact edge (x3=44 mm)
where the crystal was glued to the thermal bath. At stationary heat flow conditions, the residual temperature
difference ΔT = T3−TRwas observed to be negligible as ΔT/TR< 1 % in the overall temperature range.
In the steady state technique, the temperature within the sample does not change with time and, sub-
sequently, for stationary processes the linearized field equations (14) are reduced to the heat conservation
divq = 0 and Fourier’s law:
For temperatures well below the Debye temperature of a crystal, equation (10) can be used to determine the
specific heat of the phonons cR
ph(TR) and the thermal conductivity κ(TR) is then expressed in the form:
will be presented in section 4.
3.3 Heat Pulse Propagation
Heat pulse transport measurements have been performed using the same sample set-up as for the thermal
conductivity measurements (see Fig. 4). In a temperature range from 0.3 K to 4 K, short thermal pulses of
power˙QSand width ΔtHfrom 1 ms up to 5 ms have been generated by energizing the SMD resistance heater
with a programmable precision voltage source. On a time scale from 1 ms up to 1000 s, the subsequent
temperature responses? T(x,t) have been recorded by thermometers T1and T2using a DC–100 kHz pre-
was determined by measuring the sample base temperature T3with respect to the reference temperature TR.
In order to justify a linearized theoretical description of the experiments, the maximum temperature rise Tmax
was limited to (Tmax−TR)/TR≤ 5 % by adjusting the heat pulse energy Q =˙QS·ΔtHappropriately.
The thermodynamic field equations (14) are taken as a basis for the complete description of the non-
stationary heat pulse propagation in a body under inhomogeneous spatial conditions. According to the ex-
perimental set-up (Fig. 4) one-dimensional heat flow can be assumed. In this one-dimensional, rotationally
symmetric geometry, all thermodynamic variables are functions of time and a single spatial coordinate x:=x1.
The symmetric traceless tensor N<ij>specializes to N11=−2N22=−2N33and contains no off-diagonal com-
ponents. Introducing the notation N := N11and q := q1the one-dimensional field equations (14) are reduced
to the 3+n equations for (? T,q,N,? eν):
amplifier and signal averaging electronics. The conductance KR(T) of the thermal contact to the bath at x = L
The system (18) is a linear system of partial differential equations, which can be solved numerically
for given initial and boundary conditions. For the solution of the problem we take into account the initial
conditions? T(x,0) = q(x,0) = N(x,0) = ? eν(x,0) = 0 and the boundary conditions? T(L,t) = N(L,t) = 0.
q(0,t) = 0 otherwise, is termed the heat pulse problem.
It could be assumed that the numerical solution of the heat pulse problem can be obtained with basically
no other parameters, appearing in the field equations (18), than those we have already fixed from the analysis
of thermal relaxation calorimetry and steady state heat transport experiments. From this point of view, the
analysis of heat pulse propagation constitutes a check of both the thermodynamic modelling of heat transport
in Dy2Ti2O7and the preassigned data of specific heat and thermal conductivity.
For sufficiently small values of relaxation times τR
νthe energy densities of magnetic excitations will de-
pend on temperature according to the relationship ? eν= cR
the given initial and boundary conditions. In fact, for temperature above 1 K, the temperature profiles? T(x,t)
κ(TR)/c(TR). The heat capacity per unit volume c(TR) is thereby composed of the contributions of phonons
ν, whereas the thermal conductivity κ(TR) is given by equation (17). Below
1 K, the single variable? T no longer exclusively determines the thermodynamic state of the spin ice material
propagation in this material.
It must be pointed out that the time resolution (< 10−3s) achieved in the heat pulse experiments is much
better than that of the thermal relaxation experiments. Therefore, a comparison of the results obtained in both
experiments is only valid on the same time scale. On the other hand, the better time resolution in heat pulse
experiments permits us to investigate in detail the fast relaxation of magnetic excitations.
This problem, for which the additional boundary condition is given by, q(0,t) =˙QS/A for 0 ≤ t ≤ ΔtHand
ν? T. In this case, the field equations (18) reduce to
the classical Fourier equation of heat conduction in one dimension, which can be also solved analytically for
recorded in the heat pulse experiments are well described by Fourier’s equation with the thermal diffusivity
phand magnetic excitations cR
Dy2Ti2O7, and additional internal variables of magnetic excitations ? eνare required to describe the heat pulse
4 Results and Discussion
In the following subsections we present the results on specific heat, thermal conductivity, heat pulse experi-
ments, and relaxation times.
In order to describe the specific heat of Dy2Ti2O7measured by relaxation calorimetry we have to distin-
guish between various contributions cR
νof magnetic excitations identified by data analysis on the basis of the
thermodynamic field theory. For magnetic fields along the  axis, the Dysprosium spins can be considered
to form two orthogonal sets of spin chains: parallel and perpendicular to the field direction, called α-chains
and β-chains, respectively. This conception based on neutron diffraction experiments was given by Fennell et
al.  and verified from specific heat measurements by Higashinaka et al.  and Hiroi et al. . Following
the concept of α-chains and β-chains, we are able to identify a magnetic field dependent contribution cR
relaxation time τR
β≈ 10 s at TR= 0.6 K). Beyond that, there are further magnetic contributions (in summary denoted
γ) with considerably shorter relaxation times. It is obvious, that the number of magnetic contributions
γ2, ...), which can be identified in relaxation calorimetry and heat pulse experiments, cor-
responds to the number of magnetic excitations ? eν(or internal variables) needed to describe the experiment
α≈ 100 s at TR= 0.6 K) and a magnetic field independent contribution cR
on a given time scale.
4.1 Specific Heat
The total specific heat of Dy2Ti2O7determined by thermodynamic modelling of the temperature response
obtained in relaxation experiments is shown in Fig. 5.
specific heat (J/mol K)
B || 
Fig. 5 Total specific heat as function of temperature TRand magnetic field B as determined from thermodynamic field theory by
equation (15) (open circles). The brown line denotes the phonon specific heat cphaccording to equation (19). The inset displays
the magnetic contribution cmagwhere cphhas been subtracted from the total specific heat.
At higher temperature (TR> 10 K) the specific heat is mainly dominated by the contribution of phonons
cph(TR). In the Debye model cph(TR) is given by
cph(TR) = 9nDykBNA·
specific heat data between 10 K and 30 K, θDwas calculated to be 283 K corresponding to a Debye velocity
cD= 2165 m/s. Similar values have been reported by Hiroi et al. . For low temperatures (TR? θD/20) the
Subtracting the specific heat of phonons according to equation (19) from the total specific heat, the mag-
netic contribution cmag(TR,B) is obtained (as shown in the inset of Fig. 5). The cmagdata presented here
disagree somewhat with previously published data [5, 8–10]. Whereas for TR> 1 K all data fairly agree by
±5 %, the deviation of our data from the previously published ones increases with decreasing temperature
below 1 K, and attains values up to 400 % at TR=0.4 K. Thus, we obtain a slightly changed value of the mag-
netic entropy. Integration of cmag(TR,0)/TRfrom 0.3 K to 15 K was performed to give the entropy difference
Smag(15 K,0)−Smag(0.4 K,0) = 4.069 J/mol-DyK. The residual entropy at zero field Smag(T → 0 K,0) is
then calculated to be NAkBln(2)−4.069 J/mol-DyK = 1.694 J/mol-DyK, in remarkably close agreement
with Pauling’s prediction of 0.5NAkBln(2) = 1.686 J/mol-DyK. The magnetic specific heat consists of the
three parts cR
which is 78.14 cm3/mol for Dy2Ti2O7.
The early heat capacity measurements by Higashinaka et al.  and Hiroi et al.  have revealed that the
magnetic excitations due to α-chains behave paramagnetic. A magnetic field applied along the  direction
easily aligns the spins on the α-chains owing to the large Zeeman energy. The paramagnetic behaviour is
clearly visible as a broad peak in the specific heat data for magnetic fields B ≥ 0.5 T. (as shown in Fig. 5).
The peak shifts to higher temperature with increasing magnetic field. We are able to determine directly the
specific heat data cmα=VmolcR
for B = 0 T and B = 0.2 T are presented, which result from the thermodynamic analysis of thermal relaxation
ν(ν = α,β,γ), which add up to cmag=Vmol(cR
γ). HereVmoldenotes the molar volume
α, at least for temperatures TR<1 K. In Fig. 6 the specific heat data cmα(TR,B)
Fig. 6 Magnetic
excitations as function of temperature TR for
magnetic fields of B = 0 T and 0.2 T as calcu-
lated from thermodynamic field theory (open
circles). The full lines were calculated for para-
magnetic behaviour according to equation (20).
The inset displays the corresponding relaxation
specific heat cmα(J/mol K)
0.5 T0.5 T0.5 T0.5 T 0.5 T
1 T 1 T1 T1 T1 T
1.5 T1.5 T 1.5 T1.5 T 1.5 T
B || 
0.40.4 0.60.60.8 0.8
In the case that the magnetic α-excitations exhibit paramagnetic behaviour, the specific heat cmα(TR,B)
should be described by
cmα(TR,B) = NαkBy2dBJ
where BJ(y) is the Brillouin function for the angular momentum quantum number J. Nαis the number of
α-excitations per mol of Dy2Ti2O7, and g is the effective Land´ e factor.
From a best-fit procedure on the magnetic specific heat data for B = 1.5 T, where the α-contribution
dominates the specific heat (see broad peak at 7 K in the inset of Fig. 5), the number of α-excitations and
the effective magnetic moment were determined to be Nα= 0.5NAand gJµB= 10·cosθ ·µB= 10?2/3µB,
respectively. The magnitude of the magnetic moment is here 10µB(corresponding to the value of the bare ion
Dy3+), and its direction forms an angle of θ = 35.3◦with the  magnetic field. The effective angular
momentum was obtained to be J ≈ 1/2 (so that g ≈ 20). Thus, for high magnetic fields B ≥ 1.5 T along the
 direction, the specific heat of the α-excitations can be described by a paramagnetic Ising spin system,
where only half of the spins (namely two of four spins on a tetrahedron) couple to the field. This paramagnetic
model should only be valid in the high-field limit, where interspin coupling can be neglected. Nevertheless,
we where able to fit well all measured data cmα(TR,B) (for B = 0 T and B = 0.2 T) to equation (20) using the
same parameters as determined for B = 1.5 T. With decreasing field, however, the effective angular quantum
number decreases and approaches, in the limit of the internal field B = Bint= 0.07 T, the value J = 0.11
(corresponding to a large Land´ e factor of g = 88).
In summary, the α-excitations with relaxation times 1 s ≤ τR
spin ice state of Dy2Ti2O7for TR< 1 K, can be interpreted as paramagnetic defects with renormalized values
J and g. It is no doubt interesting that in geometrically frustrated systems the strong coupling between the
spins can lead to paramagnetic defects with a large magnetic moment g·J (in unit of the Bohr magneton µB)
but vanishing small values of J (in the limit J → 0∧g → ∞).
α≤ 100 s (see inset of Fig. 6), observed in the
In addition to the α-excitations, further magnetic excitations with relaxation times 0.1 s ≤ τR
TR< 1 K were identified by thermodynamic analysis of the relaxation experiments. The measured specific
heat cmβ= VmolcR
Fig. 7 for B = 0 T and B = 0.2 T.
β≤ 10 s for
βof these so-called β-excitations and the corresponding relaxation time τR
βare shown in
specific heat cmβ(J/mol K)
B || 
0.40.40.6 0.60.8 0.8
Fig. 7 Magnetic specific heat of theβ-excitations
as function of temperature TRfor magnetic fields
B = 0 T and 0.2 T as calculated from thermody-
namic field theory (open squares). The full line
was calculated according to a Schottky-type exci-
tation, equation (21). The inset displays the corre-
sponding relaxation time τR
As suggested by Hiroi et al.  the β-spin chains in the pyrochlore spin system are expected to be isolated
effectively from the α-spin chains, and can be treated like an Ising spin chain model with nearest-neighbour
interaction. The specific heat of the β-spin chains should then be described by a Schottky-type excitation of
cmβ(TR) = NβkB
where Nβdenotes the number of β-excitations per mol of Dy2Ti2O7. Using equation (21) we are able to fit
the nearly field independent specific heat of the β-excitations for magnetic fields B < 0.4 T fairly well (as
shown in Fig. 7). The best-fit values for both the number of β-excitations and the energy Δ were obtained
to be Nβ= 0.24NAand Δ/kB= 3.5 K, respectively. Δ can be interpreted as the nearest-neighbour effective
bound energy between β spins, and its value of 3.5 K agrees with the zero-field value obtained by Kadowaki
et al. . It has to be pointed out, however, that only a part (just the half) of the β spins make a Schottky-type
contribution to the magnetic specific heat. This means that, at least in the low field range (B < 0.5 T), further
excitations due to interaction effects contribute to the specific heat. These excitations were really detected in
thermal relaxation experiment, and in summary called γ-excitations.
Fig. 8 Magnetic specific heat of the γ-
excitations as function of temperature TR
and for B = 0 T and 0.2 T as calculated
from thermodynamic field theory (open
triangles). The full lines were calculated
by subtracting cmαand cmβfrom cmag.
specific heat cmγ(J/mol K)
B || 
In Fig. 8 the magnetic specific heat of the γ-excitations cmγ= VmolcR
presented, as determined from thermodynamic analysis of the thermal relaxation experiments. The relaxation
γ(or the individual relaxation times of the γ-excitations) is much smaller than the time resolution of
the thermal relaxation experiments of 0.1 s. Details on both the magnitude of the shortest relaxation time and
the evidence for a distribution of relaxation times will be given subsequently by analysing the steady state and
heat pulse experiments.
γfor B = 0 T and B = 0.2 T are
At stronger magnetic fields B ≥ 0.5 T the specific heat cmγincreases and a relatively sharp peak becomes
apparent at the critical temperature Tc= 1.05 K. As can be seen from Fig. 9, the contributions cmγ and
cmβfollow a similar temperature dependence for all fields from 0.5 T to 1.5 T. The temperature and field
specific heat (J/mol K)
B || 
0.4 0.6 0.8 1.0 1.20.4 0.6 0.8 1.0 1.2
Fig. 9 Magnetic specific heat cβ (open
squares) and cγ (open triangles) as func-
tion of temperature TRfor magnetic fields
B ≥ 0.5 T. The inset shows the corre-
sponding relaxation times τR
dependencies of the summarized specific heat as shown in Fig. 5 already have been reported from the specific
heat experiments by Hiroi et al.  and subsequent Monte-Carlo-studies of Yoshida et al. .
4.2 Relaxation Times
The thermal relaxation behaviour of the spin ice material Dy2Ti2O7is illustrated in Fig. 10 by the magnetic
relaxation times τR
β(TR,B = 0 T) as function of temperature TR. The data obtained by
thermodynamic analysis of the thermal relaxation experiments are presented together with experimental data
from susceptibility measurements by Snyder et al. .
It is obvious to compare the thermal relaxation of Dy2Ti2O7to that of the spin-lattice relaxation in rare-
earth salts. According to Orbach  the spin-lattice relaxation rate in Krames salts is determined by the
Raman and thermally activated processes, and given by
α(TR,B = 0 T) and τR
In fact, the temperature dependence of the relaxation time τR
peratures above 1.3 K (see blue line in Fig. 10). Below 20 K τR
to Raman process, entering a plateau region below 12 K, before passing through the Arrhenius temperature
dependence with an activation energy Eα. By fitting the data to equation (22) the activation energy was de-
termined to be Eα= 2.65 K. This value corresponds to the creation energy of a topological defect in the
paramagnetic state of Dy2Ti2O7[4, 29].
In the spin ice phase at TR< 1.1 K, the temperature dependence of the relaxation time is completely
changed. As illustrated in Fig. 10 the relaxation time of both τR
and decreases again towards lower temperatures. This non-monotonic behaviour is assumed to be associated
with the screening of the charged defects , and can be analytically described by a temperature dependent
αcan be described by equation (22) for tem-
αincreases with decreasing temperature due
βattains a maximum at TR≈ 0.5 K