Mobile copper ions as heat carriers in polymorphous copper sulfide superionic conductors

Article abstract of DOI:10.1063/1.4997501

Liquid-like mobile Cu⁺ ions are generally considered as phonon scatters in copper chalcogenide superionic conductors, but this is recently disproved by a neutron spectroscopy study on lattice dynamics of Cu₂Se compound [Voneshen et al., Phys. Rev. Lett. 17, 118 (2017)]. In this work, we provide a different perspective by investigating the thermal transport of three transformable polymorphs of Cu₂S compounds with varying Cu⁺ content. We show that the disordered and mobile Cu⁺ ions are not the primary factor for suppressing the heat transport. A notable dependence of thermal conductivity on Cu⁺ content is observed. By correlating the electrically deducted thermal conductivity with the ion motion behavior for β-Cu₂S superionic phase, we reveal that these fast ionic species Cu⁺ are heat carriers instead, which make an appreciable contribution to thermal conduction.

Full text of DOI:10.1063/1.4997501

Mobile copper ions as heat carriers in polymorphous copper sulfide superionic
conductors
Xin Liang
Citation: Appl. Phys. Lett. 111, 133902 (2017); doi: 10.1063/1.4997501
View online: http://dx.doi.org/10.1063/1.4997501
View Table of Contents: http://aip.scitation.org/toc/apl/111/13
Published by the American Institute of Physics
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APPLIED PHYSICS LETTERS 111, 133902 (2017)
Mobile copper ions as heat carriers in polymorphous copper sulfide
superionic conductors
a)
Xin Liang
School of Materials Science and Engineering, Changzhou University, Changzhou, Jiangsu 213164, China
(Received 25 July 2017; accepted 15 September 2017; published online 26 September 2017)
þ
Liquid-like mobile Cu ions are generally considered as phonon scatters in copper chalcogenide
superionic conductors, but this is recently disproved by a neutron spectroscopy study on lattice
dynamics of Cu Se compound [Voneshen et al., Phys. Rev. Lett. 17, 118 (2017)]. In this work, we
2
provide a different perspective by investigating the thermal transport of three transformable
þ
polymorphs of Cu S compounds with varying Cu content. We show that the disordered and mobile
2
þ
Cu ions are not the primary factor for suppressing the heat transport. A notable dependence of
þ
thermal conductivity on Cu content is observed. By correlating the electrically deducted thermal
conductivity with the ion motion behavior for b-Cu S superionic phase, we reveal that these fast
2
þ
ionic species Cu are heat carriers instead, which make an appreciable contribution to thermal
conduction. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4997501]
Chalcogenide based superionic conductors demonstrate
great potential in thermoelectric applications due to their
most superionic conductors. k is Boltzmann constant, and
B
T is the absolute temperature. This model predicts that ionic
1–3
ꢁ3
extremely low lattice thermal conductivity. It has attracted
intensive research attention on the thermal transport proper-
ties of these superionics, with the key question on under-
standing the role of fast ionic species. It has been suggested
thermal conductivity can be quite low, jion ꢂ 10 W/m K,
which is insignificant as compared to the phonon or lattice
contribution jph that is in the order of unit W/m K.
However, by working on the Hamiltonian, current operators
and transport coefficients based on the lattice gas model,
the ionic thermal conductivity jion is estimated to be ꢂ2
that the dramatically low thermal conductivity of Cu Se is
2
attributed to the liquid-like diffusion of copper ions which
effectively suppress the transverse phonons and reduce the
phonon mean free path close to the minimum wavelength in
ꢃ
W/m K at 200 C for a superionic conductor with electrical
7
conductivity of ꢂ2 S/cm. This suggests that the contribu-
1,4
5
glass. However, a very recent work disagrees with this
point of view and ascribes the ultralow thermal conductivity
of Cu Se to the anharmonicity by finding that the copper
tion of mobile ions to thermal conductivity is in the same
order as that of phonon contribution, making up a non-
negligible portion of the total thermal conductivity. The
conclusions drawn from the two theoretical work lead to a
controversial issue on whether the amount of heat carried
by fast ionic species is appreciable. A recent molecular
2
hopping time scales are too low to influence the lattice vibra-
tions and that transverse phonons persist in both ordered and
disordered phases.
þ
dynamic (MD) simulation work on Ag Te superionic com-
2
Regardless of the fact that mobile ionic species like Cu
þ
pound shows that thermal conductivity varies as the com-
þ
and Ag can or cannot be the primary reason for the dramati-
8
pound changes from being Ag deficient to excessive,
cally reduced thermal conduction, these charged ions with
remarkably high mobility, are potentially heat carriers at the
same time, as the same analogy to electrons and holes. The
contribution to total thermal conductivity by these “free
ions” can be similarly expressed via an “ionic Lorenz
which implies the influence of mobile ions on thermal con-
ductivity of superionics. As motivated, the present work
experimentally probes into this problem.
Certain phases of binary silver and copper chalcogenides
þ
þ
6
are well-known superionics with Ag and Cu as fast ionic
species, which are potential model systems for experimental
investigations. We note that reasonable allowance for self-
nonstoichiometry is desired so that the concentration of fast
ionic species can be tuned in a relatively appreciable range
while the system still remains the same crystal structure with-
out impurity phases. The other requirement for the model sys-
tem is that electrons or holes dominated transport is
sufficiently low so that the electronic contribution to thermal
number” Li
ꢀ
ꢁ
ꢀ
ꢁ
9
aL0
ꢀ0
jion ¼ LiriT ¼
;
(1)
p2Z2
2
kBT
where r is the electrical conductivity due to the ionic trans-
i
port, L0 is the Sommerfeld electron Lorentz number L0
ꢂ ꢃꢂ ꢃ
2
B
2
2
k
p
¼ 2:45 ꢀ 10ꢁ8 J K
2
ꢁ2 ꢁ2
C , and ꢀ0 is the bind-
¼
3
e
ing energy of the localized ion and is regarded as “energy
gap” in the free ion model above which free-ion like states
exist. Z is the charge carried by conducting ionic species,
and a is a defined term related to the energy dependence of
ion mean free path, which has a typical value of ꢂ1 for
conductivity j is negligible as compared to the total thermal
el
9
10
conductivity j . As a consequence, Ag Te, Ag Se,
tot
2
2
1
11
Cu Se, and Cu Te binary system which possess electrical
2
2
3
conductivity in the order of ꢂ10 S/cm are not appropriate
systems for the present purpose. Ag S system, albeit with
2
smaller electrical conductivity (ꢂ200 S/cm at room tempera-
a)
12
ture and ꢂ1000 S/cm at elevated temperatures), is found to
Author to whom correspondence should be addressed: liangxin@cczu.edu.cn
0003-6951/2017/111(13)/133902/5/$30.00
111, 133902-1
Published by AIP Publishing.

133902-2
Xin Liang
Appl. Phys. Lett. 111, 133902 (2017)
FIG. 1. Illustration of atomic models for three Cu
high temperature a phase (cubic). The blue and yellow balls represent Cu and S atoms, respectively. The fractional filling of Cu balls illustrates that Cu ions
have partial occupancy and are disordered in b and a phases.
2
S polymorphs: low temperature c phase (monoclinic), intermediate temperature b phase (hexagonal), and
þ
þ
have low tolerance for Ag content variation. Among all the
copper and silver based binary chalcogenides, Cu S superi-
scanning calorimetry apparatus (Netzsch STA 449 F3 Jupiter,
Germany). Mass densities q were measured using the
Archimedes method. All sintered samples were well densified
with relative mass densities above 96% as compared to the
2
onics, with electrical conductivity ꢂ10 S/cm from room tem-
2
perature to 800 K, appears to be a suitable model system for
3
the present work. In addition, Cu-S binary phase equilibria
suggests a reasonable range of stoichiometric modulation for
theoretical value of 5.787g/cm .
All Cu26dS samples remain the pure low temperature
LC c phase (PDF#33-0490), as seen from the room tempera-
ture X-ray diffraction patterns in Fig. 2. Thermal diffusivity
of Cu26dS superionic compounds measured from room tem-
perature to 923 K is presented in Fig. 3(a). There is an abrupt
change of thermal diffusivity across the phase boundary at
ꢂ373 K, while the one at 700 K is less remarkable. These
two phase changes are well recorded by measured specific
heat capacity with temperature, which is shown in Fig. 3(b).
The thermal conductivity of Cu26dS compounds is readily
obtained, as presented in Fig. 3(c), which has a weak depen-
dence on temperature except across phase boundaries. In
each phase region, there is an appreciable variation of ther-
mal conductivity with Cu content. To investigate the ionic
13
þ
Cu S compound, which provides the room for tuning Cu
2
content.
Stoichiometric Cu S compound has three transformable
2
polymorphs, as shown in Fig. 1. Below 370 K, it crystallizes
into low chalcocite (LC) c phase with a monoclinic lattice,
and stabilizes to a high chalcocite (HC) b phase with a hex-
agonal structure between 370 and 700 K. Above 700 K, Cu2S
transforms to a FCC cubic a phase, which has been long well
known as a classic superionic compound that is characteris-
þ
14
tics of disordered and fast mobile Cu ions. Also in HC b
compound, Cu are distributed over the sites of 2b, 4f and
þ
6
g in disorder and mobile through the interstices of hexago-
1
5
nal closed packed S atoms. Recently, it is found that Cu
þ
sublattices in b compound (HC) are indeed in liquid phase
contribution of mobile Cu ions to thermal conduction, it is
1
6
whereas S sublattices remain solid framework. The LC c
phase, where both cations and anions are ordered in rigid
framework, is a typical ionic compound without any superi-
onic characteristics.
necessary to carefully deduct the electronic contribution
from the measured thermal conductivity which consists of
contributions from all types of heat carriers.
The electronic contribution to thermal conductivity jel
can be obtained from the Wiedemann-Franz law
Cu26dS powders with varying Cu content (Cu1.98S,
Cu1.99S, Cu S, Cu2.03S, and Cu2.05S) were prepared by melt-
ing the pure substance Cu (shot, 99.9%, Alfa) and S (pieces,
2
jel ¼ LrT;
(2)
9
9.99%, Aladdin) in quartz tubes sealed under vacuum. The
mixtures were heated to 673 K in 10 h, and then ramped to
383 K with a rate of 3.5 K/min. After being thermally equil-
where r is the electrical conductivity. In Fig. 3(d), we present
the measured electrical conductivity from room temperature
1
ibrated for 10 h, the powders were furnace cooled to room
temperature. The samples were ground into fine powders and
then densified into solid pellets on a spark plasma sintering
V
R
system (SPS LABOX-325, Sinter Land , Japan), under an
axial compressive stress of 65 MPa in vacuum and held at
713 K for 5 min. X-ray diffraction patterns were taken at
room temperature under a Rigaku D/max 2500 PC X-ray dif-
fractometer using Cu Ka radiation. Temperature dependence
of electrical conductivity (r) and Seebeck coefficient (S)
were measured in the argon atmosphere on a Netzsch SBA
458 Nemesis system (Germany). Thermal conductivity (j)
was obtained using the standard relation j ¼ DCpq, where D
is the thermal diffusivity, C is the specific heat capacity,
p
and q is the mass density. Thermal diffusivity (D) was mea-
sured using the laser flash method on a Netzsch Micro
V
R
Flash LFA 457 (Germany) with flowing argon gas, and
then analyzed using a Cowan model with pulse correction.
The heat capacity Cp was measured on a differential
FIG. 2. Room temperature X-ray diffraction patterns of Cu26dS compounds
with varying Cu content.

133902-3
Xin Liang
Appl. Phys. Lett. 111, 133902 (2017)
FIG. 3. Thermal-physical (a)–(c) and
electrical transport properties (d)–(f) of
Cu26dS as a function of temperature
covering over three polymorphs.
Thermal diffusivity (a), specific heat
capacity (b), electrical conductivity
(
d), and Seebeck coefficient (e) mea-
sured from room temperature up to
23 or 923 K. The total thermal con-
8
ductivity (c) is obtained based on the
measured thermal diffusivity and spe-
cific heat capacity. The Lorenz number
(
f) is calculated from the measured
temperature-dependent Seebeck coeffi-
cient data using a single parabolic
model. The dotted orange lines indi-
cate the phase transition temperatures
with each phase regime labelled on the
top of the figure.
to 823 K. Room temperature electrical conductivity monotoni-
þ
the obtained L values are generally smaller than the free
2
ꢁ8
ꢁ2 ꢁ2
cally varies with nominal Cu content, which evidences the
electron limit of 2:45 ꢀ 10 J K
C .
changing composition that covers both Cu deficient and
excessive ones. The temperature-dependent Lorenz numbers
can be calculated using the single parabolic model (SPB)
In Fig. 4(a), we show the electronic thermal conductivity
jel as a function of temperature, which is contributed by holes
in these Cu26dS compounds. Subtracting the electronic contri-
bution j from the total thermal conductivity j , we obtain
1
7
based on solving the Boltzmann transport equations
el
tot
the jtot ꢁ jel values, which consists of contributions only
2
2
2
B
ð1 þ kÞð3 þ kÞF g F
g ꢁ ð2 þ kÞ F
g
kð Þ 2þkð Þ
1þkð Þ k
from phonons and ions, as also presented in Fig. 4(a). The
electronic thermal conductivity j of Cu26dS is in the range
L ¼
;
2
2
2
e
ð1 þ kÞ F g
kð Þ
el
ꢁ4
ꢁ2
of 10 –10 W/m K, which is orders of magnitude smaller
than jtot ꢁ jel, being negligibly small. The Cu26dS system we
chose then proves to be a proper model system for the present
(
3)
where the Fermi integral is given by
study. Intriguingly, the thermal conductivity jtot ꢁ j of
el
ð
1
xjdx
þ expðx ꢁ gÞ
Cu S monotonically increases as Cu content decreases, by
2
FjðgÞ ¼
:
(4)
1
20%–40% in a narrow range of non-stoichiometry from 2.5%
Cu excess to 1% Cu deficiency for three polymorphs. Point
0
þ
The reduced Fermi energy g is a function of the Seebeck
coefficient and for p-type semiconductor
defect scattering induced by excessive or deficient Cu ions
1
8
cannot be the reason, since there is no need to the increase the
number of defects for such a narrow non-stoichiometric range.
The Cu atomic sites are only partially filled in b and a phases,
and there are more possible occupations sites than the number
"
#
kB ð2 þ kÞF1þkðgÞ
Sh ¼ e
ꢁ g ;
(5)
ð1 þ kÞF ðgÞ
k
16
of Cu atoms. For instance, the occupancy for the two-fold
15
where the Seebeck coefficient measured from room tempera-
ture to 823 K is shown in Fig. 3(e). Assuming that carrier
relaxation time is limited by acoustic phonon scattering
with carrier scattering factor k ¼ 0, temperature-dependent
Lorenz numbers are obtained by solving Eqs. (3)–(5) based
on the measured Seebeck coefficient. As shown in Fig. 3(f),
coordinated 6g Cu sites in b phase is only 0.19. Since the
Cu ions exist in a completely disordered and liquid-like state
þ
in b and a phases which possess intrinsically strong anharmo-
14
nicity, the impact of small stoichiometric modulation on
phonon-phonon scattering should be insignificant as well.
Eventually, understanding of these experimental observations

133902-4
Xin Liang
Appl. Phys. Lett. 111, 133902 (2017)
dðln rTÞ
Ea ¼ ꢁ
;
(7)
dð1=k TÞ
B
l is the average jumping distance of mobile ions that is given by
X
2
f s v
i
i
i
i
l ¼
X
;
(8)
2
fi
i
in which fi, vi, and si are the weight factor, group velocity
and relaxation time of the ion mode with frequency x . l can
i
thus be considered as ion “mean free path.” The heat of
transport Q is related to the Seebeck coefficient S (also so-
called thermopower) according to the equations of irrevers-
21
ible thermodynamics
Q
S ¼ ꢁ þ H;
eT
(9)
where e is the charge of mobile ion species and H is a correc-
tion term due to electrode contact potential. According to
Eq. (9), the heat of transport Q can be estimated from the
slope of the plot of S against ꢁ1=T.
Among the three polymorphs of Cu S, the low tempera-
2
ture c (LC) phase is well-known as a non-superionic conduc-
tor, whereas thermally excited carriers can participate in the
conducting process of high temperature a (cubic) phase. In
view of that, the intermediate temperature b (HC) superionic
phase is the most suitable system for understanding the role
FIG. 4. (a) Electronic thermal conductivity jel and electronically deducted
thermal conductivity jtot ꢁ jel as a function of temperature for Cu26dS com-
þ
pounds. (b) Variation of jtot ꢁ jel as a function of Cu content for three
þ
of mobile Cu ions in heat conduction. As shown in Fig.
Cu
2
S polymorphs. The variation of the ion “mean free path” l (right axis)
þ
against Cu content for b superionic phase is also superimposed on the fig-
ure. The connecting lines are guides to the eye.
4(b), the thermal conductivity jtot ꢁ j varies by almost
el
4
assessed from the electrical measurements, the ion “mean
0% in a narrow nonstoichiometric range. With the Q and Ea
free path” l for b-Cu26dS can be estimated according to Eq.
þ
requires the examination on the role of Cu ions in thermal
conduction.
þ
(
on Fig. 4(b). Physical intuitively, large values of l (or the
6), which is plotted against Cu content and superimposed
þ
If fast ionic species Cu act primarily as phonon scat-
ters, it is then expected that the thermal conductivity jtot
j would be the lowest for a (cubic) phase where Cu ions
are completely disordered in a liquid-like state, followed by
b (HC) phase, and then be the highest for c (LC) phase where
Cu ions are ordered in rigid solid sublattices. However, this
19
ratio Q=E ) suggests an excellent ionic conductor. Since
a
energy barrier for ion jumping basically sits in-middle-
between two adjacent equivalent atomic sites, an ion “mean
free path” with a half of the interatomic distance suggests
that the ionic flow can readily occur. For stoichiometric b-
þ
ꢁ
el
þ
Cu S, we find l ꢄ 0:45a0, which is consistent with the fact
2
is not the trend we observed, and a (cubic) phase holds the
highest jtot ꢁ jel among three polymorphs across the non-
stoichiometric range, as seen from Fig. 4(b). Our observa-
tions, from a different perspective, are consistent with the
15,16
that b-Cu S is a good ionic conductor.
As shown in Fig.
2
4
Cu content decreases, which suggests that the mobility of
(b), the ion “mean free path” l dramatically increases as
þ
þ
þ
liquid-like Cu ions is largely improved in Cu deficient
samples. It is striking to find that jtot ꢁ j and l vary virtu-
þ
recent findings that liquid-like diffusion of mobile Cu ions
5
does not significantly suppress phonon transport.
el
þ
ally similarly as a function of Cu content for b phase,
Based on the hopping mechanisms in ionic conductors
and solving the Boltzmann transport equation to ionic flow,
and compared with the phenomenological relation according
to the laws of irreversible thermodynamics, the heat of trans-
port by ions Q is related to the activation energy for transport
which implies the correlation of the thermal conductivity
with the ion motion in superionic conductors. Notably, the
samples with larger l values have higher thermal conductiv-
ity jtot ꢁ jel, which suggests that the ionic contribution to
þ
thermal conduction is enhanced as the mobility of Cu ions
19,20
Ea by
þ
is improved, and that mobile Cu ions flow as heat carriers
in these superionic compounds.
2
l
Q ¼ Ea;
(6)
In summary, we have experimentally investigated the
þ
a0
role of mobile Cu ions in heat conduction of Cu S polymor-
2
þ
where a0 is the interatomic distance. Ea can be estimated
from the measured temperature-dependent electrical
phous superionic compounds by varying Cu content. The
þ
high temperature a phase where Cu ions are completely
disordered in a liquid-like sublattices holds the highest
6
conductivity

133902-5
Xin Liang
Appl. Phys. Lett. 111, 133902 (2017)
5
electronically deducted thermal conductivity among the
three polymorphs, suggesting that the ultralow thermal con-
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