Acta Materialia 206 (2021) 116636
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Acta Materialia
journal homepage: www.elsevier.com/locate/actamat
Lattice evolution, ordering transformation and microwave dielectric
properties of rock-salt Li3+x Mg2–2x Nb1-x Ti2x O6 solid-solution system: A
newly developed pseudo ternary phase diagram
Xing Zhang a,b, Zixuan Fang a,b,∗, Hongyu Yang a,b, Peng Zhao a,b, Xiao Zhang c, Yuanpeng Li a,
Zhe Xiong a,b, Hongcheng Yang a,b, Shuren Zhang a,b, Bin Tang a,b,∗
a
State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu, 610054, China
National Engineering Research Center of Electromagnetic Radiation Control Materials, University of Electronic Science and Technology of China, Chengdu,
610054, China
c
China Key System & Integrated Circuit Co., Ltd. Wuxi, 214035, China
b
a r t i c l e
i n f o
Article history:
Received 3 August 2020
Revised 16 November 2020
Accepted 6 January 2021
Available online 10 January 2021
Keywords:
Microwave dielectric ceramics
Orthorhombic-cubic-monoclinic phase
transition
Ordering transformation
Reconstructed superlattice
Low dielectric loss
a b s t r a c t
New types of multi-component Li3+ x Mg2–2 x Nb1- x Ti2 x O6 (0 ≤ x ≤ 1) solid-solution ceramics were designed
based on the Li2 TiO3 −Li3 NbO4 −MgO pseudo ternary phase diagram and studied for microwave dielectric applications. As the substitution amount (x) increased, we detected the phase transitions among the
orthorhombic, cubic, and monoclinic phase driven by the compositional changes, as well as accompanied by an order-disorder-order transformation. A full range of solid solutions was formed between the
Li3 Mg2 NbO6 and Li2 TiO3 endmembers, with no trace of other impurities. In the sample with the low
substitution concentration (x = 0.2 mol), a coherent phase interface (CPI) between the cubic and orthorhombic lattices was formed with no obvious misfit dislocation or stacking fault, indicating the small
differences in crystal configuration, chemical bonding properties, and subcell lattice parameters between
the two phases. Besides, there were observed diverse reconstructed superlattices, one kind is that possessed a “transition form” of the two phases and was formed nearby the CPI, and the other kind was
formed based on the cubic or orthorhombic lattices independently and was observed on a larger scale
nearby the CPI. The preferential substitutions of the non-equivalent cations, which were determined by
ionic radius, electronegativity, and local electroneutrality, and the interfacial strains would together act on
the formation of these superlattices. The Q × f values measured in the microwave range increased considerably around the compositional range where the superlattices were formed, indicating that the effect
of reconstructed superlattices on the intrinsic loss should not be overlooked. As proven by the dielectric
response in the high-frequency range (0.5 − 1 THz), the x = 0.2 sample indeed showed extremely higher
Q × f values than other ones, which illustrated that the sample with the reconstructed superlattices was
related to a small lattice vibrational anharmonicity that is favorable for the low dielectric loss.
© 2021 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
1. Introduction
The ever-growing traffic explosion in mobile communications
has recently drawn increasing interest in the designs of new microwave dielectric components for high data rates. Certain frequency range near the bottom of the millimeter-wave spectrum
(30−80 GHz) is being used in the 5th generation cellular networks because of the enhanced spectrum bandwidth, which enables high-speed signal transmission [1]. Because the relative per-
∗
Corresponding author.
E-mail addresses: zixuanfang@uestc.edu.cn (Z. Fang), tangbin@uestc.edu.cn (B.
Tang).
https://doi.org/10.1016/j.actamat.2021.116636
1359-6454/© 2021 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
mittivity is inversely proportional to the signal propagation velocity through the medium, materials with low permittivities (ε r ≤25)
are considered fitting for the millimeter-wave applications [2]. Besides, a high quality factor (Q) and a near-zero temperature coefficient of the resonant frequency (τ f ) are essential factors for the
practical usage of dielectric ceramics [3,4].
In recent years, numerous ceramics with the rock-salt structure were explored with properties well-suited for millimeterwave applications because of their low relative permittivities and
low dielectric losses, such as Li2 TiO3 , Li3 NbO4 , Li3 Mg2 NbO6 , and
Li2 Mg3 TiO6 , etc. [5-8]. Among these matrix ceramics, the Li2 TiO3
ceramic attracted much attention because of its unique positive τ f
(τ f =30−36 ppm/ °C) within the group of low-permittivity dielec-
X. Zhang, Z. Fang, H. Yang et al.
Acta Materialia 206 (2021) 116636
tric ceramics. According to the mixing rule of dielectrics, Li2 TiO3
was a useful dopant to composite with another phase with a negative τ f to get a near-zero τ f value [9-11]. The dielectric loss will
be inevitably increased if the impurity phase is introduced into
the matrix, but this effect can be avoided by forming a solid solution in the Li2 TiO3 -based ceramic system, where a high Q together
with a near-zero τ f value can be jointly achieved. For instance,
Bian et al. utilized Mg2+ ions to co-substitute Li+ /Ti4+ ions to
form the (1-x)Li2 TiO3 -xMgO solid solutions, where the replacement
mechanism is 3Mg2+ 2Li+ +Ti4+ , and the excellent microwave dielectric properties were achieved in the x = 0.24 sample with
εr =19.2, Q × f = 106,226 GHz, and τ f =3.56 ppm/ °C [12]. Interestingly, a continuous monoclinic-cubic phase transition accompanied by an order-disorder transformation occurred in the Li2 TiO3 rich end of the Li2 TiO3 −MgO solid-solution system, and the transformed phase corresponded to the symmetry of the MgO cubic
structure with Fm3̄m space group (S.G.). The Li2 Mg3 TiO6 ceramic
adopted a disordered cubic structure and exhibited ultra-high Q × f
values (ε r =15.2, Q × f = 152,0 0 0 GHz, and τ f =−39 ppm/ °C),
and it belongs to the Li2 TiO3 −MgO solid-solution system [8]. Besides, phase-transition phenomena appeared to be common in
the solid solutions formed by the endmembers with a rock-salt
crystal configuration but different symmetries. For other examples of the two-endmember systems, an ordered cubic (S.G. I43 m)-disordered cubic (S.G. Fm3̄m)-monoclinic (S.G. C2/c) phase
transition was reported in the Li2 TiO3 −Li3 NbO4 system, and an
orthorhombic (S.G. Fddd)-cubic (S.G. Fm3̄m) phase transition was
found in the Li3 NbO4 −MgO pseudo-binary system [13,14]. Among
these solid solution systems, a wide range of desired properties
was tunable by compositional modifications. On the whole, the
above-mentioned studies indicated that the Li2 TiO3 , Li3 NbO4 , and
MgO could form solid solutions in pairs. We supposed that the
three-endmember solid solution systems would offer more degrees
of freedom to obtain promising properties that cannot be achieved
in the two-endmember systems. As we have preliminarily studied in our earlier work, the compound of the central section of
the Li2 TiO3 −Li3 NbO4 −MgO phase diagram, Li5 MgTiNbO8 , adopted
a pure cubic phase and exhibited excellent properties of ε r =17.55,
Q × f = 109,700 GHz, and τ f =−32.5 ppm/ °C, which has shown
great potential for the development of the materials with excellent
performance in the Li2 TiO3 −Li3 NbO4 −MgO pseudo ternary system
[15]. To make it easier to be comprehended the structural evolutions among the Li2 TiO3 , Li3 NbO4 , and MgO ceramics, we plotted
the pseudo ternary phase diagram of the Li2 TiO3 −Li3 NbO4 −MgO
system according to the data from the abovementioned literature
and the other studies concerning the Li2 MgTiO4 , Li3 Mg2 NbO6 , and
Li4 Mg3 Ti2 O9 compounds [7,16,17], as shown in Fig. 1.
Apart from the solid solutions in the Li2 TiO3 −Li3 NbO4 −MgO
pseudo ternary system, there were many other two-endmember
systems with large solid solubilities and excellent microwave dielectric properties, such as the Li2 SnO3 −MgO [18], Li2 ZrO3 −MgO
[19], and Li2 ZrO3 −Li3 NbO4 [20] systems. The universal large solid
solubilities in the rock-salt solid solutions may be attributed to
the loose constraint of the ion radius for the rock salt structure,
where the radii of the cations (RA ) and the radii of the anions (RX )
should meet the scope of 0.42≤RA /RX ≤0.72 [21]. A large solid solubility of a rock-salt system represents that there is a broad compositional range that can be adjusted to seek desired properties
without worsening them due to impurities. On the other hand,
it appeared that the composition-driven phase transitions in the
abovementioned solid solutions were induced by the complex substitutions of the non-equivalent ions, and what could be determined was that the change of cation ordering and crystal symmetry would inevitably affect the intrinsic loss of the dielectrics
[22,23]. But the studies concerning the mechanisms of the phase
transitions and the origins of the ultra-low loss of these rock-salt
Fig. 1. Pseudo phase diagrams of the Li2 TiO3 –Li3 NbO4 −MgO ternary system.
systems were limited until now. It is noticeable that there has
been a phenomenological correlation between the high ordering
degree and high Q value widely studied in complex perovskites
[24,25]. Besides, the high Q values were also found in the complex perovskites that were substituted by the non-equivalent ions,
and the extremely low losses were owed to the formation of the
ordering-induced domains in the samples with the low dopant or
substituent concentration [26]. It is vital and necessary to understand the intrinsic loss affected by structural changes and the most
promising approach is to study the high-frequency response of materials, including the whole submillimeter (0.3 − 3 THz) and part
of the far-infrared (1.5 − 36 THz) range. The intrinsic dielectric
properties of materials are overwhelmingly stronger than the extrinsic ones in the dielectric response at tremendously high frequency due to the proximity of phonon eigenfrequencies (generally
in the order of 1011 −1012 Hz) [27,28]. The classic damped oscillator
model which fits the far-infrared reflectivity data were widely used
to extrapolate the dielectric properties from far-infrared down to
the microwave range, but it is not accurate for materials with permittivities below 20 [28]. Direct extrapolation of dielectric properties from submillimetre to microwave range is valid because the
proportionality ε ’’ (imaginary part of dielectric function) ∝ f (frequency) is roughly obeyed in the whole range from the microwave
to submillimeter wave [27-29].
In this paper, we are aiming to clarify the relationship among
the structural changes, phonon vibrations, and microwave dielectric properties of the Li3+x Mg2–2x Nb1-x Ti2x O6 (0 ≤ x ≤ 1) ceramics in the Li2 TiO3 −Li3 NbO4 −MgO pseudo ternary system. The
Li3 Mg2 NbO6 endmember was chosen due to its promising properties (ε r =16.8, Q × f = 79,600 GHz and τ f =−27.2 ppm/ °C), and
several studies have proved that the properties could be effectively
improved by the non-equivalent ion substitutions [7,30-32]. The
Li2 TiO3 endmember was selected because the temperature-stable
sample was expected in the Li2 TiO3 -rich end of the phase diagram.
2. Methods
2.1. Material preparation
Li3+ x Mg2–2 x Nb1- x Ti2 x O6 (0 ≤ x ≤ 1) ceramics were prepared
via a high-temperature solid-state reaction method. Reagent-grade
raw powders of Li2 CO3 (99.99%), Nb2 O5 (99.99%), TiO2 (99.99%),
and Mg(OH)2 •4MgCO3 •5H2 O (99.95%) were baked at 150 #x00B0;C
2
X. Zhang, Z. Fang, H. Yang et al.
Acta Materialia 206 (2021) 116636
indicative by the global instability index (GII) [38]:
GII =
Vi(obs) − Vi(theo)
2 1 / 2
(4)
where Vi(obs) is the valence sum of the experimentally observed
bond valence vi j (obs) , and Vi(theo) is the sum of the theoretical bond
valence. Brown has described that the structure is strained when
the BSI value is more than 0.05 vu (valence unit), and the crystal
is unstable when the GII value is greater than 0.20 vu [35].
Applying a statistical method, the configurational entropy of a
certain macrostate can be calculated by the Boltzmann entropy
equation [39]:
Fig. 2. Schematic of the device that was used to prevent lithium volatilization in
high temperatures.
Scon f ig = kB ln W
(5)
where kB is the Boltzmann constant and W is the number of microstates possible for the given macrostate. Considering a crystal with N atoms and k crystallographic orbits, when the atoms
are completely disordered, the configurational entropy is obtained
[39]:
overnight to remove the moisture. The raw powders were weighed
according to their stoichiometric ratio and ball milled with zirconia media in ethyl alcohol for 8 h and then were dried for 12 h.
The dried slurries were calcined at 900 #x00B0;C − 1000 °C in air
for 4 h. A second grinding using the same method as above for 3 h
then was conducted, and the reground and dried powders were
granulated by adding an 8 wt% PVA as a binder, then pressed into
cylinders with a dimension of 12 mm × 6 mm under a pressure
of 16 MPa. The ceramic green bodies were buried in high-purity
inert oxide powders with Li2 CO3 powders covered to prevent the
Li element from evaporating in high temperatures, where the inert
oxide powders (such as ZrO2 and Al2 O3 ) were used as a protective
agent and the Li2 CO3 powders were used for providing a Li-rich atmosphere, as shown in Fig. 2. The samples were sintered with the
atmosphere controlling from 1210 °C to 1290 °C for 4 h.
Scon f ig = −kB N
k
ni
ni
ln
N
N
(6)
i=1
where the ni is the number of atoms from the ith orbit. Then, considering a specifical case that an ordered crystal has N atoms to be
preferentially substituted by ni atoms, and the N atoms have exn
actly 2 crystallographic orbits, in this context, the W=CNi , and the
configurational entropy is in the form:
Scon f ig = kB ln CNni = kB ln
N!
ni !(N − ni )!
(7)
n
where CNi represents the combinatorial number of the ni atoms
at the N sites in the crystals. After applying the Stirling formula
ln n! = n(ln n − 1 ), we obtain:
2.2. Crystal structure refinement and calculation
The powder X-ray data set was collected in the 2θ range
from 13°−90° with step size 0.0131° employing an X’Pert ProMPD (Philips, Netherlands) X-ray diffractometer (XRD), offering CuKα 1 radiation excited by a monochromatic incident beam of wavelength 1.540598 Å. Rietveld refinement method was performed using the General Structure Analysis System (GSAS) software together
with the EXPGUI program [33,34].
The bond parameters were obtained according to the bondvalence theory. The bond valence Vi of an atom is obtained by the
valence-sum rule [35]:
Vi =
νi j
Scon f ig = −kB N
High-resolution transmission electron microscopy (HRTEM) and
selected area electron diffraction (SAED) images were obtained
on a Tecnai G2 F20 S-Twin TMP (the United States) microscope,
which were operated at 200 kV. A crushing method was used to
prepare the specimens for the TEM analysis, and the obtained superfine samples were then dispersed in acetone and the suspension liquid was taken onto a 200 mesh Cu grid with carbon-coated
holey films for observation. The Crystal Maker together with Single Crystal software was used to simulating the standard electron diffraction models for the ideal crystals. Fast Fourier transform (FFT) and inverse FFT for the HRTEM images were conducted
by the Digital Micrograph software [40]. Raman spectroscopy was
conducted by a Horiba Jobin-Yvon HR800 UV (France) Raman spectrometer offering an incident beam with 514.5 nm wavelength, and
the peaks were analyzed by Peakfit software. The surface morphology of the grains was observed using an FEI Inspect F (England)
scanning electron microscopy (SEM), and the average grain sizes
were calculated by the Image J software.
(1)
vi j = exp
Ri j − di j
b
(2)
where dij is the length of the bond, Rij is the bond valence parameters from Brown’s report [36], and the b is a constant taken
to be 0.37 Å. As that lattice strain effects can cause excessive
bond stretching or compression, which leads to the mismatches
of cation-anion bond lengths, the so-called bond strain index (BSI)
gives the average deviation of the bond-valence values and was defined by the equation [37]:
BSI =
vi j (obs) − vi j (theo)
2 1 / 2
(8)
2.3. Characterization
where vij is the valence of each bond that the atom forms. The
bond valence of a chemical bond can be calculated from its bond
length [35]:
ni
ni
N − ni
N − ni
ln
+
ln
N
N
N
N
2.4. Microwave and terahertz wave dielectric properties
measurements
(3)
Bulk density was measured in deionized water by the
Archimedes’ principle. Relative density was obtained by the equation:
where vi j (obs) is the experimentally observed bond valence, and
vi j (theo) is the theoretical bond valence. The angle brackets indicated an average taken over all bonds in the formula unit. The
chemical strain induced lattice strain over a whole structure was
ρrel =
3
ρobs
ρtheo
(9)
X. Zhang, Z. Fang, H. Yang et al.
Acta Materialia 206 (2021) 116636
Fig. 3. (a) XRD patterns of the Li3+ x Mg2–2 x Nb1- x Ti2 x O6 (0 ≤ x ≤ 1) microwave dielectric ceramics sintered at 1250 #x00B0;C for 4 h. (b) Amplified spectra of the XRD
patterns from 15° to 30° (c) Amplified spectra of the XRD patterns from 41.5° to 45° (For interpretation of the references to colour in this figure legend, the reader is referred
to the web version of this article.)
where ρ obs is observed density, and ρ theo is theoretical density.
Concerning the materials consisting of two phases, the theoretical
density ρ theo can be calculated by:
ρtheo =
W1 + W2
W1
W
2
ρ1 + ρ2
to the pure orthorhombic phase (Li3 Mg2 NbO6 : JCPDS no.36–1018)
in the Fddd space group (No. 70). Yet an extra diffraction peak
emerges on the left side of (026) peak in the pattern of x = 0.1,
and the relative intensity of this peak increases with the increase
of the x value in the range of 0.1 ≤ x ≤ 3.0. The specimens in
the range of x = 0.3 to 0.8 crystallize to the pure cubic structure
with the Fm3̄m (No. 225) space group, which resembles the MgO
phase (JCPDS no. 01–1235). The abovementioned additional peaks
in the specimens ranging from x = 0.1 to 0.3 are the diffraction
peak of (200) planes of the cubic structure, and there is no trace
of other impurity peaks observed in these specimens; therefore,
there is an orthorhombic-cubic phase transition within the range
of x = 0.1 − 0.3. It is noticeable that the widths of diffraction lines
of the x = 0.28 pattern were much large than that of the x = 0.30
sample, indicating that when the phase transition progressed to
a late stage, there was a remarkable nonuniformity of crystallite
size, which was ascribed to the difference in crystal parameters
between the two phases, together with large crystal distortions
induced by the internal strains associated with the mismatches in
bond lengths [42]. On the other hand, the patterns of the specimens of x = 0.92−1 correspond to the monoclinic phase (Li2 TiO3 :
JCPDS no.33–0831) in the C2/c space group (No. 15), and, similar
to the structural changes between the orthorhombic and cubic
phases, there is a continuous cubic-monoclinic phase transition
within the range of x = 0.8 − 0.92. No other diffraction peak of
impure phases was observed in all the specimens, indicating that
all of the starting materials have formed solid solutions. Within
the phase transition stage, namely the two-phase solid solution,
the chemical compositions were identical for different phases, this
characteristic is different from the two-phase composites.
All of the three kinds of phases have the rock-salt type structural characteristics, where the cations and anions are octahedrally
coordinated by each other. The differences among them are the
coordination manner of the cations, as shown in Fig. 4. The or-
(10)
where W1 and W2 are the weight fractions and ρ 1 and ρ 2 are the
theoretical densities of the individual phases. Microwave dielectric
properties were measured by Hakki–Coleman method [41] with a
network analyzer (Agilent Technologies E5071C, the United States)
in TE011 mode together with a temperature chamber (DELTA 9023,
Delta Design, USA), and the measured frequencies of the specimens
herein were ranging from 7.6 − 9.6 GHz. The temperature coefficient of resonant frequency was calculated according to the variation of the resonant frequency from 25 °C to 85 °C:
τf =
f t2 − f t1
ft1 (t2 − t1 )
(11)
where ft1 and ft2 are the resonant frequencies at t1 = 25 °C and
t2 = 85 °C, respectively. The dielectric response at the terahertz
range was measured by a commercial THz time-domain spectroscopy (TDS) machine (Zomega FiCO) at room temperature. The
applied system comprises a set of advanced optoelectronic devices,
where the pulsed terahertz signals were generated by a femtosecond laser and a photoconductive antenna, and an oscillating optical delay line enabled fast scanning of the pulsed signals.
3. Results and discussions
3.1. Structural evolution and ordering transformation
The XRD patterns of the representative specimens of
Li3+ x Mg2–2 x Nb1- x Ti2 x O6 (0 ≤ x ≤ 1) ceramics are shown in
Fig. 3. Specifically, the patterns of x = 0 and x = 0.08 correspond
4
X. Zhang, Z. Fang, H. Yang et al.
Acta Materialia 206 (2021) 116636
Fig. 4. Schematic delineation of the (a) orthorhombic, (b) cubic, and (c) monoclinic structure.
thorhombic Li3 Mg2 NbO6 phase exhibits a rock salt superstructure,
where the Nb locates in one set of the octahedral sites (8a Wyckoff sites), and the Li/Mg occupies over three other sets of the octahedral sites in a partially ordered way (16 g, 16 g, and 8b Wyckoff sites). The Li2 TiO3 phase consists of the monoclinic rock salt
superstructures, of which the Ti occupies the 4e Wyckoff sites
and the Li ions occupy three different positions including 4e, 4d,
and 8f Wyckoff sites. As for the cubic phase, it consists of facecentered cations and anion lattices that are octahedrally coordinated by each other, and the different kinds of cations all together occupy the 4a Wyckoff positions randomly. Generally, the
superlattice diffraction caused by the cation ordering in a supercell
should appear at the low Bragg angles in the XRD patterns, such as
the (111) and (004) reflections in the orthorhombic structures and
the (002) reflections in the monoclinic structures. However, there
are no superlattice reflections in the disordered cubic structures.
Hence, an order-disorder-order transformation was occurring along
with the orthorhombic-cubic-monoclinic phase transition with the
increase of the x value in the Li3+ x Mg2–2 x Nb1- x Ti2 x O6 (0 ≤ x ≤ 1)
system.
Rietveld refinement method was conducted on characterizing
the phase weight fractions and cell parameters, the goodness-of-fit
values (Rwp , Rp , and χ 2 ) show high reliability, as seen in Fig. 5; and
the refined atomic fractional coordinates and lattice parameters
for the representative specimens of the Li3+ x Mg2–2 x Nb1- x Ti2 x O6
(0 ≤ x ≤ 1) ceramics are shown in Table S1. Fig. 6(a) displays the phase weight fractions in a pseudo-binary phase diagram as a function of the x values, the pure orthorhombic and
monoclinic phases are in relatively narrow compositional ranges,
while the cubic phase is in a wide compositional range. Fig. 7
shows the compositional ranges with different structural characteristics of the Li3+ x Mg2–2 x Nb1- x Ti2 x O6 (0 ≤ x ≤ 1) ceramics in
the Li3 NbO4 −MgO−Li2 TiO3 pseudo-ternary phase diagram, and it
is easy to identify the ratio of the three rock-salt matrixes by
this phase diagram. All of the cell volumes of the three structures
show non-linear decreasing trends with the increasing x values, as
shown in Fig. 6(b). In the meanwhile, as shown in Fig. 3, the movements of diffraction peaks toward the higher angles also indicate
the decrease of the cell volumes. On the other hand, the ion replacement mechanism of the Li3+ x Mg2–2 x Nb1- x Ti2 x O6 system can
be generally written as Li+ +2Ti4+ 2Mg2+ +Nb5+ , where the average effective ion radius of the (Li1/3 Ti2/3 )3+ (r = 0.657 Å) clusters is smaller than that of the (Nb1/3 Mg2/3 )3+ (r = 0.693) clusters. Besides, because of the formation of solid solutions in the
whole range of the Li3+ x Mg2–2 x Nb1- x Ti2 x O6 (0 ≤ x ≤ 1) system,
the substitutions by the group of ions that have the smaller average effective ion radius lead to the decrease of the cell volumes
of whichever phases. Specifically, Goldschmidt’s rules give detailed
descriptions of the ion substitution mechanisms [43]:
R=
|RA − RB |
RB
× 100%
(12)
which demonstrate that (i) a full substitution can achieve if the ion
radius difference R is less than 15%, and a limited replacement
happens if the R differs between 15% and 30%; (ii) the elements
can be replaced in case of the charge neutrality that is maintained
by the charge balance of the other contributing ions in the material. Afterwards, Ringwood complemented the substitution laws
with the difference in electronegativity of the ions [44]. Table 1
shows the possible substitution types and degree of substitutions
of the Li3+ x Mg2–2 x Nb1- x Ti2 x O6 (0 ≤ x ≤ 1) system. Therefore, according to the replacement laws, in the Li3 Mg2 NbO6 -rich end of
the pseudo ternary phase diagram, where the structure maintains
the orthorhombic type, the defect equation appears as:
2Li2 TiO3
Li3 Mg2 NbO6
→
3LiLi + LiMg + T i··Mg + T iNb + 6OO
(13)
As for the Li2 TiO3 -rich end of the phase diagram, the
Li3 Mg2 NbO6 can be considered as the substituent, hence the defect equation is:
Li3 Mg2 NbO6
2Li2 TiO3
→
3LiLi + Mg·Li + MgTi + Nb·Ti + 6OO
(14)
Besides, in the pure cubic phase, the defect equation cannot
be specifically written because the cations are occupied in a completely disordered manner. The general forms of the atomic coordinates and atomic occupancy ratio for the pure phases are given
in Table 2. Though the global electrical neutralities were satisfied
in the abovementioned defect equations, the specific cation distribution in the crystals was commonly complex and the cation substitution should occur in the form of the conservation of the local electroneutrality according to Pauling’s rule of electroneutrality
[45]. After considering the aforementioned Goldschmidt’s rules of
substitution, the principle of electronegativity, and Pauling’s rule
of electroneutrality simultaneously, the substitution type with the
lowest possible energy state in a complex structure could be preliminarily deduced without regard to the distortions of crystals. As
shown in Fig. 8(a) and (b), the cations tend to substitute the NbMg-Mg clusters of which the cation-oxygen octahedrons were edge
shared in the single or stable Li3 Mg2 NbO6 phase, because only
in this configuration that the local electroneutrality of the system
could be attained. Similarly, as indicated by Fig. 8(c) and (d), the
5
X. Zhang, Z. Fang, H. Yang et al.
Acta Materialia 206 (2021) 116636
Fig. 5. Representative Rietveld refinement results with experimental (red circles) and calculated (black line) X-ray powder diffraction profiles for the Li3+ x Mg2–2 x Nb1- x Ti2 x O6
(0 ≤ x ≤ 1) ceramics sintered at 1250 #x00B0;C for 4 h. The short vertical lines below the patterns mark the positions of Bragg reflections. The bottom continuous line is
the differences between the observed and the calculated intensity. “R” values (χ 2 , Rwp, and Rp) relate to the goodness of fit. (For interpretation of the references to colour
in this figure legend, the reader is referred to the web version of this article.)
Table 1
Ion radius and electronegativity of the elements, the
the degree of the substitutions.
substituent
matrix
matrix
matrix
substituent
matrix
matrix
matrix
substituent
matrix
matrix
substituent
matrix
matrix
R between the substituent and the matrix ions, and
Ion type
Electronegativity
Ion radius (CN=6) (Å)
Li
Mg
Nb
Ti
Ti
Mg
Nb
Li
Mg
Li
Ti
Nb
Li
Ti
0.98
1.31
1.59
1.54
1.54
1.31
1.59
0.98
1.31
0.98
1.54
1.59
0.98
1.54
0.76
0.72
0.64
0.605
0.605
0.72
0.64
0.76
0.72
0.76
0.605
0.64
0.76
0.605
cations are more likely to substitute Li-Ti-Ti clusters in the single
or stable Li2 TiO3 phase.
The bond strain index (BSI) and global instability index (GII) of
the disordered cubic phase are much larger than that of the ordered orthorhombic and monoclinic phases, as shown in Fig. S1,
which indicates large bond mismatches in the disordered structure [46]. It was clear that the large amounts of the non-equivalent
R
–
5.6%
18.8%
25.6%
–
16.0%
3.5%
20.4%
–
5.3%
19.0%
–
15.8%
5.8%
Degree of substitution
–
Full
Limited
Limited
–
Limited
Full
Limited
–
Full
Limited
–
Limited
Full
substitutions in the crystals would give rise to the lattice strain
that was associated with the mismatch in the cation-anion bond
length, and the increased strain would give rise to the destabilization of the ordered structure. The transformation from the ordered
to the disordered phases appeared to be an energetic compromise
in which the minimization of the coulombic cationic repulsions
was attained in the system at the expense of the cation ordering.
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X. Zhang, Z. Fang, H. Yang et al.
Acta Materialia 206 (2021) 116636
Table 2
General forms of the atomic fractional coordinates data and atomic occupancy ratio for the structures of the
Li3+ x Mg2–2 x Nb1- x Ti2 x O6 (0 ≤ x ≤ 1) ceramics.
x (mol)
Structure
Atom
Wyckoff Position
x
y
z
Occupancy
0 − 0.08
orthorhombic
8a
16 g
16 g
16 g
16 g
8b
8b
16 g
16 g
8a
16f
32h
0.125
0.125
0.125
0.125
0.125
0.125
0.125
0.125
0.125
0.125
0.125
0.116
0.125
0.125
0.125
0.625
0.625
0.625
0.625
0.625
0.125
0.125
0.354
0.379
0.125
0.293
0.293
0.289
0.289
0.125
0.125
0.289
0.293
0.125
0.125
0.295
1-x
0.72
0.28-x
0.38
0.62-x
0.79
0.21
x
x
x
1.00
1.00
0.1 − 0.28
0.3 − 0.8
orthorhombic-cubic transition
cubic
4a
4a
4a
4a
4b
0.000
0.000
0.000
0.000
0.500
0.000
0.000
0.000
0.000
0.500
0.000
0.000
0.000
0.000
0.500
(3 + x)/6
(2–2x)/6
(1-x)/6
x/3
1.00
0.8 − 0.92
0.92−1
cubic-monoclinic transition
monoclinic
Nb
Li1
Mg1
Li2
Mg2
Li3
Mg3
Li4
Ti1
Ti2
O1
O2
–
Li
Mg
Nb
Ti
O
–
Li1
Li2
Li3
Ti1
Ti2
Mg1
Mg2
Nb
O1
O2
O3
8f
4d
4e
4e
4e
4e
4e
4e
8f
8f
8f
0.238
0.250
0.000
0.000
0.000
0.000
0.000
0.000
0.141
0.102
0.138
0.077
0.250
0.045
0.415
0.747
0.045
0.415
0.747
0.265
0.586
0.906
0.000
0.500
0.250
0.250
0.250
0.250
0.250
0.250
0.138
0.138
0.135
1.00
1.00
(1 + x)/2
(1 + x)/2
(1 + x)/2
(1-x)/2
(1-x)/2
(1-x)/2
1.00
1.00
1.00
Fig. 6. (a) Pseudo phase diagram of the Li3+ x Mg2–2 x Nb1- x Ti2 x O6 (0 ≤ x ≤ 1) ceramics sintered at 1250 #x00B0;C for 4 h. (b) Variation of the cell volumes of the
orthorhombic, cubic, and monoclinic crystals in different phase stages. (For interpretation of the references to colour in this figure legend, the reader is referred to
the web version of this article.)
Fig. 7. Pseudo ternary phase diagram of the Li3 NbO4 −MgO−Li2 TiO3 system. The
studied compounds in different phases are shown in different symbols.
where
N
is
Avogadro
constant,
NNb(matrix) =N/6,
(NTi(substituent) )/2=xN/6. For the cubic phase, the cations are in
the state of complete disorder, hence the configurational entropy
of the cation replacements is in the form:
In this case, the entropic contributions to the overall free energies
should be significant. According to the abovementioned, in a single
or stable Li3 Mg2 NbO6 phase, the substituent ions tend to enter into
the Nb-Mg-Mg complex sites. The number of crystallographic orbits of this substitution pattern is exactly 2, and the configurational
entropy (Sconfig ) of the cation substitutions in the orthorhombic
phase is shown as a function of the substitution amount (x), where
for 1 mol of the compounds:
+
Besides, according to the aforementioned, the substituent
cations tend to enter into the Li-Ti-Ti complex sites of the single
or stable monoclinic phase, thus the number of crystallographic
orbits of the substitution pattern is still 2. The configurational entropy of the cation substitutions for the monoclinic phase can be
(NTi(substituent) )/2
Sconfig_(orthorhombic) = kB lnCN
Nb(matrix )
= −kB N [x ln x + (1 − x )ln(1 − x )]
3+x
3+x
1−x
1−x
ln
+
ln
6
6
3
3
1−x
1−x
x
x
ln
+
ln
(16)
6
6
3
3
Sconfig_(cubic) = −kB N
(15)
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X. Zhang, Z. Fang, H. Yang et al.
Acta Materialia 206 (2021) 116636
Fig. 8. Structure and atomic distributions of (a) Li3 Mg2 NbO6 and (c) Li2 TiO3 . Schematic representation for the mechanisms of the preferential cation substitutions in the
single or stable (b) Li3 Mg2 NbO6 and (d) Li2 TiO3 structure.
obtained:
Sconfig_(monoclinic) = kB lnC
same range as the free energies for cation ordering, which was expected to be very high in the system of this study.
The transformation of the lattice structures can be more clearly
studied by the electron diffractions (ED) techniques. Fig. 10 shows
the selected area electron diffraction (SAED) patterns and the
corresponding high-resolution transmission electron microscope
(HRTEM) images of the samples of x = 0, x = 0.5, and x = 1
taken along the main zone axes. The lattice transition from the orthorhombic to the cubic structure can be verified by Fig. 10(a) and
Fig. 10(b) or Fig. 10(d) and Fig. 10(e), of which the patterns were
taken along the same [100] direction ([100]O [100]C ). Besides, the
alternately dark and bright spots of the SAED pattern of the x = 1
sample, as seen in Fig. 10(c), are caused by the secondary electron
diffraction [47]. The secondary electron emission seems inevitable
in the x = 1 sample because of the strong preferred (002) orientation of the pure Li2 TiO3 compound (x = 1).
Moreover, to specifically study the lattice characteristics during
the orthorhombic-cubic phase transition process, a sequence of the
electron diffractions were conducted on the x = 0.2 sample. The
HRTEM image of the region Ⅰ of the filmy particle is shown in
Fig. 11(b), it can be seen that there is a crystallographic phase tran-
NNb(substituent)
(NTi(matrix) )/2
= −kB N ( (1 − x ) ln (1 − x ) + x ln x )
(17)
where (NTi(matrix) )/2=N/6, NNb(substituent) =(1-x)N/6. The values of the
configurational entropies for the cation substitutions in the three
types of phases are shown in Fig. 9. The disordering of the cubic
phase produces much larger Sconfig compared to the ordered orthorhombic and monoclinic phases. The largest Sconfig of the cubic phase (x = 0.3, Sconfig _( cubic ) =9.56 J/(mol•k)) is achieved in
x = 0.3 sample, where the orthorhombic-cubic phase transition
is completed, and it is almost two times larger than the ordered
orthorhombic phase (x = 0.3, Sconfig _( orthorhombic ) =5.08 J/(mol•k)).
Hence the disordering introduces a significant favorable contribution to the free energy and is more likely for the strained structures to reach a stable state. It is exactly for this reason, the disordered phase occupied a wide compositional range in the phase
diagram, as shown in Fig. 6(a). The excess energies associated with
the disruption of the cation order would be approximately in the
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X. Zhang, Z. Fang, H. Yang et al.
Acta Materialia 206 (2021) 116636
low, and the interfacial energy for this CPI would be at a low level
as well [49]. The cubic rock-salt structure is actually the subcell
of the orthorhombic rock-salt supercell, with a relationship√of the
cell parameters between
√ the subcell and supercell: asup = 2asub ,
bsup =2bsub , and csup =3 2csub [50]. In this context, on one hand,
the abovementioned phase separation behavior showed the two
kinds of lattices were continuous across the interface, and on the
other hand, there formed different kinds of reconstructed superlattices. First, as shown in Fig. S2, there are two kinds of spontaneously formed superlattices nearby the region of the CPI. As indicated by the FFT patterns (Fig. 12(d) and (e)) and HRTEM images
(Fig. 12(h) and (i)) of the areas where the superlattices formed (the
region C and D of Fig. 12(a)), the large supercells of the reconstructed superlattices contain the atomic configuration characteristics of both the cubic and orthorhombic lattices, which indicates
that these superlattices were the “transition form” between the orthorhombic and cubic lattices. The so-called “superlattice” is with
a periodic atomic arrangement and that the structural modulation
occurs to have a longer period (normally an integral multiple of
the original period) than that of the original crystal structure. Besides, another kind of superlattices was formed based on the cubic
or orthorhombic lattices independently, which was observed on a
larger scale nearby the CPI, as proved by the HRTEM images of the
region Ⅱ (Fig. 11(d)) and region Ⅲ (Fig. 11(g)) of the particle and
their corresponding SAED patterns (Fig. 11(e) and Fig. 11(h)). As
for Fig. 11(e), the pattern basically relates to the cubic structure,
but as compared to the SAED pattern for the pure cubic structure in Fig. 10(b) together with the simulated standard ED pattern
for the Fm3̄m cubic structure in Fig. 11(f), there are extra reflections observed within the fundamental spots. These extra reflections are the so-called “superlattice reflections” (SRs), the patterns
herein can be indexed to the cubic structure and they indicate a
tripling of the lattice parameters of the original cubic structure. As
proved by Fig. 11(d), the interplanar spacings of d1 and d2 are the
tripling of the d-spacing of the (002) and the (022) lattice planes
in the original cubic crystal, respectively. Besides, concerning the
SAED pattern for region Ⅲ (Fig. 11(h)), it basically associates with
the orthorhombic structure, and it likewise contains the SRs. In
this pattern, the superlattice reflections represent a doubling of
the d-spacing of the {022} planes of the original orthorhombic
Fig. 9. Configurational entropy (Sconfig ) of the cation substitutions for the three different types of phases in the Li3+ x Mg2–2 x Nb1- x Ti2 x O6 (0 ≤ x ≤ 1) system as a function of the substitution amount (x).
sition interface (PTI) between the orthorhombic and cubic structure. Fig. 11(c) shows the FFT pattern of the blue boxed region of
Fig. 11(b), which contains the two neighboring phases as well as
the PTI. The pattern indicates a regular orientation relationship between the two phases, where the (004)O planes of the orthorhombic phase are parallel to the (020)C planes of the cubic phase,
and this orientation relationship can be as well derived from the
atomic configuration of the interface, as shown in the amplified
HRTEM images of the PTI (Fig. S2). Besides, scarcely any misfit dislocations or stacking faults are observed in this interface, which
indicates that a coherent phase interface (CPI) is formed because
the two phases have a similar rock-salt type lattice configuration,
similar chemical bonding properties, and small mismatches in subcell lattice parameters [48]. Predictably, the coherency strains induced by such consistent structures with a good “match” would be
Fig. 10. SAED patterns of Li3+ x Mg2–2 x Nb1- x Ti2 x O6 (0 ≤ x ≤ 1) ceramics for (a) x = 0 sample taken along [100]O zone axis, (b) x = 0.5 sample along [100]C zone axis, and (c)
x = 1 sample along [001]M zone axis. (d − f) The correspongding HRTEM images of the selected areas of the above samples.
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Acta Materialia 206 (2021) 116636
Fig. 11. (a) Bright-filed TEM image of the x = 0.2 sample. (b) HRTEM image of the region Ⅰ of the particle. (c) FFT image for the blue boxed area of the (b) image. HRTEM
images for (d) region Ⅱ parallel to [100]C zone axis and (g) region Ⅲ taken along [100]O zone axis. SAED patterns for (e) region Ⅱ along [100]C zone axis and for (h) region
Ⅲ parallel to [100]O zone axis. Simulated standard ED model of the (f) cubic lattice with Fm3̄m space group and of the (i) orthorhombic lattice with Fddd symmetry. (For
interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
structure. As indicated in Fig. 11(g), the interplanar spacing of d3
is the double value of the original (022) planes. Overall, the observed superlattices are all nearby the phase transition interface
and are on a relatively short-range scale. The formation of these
reconstructed superlattices may be driven by the interfacial strains
between the two phases together with the aforementioned preferential substitution manners of the nonequivalent ions. The occurrence of the superlattices lowers the entropy of the local region by
introducing a higher level of ordering, which contributes an unfavorable part to the free energy. Hence, the formation energy of
the reconstructed superlattices would be high and the superlattices
would be formed in a small area; and once the internal strain of
the lattices was too large to keep the ordering, the structure was
likely to change into a more stable disordered cubic subcell. On the
other hand, there were not distinct monoclinic-cubic phase transition interfaces found in this study. However, as indicated by the
study of Leu-et al. [51], the superlattice structures were formed
and observed in the Li2 TiO3 −MgO system, where the monocliniccubic phase transition together with the order-disorder transformation occurred. Therefore, it is reasonable to conclude that the
Mg substituting in the Li sites together with the Mg and Nb cosubstituting in the Ti sites of the Li2 TiO3 compound, as shown in
Eqn. 10, can also induce the formation of superlattices during the
monoclinic-cubic phase transition process.
Raman spectrum was conducted to examine the detailed
phonon vibration modes in the Li3+ x Mg2–2 x Nb1- x Ti2 x O6 (0 ≤ x ≤ 1)
solid solution system, as shown in Fig. 13. Peaks are broad because of the coupling of phonons and the overlapping of sig10
X. Zhang, Z. Fang, H. Yang et al.
Acta Materialia 206 (2021) 116636
Fig. 12. (a) Amplified image of Fig. 11(b). (b − e) FFT images, (f − i) HRTEM photographs, and (j − k) schematic of the simulated configuration of the group of atoms for
different areas (A − D) marked in the (a) image.
octahedra [4,56]. It is noticeable that the Nb-O bond vibrations
are weak in the monoclinic type solid solution (x = 0.9 sample)
and they vanish in the Li2 TiO3 end-member (x = 1). Besides, the
redshift of the band from 792 cm−1 to 782 cm−1 represents the
lower Nb-O bond vibration energies of the x = 0.1 sample compared to that of the pure Li3 Mg2 NbO6 . Meanwhile, the blueshift
of the band from 788 cm−1 to 800 cm−1 indicates that the Nb-O
bond strengths are increasing and the vibration energies are rising
with the increasing substitution amount, and the phenomenon is
attributed to the decrease of the cell volume of the cubic structure
[57]. Furthermore, the peaks in the range of 137 ~ 151 cm−1 relate
to the Nb movement [15,58]. The bands located in the range of
270 ~ 430 cm−1 are associated with the O-Li-O bending and Li-O
stretching in the LiO6 octahedra [59]. On the other hand, however,
the broad bands located at around 480 cm−1 in the x = 0, 0.1, and
0.2 samples are correlated with the cation ordering, which is attributed to the weak Nb-Nb bonds symmetric vibrations (Nb-Nb
reverse vibrations: B1g and B2g ; Nb-Nb stretching vibrations: B3g )
[31]. The bands in the region of 650 ~ 750 cm−1 were assigned
to Ti-O stretching in the TiO6 octahedra [60]. Hence, it could be
concluded that the Ti-O stretching vibrations of the monoclinic
structure are stronger than that of the orthorhombic and the cubic structure.
The surface micro-morphology of the Li3+ x Mg2–2 x Nb1- x Ti2 x O6
(0 ≤ x ≤ 1) ceramics was studied by SEM, as shown in Fig. 14.
The measured average grain sizes for the representative specimens
are shown in Fig. 14(l) and Fig.S3 (d). It is noticeable that the average grain size (AGS) presents a sharply decrease as the substitution amount (x) increases from 0 to 0.1, and then it increases with
the increase of the substitution amount, reaches the top when the
substitution amount up to 0.8. As the specimens were crystallized
in the same sintering process, the difference in the grain growth
should be mainly attributed to the difference in the diffusivity of
ions, and the grain boundary mobility in different crystals [16,61].
When the substitution amount x is 0.1, the structure maintains the
ordered orthorhombic type. According to the preferential substitution mechanisms mentioned above, the substituent of Li+ and Ti4+
ions tends to enter into the edge shared octahedral sites of the
Nb-Mg-Mg clusters, which would cause lattice distortion. In this
case, the extra Coulombic destabilization of local distorted structures would form the barriers for ionic migration and then reduce
the diffusivity of the ions, which results in the more activation energy required for grain growth [62]. As a result, the AGS of the
Fig. 13. Raman spectra of the Li3+ x Mg2–2 x Nb1- x Ti2 x O6 (0 ≤ x ≤ 1) ceramics sintered
at 1250 #x00B0;C for 4 h. (For interpretation of the references to colour in this
figure legend, the reader is referred to the web version of this article.)
nals, as shown in the Gaussian deconvoluted Raman spectra of
the representative samples (Fig. S3). It is difficult to figure out
all these modes, however, the intense peaks arising from the
typical Raman scattering in a specific symmetry can be identified according to the literature relating to the analogical structures. The orthorhombic phase in the Fddd space group, where
the point group is D2h , presents 51 normal Raman active modes:
Fddd = 8Ag + 12B1g + 15B2g +16B3g [52]. The monoclinic structure with the space group C2/c (its point group is C2h) shows 33
Raman active modes: C2/c = 15Ag + 18Bg [52]. Besides, because
of the inversion symmetry of the disordered cubic phase (Fm3̄m
space group, Oh point group), the first-order Raman effect is forbidden in this structure and a second-order Raman scattering is allowed, which presents an irreducible representation from the twophonon processes: F m3̄m = A1g + Eg +T2g [53-55]. The Raman
spectra of the orthorhombic structures are partly similar to that of
the cubic phases because that the numerous vibration modes in
both structures are of the same types. For instance, the bands at
around 790 cm−1 correspond to the Nb-O bond vibrations in NbO6
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X. Zhang, Z. Fang, H. Yang et al.
Acta Materialia 206 (2021) 116636
Fig. 14. (a~k) SEM micrographs of the Li3+ x Mg2–2 x Nb1- x Ti2 x O6 (0 ≤ x ≤ 1) ceramics sintered at 1250 #x00B0;C for 4 h. (l) Average grain sizes of the specimens of the
Li3+ x Mg2–2 x Nb1- x Ti2 x O6 (0 ≤ x ≤ 1) ceramics. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
x = 0.1 sample decreases remarkably as compared to that of the
pristine Li3 Mg2 NbO6 , and a similar variation trend of AGS was also
observed in the Li2 TiO3 -rich end, as shown in Fig.S4. On the other
hand, because of the random distribution of the cations in the disordered structure, the cation repulsive interactions are likely less
pronounced than in the ordered phase, which can lead to a favorable cation diffusivity, and hence the larger AGS of the disordered
cubic samples compared to that of the ordered orthorhombic specimens can be explained. A similar influence of the cation ordering
on the ionic diffusivity was reported and discussed in other literature [63]. Besides, the average mass and the average effective
ion radius of the (Li1/3 Ti2/3 )3+ clusters are 34.25 g/mol and 0.657 Å
(CN=6), respectively, both of which are smaller than that of the
(Mg2/3 Nb1/3 )3+ clusters (average mass: 41.18 g/mol; average effec-
the polygonal grains in the microstructure morphology, the grain
boundary morphologies present a noticeable difference between
the ordered and disordered structures. The grain boundaries in
the ordered samples appear to be angular and faceted, while the
grain boundaries are more smoothly curved and, therefore, have a
rough structure in the disordered samples. This difference is indicated in Fig.S5 by the sketches for the “faceted” and “rough” grain
boundary morphologies. The “rough” morphology was ascribed to
the melting of the grain boundary, which was scarcely observed
in the ordered samples, indicating the grain boundary diffusion of
the disordered samples was higher than that of the ordered [65].
Therefore, it is reasonable to conclude that there are critical driving forces for grain growth and densification for the ordered structures, namely, that the grain growth is inhibited unless the driving
forces are above critical values; but there is no such restriction inhibiting the grain growth for the disordered structures, where the
continuous grain growth and densification occurred with the sintering process. This assumption can be proved by Si-Young Choi
et al’s work [61].
tive ion radius: 0.693 Å, CN=6). The larger atomic weight and ion
radius can both lead to less diffusion rate [64]. Hence, the substituent clusters should have a higher diffusion rate than the group
of ions of the matrix that were substituted, which resulted in better sintering behavior and the continuous increase of the AGS in
the pure cubic phases (0.3 ≤ x ≤ 0.8) with the increasing x value.
Furthermore, the reduced AGS for x = 0.9 and 1.0 samples can
also be explained by the lower diffusion rates of the ions in the
ordered monoclinic structure than that are in the disordered cubic structure. On the other hand, while all the specimens display
3.2. Microwave and terahertz wave dielectric properties
Fig.
15(a)
shows
the
bulk
densities
of
the
Li3+ x Mg2–2 x Nb1- x Ti2 x O6 (0 ≤ x ≤ 1) ceramics sintered from
12
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Acta Materialia 206 (2021) 116636
in this paper, Li3+ x Mg2–2 x Nb1- x Ti2 x O6 , the molecular polarization
can be calculated as follows:
αD (Li3+x Mg2−2x Nb1−x T i2x O6 )
= (3 + x )α Li+ + (2 − 2x )α Mg2+ + (1 − x )α N b5+
−
+2xα T i4+ + 6α O2
(20)
where α is the polarizability of ions from Shannon’s reports [67].
Besides, in the two-phase solid solutions (0.1 ≤ x ≤ 0.3 and
0.8 ≤ x ≤ 0.92), the relative permittivities can be calculated from
the Lichtenecker empirical rule [68]:
lnεr = V1 l nεr1 + V2 l nεr2
(21)
where Vi and ε ri are the volume fraction and the relative permittivity of the ith phase, respectively. The calculated theoretical relative permittivities (ε r-theo ) present a continuously increasing trend, as shown in Fig. 16(b). The ionic polarizability of the
(Li1/3 Ti2/3 )3+ substituent clusters (2.35 Å3 ) is higher than that of
Fig. 15. (a) Bulk densities of the Li3+ x Mg2–2 x Nb1- x Ti2 x O6 (0 ≤ x ≤ 1) ceramics
(1230~1290 °C, 4 h). (b) Relative densities of the Li3+ x Mg2–2 x Nb1- x Ti2 x O6 (0 ≤ x ≤ 1)
ceramics sintered at 1250 °C for 4 h. (For interpretation of the references to colour
in this figure legend, the reader is referred to the web version of this article.)
the (Mg2/3 Nb1/3 )3+ clusters (2.20 Å3 ) in the matrix. Hence, the increase of the ε r-exp in the range of x = 0.4 to 0.9 should be attributed to the decrease of the porosity and the increase of the ion
polarizability, and the increase of the ε r-exp in the range of x = 0.9
to 1.0 should be mainly attributed to the increase of the ion polarizability.
The Q × f values in temperatures from 1230 #x00B0;C to
1290 #x00B0;C are plotted as a function of the substitution
amount (x) in Fig. 17(a). The Q × f values of all the samples show
fluctuation changes and the variations of the Q × f values appear
to be sensitive to the changing of the phase types. Generally, the
extrinsic factors including grain size, secondary phase, and porosity, and the intrinsic factors involved in phonon vibrations, internal
strain, and crystal structures would together affect the dielectric
loss [69,70]. As dielectric loss was particularly sensitive to porosity, the variation trend of the Q × f values was partially similar
to that of the relative densities, as seen in Fig. 15(b). The effect
of the impurity phase could be excluded in the system because
the solid solutions were formed in the full compositional range.
Besides, according to Jonathan D. Breeze et al.’s study, the grain
sizes or grain boundaries would have a very limited influence on
the dielectric loss of the materials with a low relative permittivity [71]. While in the dense structures (ρ re >94%), intrinsic dielectric loss played a dominant role in the Q × f values. It is noticeable that there is a sharp increase in the Q × f value in the
Li3 Mg2 NbO6 -rich end of the phase diagram as a function of the
increasing Li2 TiO3 substituent content, and likewise, the Q × f values increase sharply as a function of the increasing Li3 Mg2 NbO6
substituent content in the Li2 TiO3 -rich end. For this reason, it can
be seen that the superior compounds with ultra-high Q × f values
(Q × f>10 0,0 0 0 GHz) in this study are in the orthorhombic-cubic
phase transition or the monoclinic-cubic phase transition parts.
Considering the reconstructed superlattices formed in the samples
with relatively low substituent concentration, which we have mentioned above, there was an assumption of a phenomenological correlation between the emergence of superlattice and low dielectric loss. The logical association of this assumption was derived
from the similar phenomena that were widely discussed in complex perovskites materials. Numerous studies have reported the
pronounced effect of the substitution of non-equivalent ions on the
B site cations in improving the Q value of the materials, such as in
the Ba(Zn1/3 Ta2/3 )O3 -BaZrO3 [72], Ba(Zn1/3 Ta2/3 )Mn0.1 O3 [73], and
Ba(Mg1/3 Ta2/3 )O3 -BaSnO3 [74] systems. Davies et al. investigated
in detailed the cation ordering reaction in the Ba(Zn1/3 Ta2/3 )O3 BaZrO3 system by TEM techniques and ascribed the very high Q of
the ceramics to the formation of the short-range ordered domains
and the stabilization of the ordering-induced domain boundaries
1230 #x00B0;C to 1290 #x00B0;C for 4 h as a function of the substitution amount (x). The bulk density shows a fluctuant decrease
trend, and the optimal sintering temperature for the densification
of the samples varies as the phase type changes. Fig. 15(b) displays the relative densities (ρ re ) of the Li3+ x Mg2–2 x Nb1- x Ti2 x O6
(0 ≤ x ≤ 1) ceramics sintered at 1250 #x00B0;C for 4 h. The
relative density varies as an M-shaped trend, peaks at 98.77% in
the x = 0.08 sample, where the orthorhombic-cubic transition
was about to occur. Then the orthorhombic-cubic phase transition
process lowers the relative density significantly because of the
deterioration of the sintering behavior and the increased porosities
as affirmed by the SEM observation (Fig. 14). Besides, when the
cubic-monoclinic phase transition was about to come to an end,
the relative density reaches its second peak at x = 0.88. Overall,
the porosity (p = 1-ρ re ) is consistent with the changing trend
of the number of micropores in the SEM images, as seen from
Fig. 14 and Fig.S4.
The relative permittivities (ε r ) of the samples sintered from
1210 °C to 1290 °C for 4 h as a function of the substitution amount
are shown in Fig. 16(a). The ε r firstly increases slightly, then decreases and then rises sharply. The variation trend of the ε r is
partially agreed with the changing trend of the relative density
(ρ re ) because the relative permittivity is sensitive to the porosity. The porosity-corrected permittivities (ε r-corr ) were calculated
by the following equation [66]:
εr−corr = εr−exp (1 + 1.5P )
(18)
where P is fractional porosity and ε r-exp is the experimentally
measured ε r . Since eliminating the effect of porosity on the ε r ,
the variation trend of the ε r-corr is partially different from that
of the ε r-exp especially in the range of x = 0 to 0.4. The result
indicated that the ε r-exp of the specimens that were during the
orthorhombic-cubic phase transition was mainly affected by the
extrinsic factor associated with the porosity. Besides, in the very
dense structure, ionic polarization is the intrinsic and dominant
factor that affects the relative permittivity. The theoretical relative
permittivities (ε r-theo ) can be calculated according to the ClausiusMosotti equation [67]:
εr−theo =
3Vm + 8π αD
3Vm − 4π αD
(19)
where Vm is molar volume and α D represents molecular polarizability. Considering the general form of the compositions studied
13
X. Zhang, Z. Fang, H. Yang et al.
Acta Materialia 206 (2021) 116636
Fig. 16. (a) Relative permittivities (ε r ) of the Li3+ x Mg2–2 x Nb1- x Ti2 x O6 (0 ≤ x ≤ 1) ceramics sintered from 1210 °C to 1290 °C for 4 h. (b) The ε r (ε r-exp ) of the
Li3+ x Mg2–2 x Nb1- x Ti2 x O6 (0 ≤ x ≤ 1) ceramics sintered at 1250 °C for 4 h with error bars, and its porosity corrected permittivities (ε r-corr ) and calculated theoretical permittivities (ε r-theo ). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 17. (a) Quality factor (Q × f) of the Li3+ x Mg2–2 x Nb1- x Ti2 x O6 (0 ≤ x ≤ 1) ceramics sintered from 1230 °C to 1290 °C for 4 h. (b) Q × f value with error bars, full
width at half maximum (FWHM) (β ) of the peaks nearby 43 two-theta degrees in the XRD patterns ((026) peaks in the orthorhombic structures, (200) peaks in the cubic
structures, and (−133) peaks in the monoclinic structures), and internal strain of d-spacing of the Li3+ x Mg2–2 x Nb1- x Ti2 x O6 (0 ≤ x ≤ 1) ceramics sintered at 1250 °C for 4 h.
(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
On the other hand, the internal strain was a vital source of intrinsic dielectric loss and would be appropriate for the characterization even when the phase model and crystal symmetry changed
[69]. Stokes and Wilson have proposed the equation that correlated
the internal strain of crystals with the broadening of the DebyeScherrer lines in x-ray photographs [42]:
via the partial segregation of the Zr ions when a small amount of
the substituent ions were added [26]. Besides, recently, Bian et al.
also reported the big rise of the Q × f values in the Li2 TiO3 −MgO
rock-salt system when a small amount of Mg ions were added [12].
Then the follow-up works by Lii-Cherng Leu-et al. investigated the
ordering changing of the Li2 TiO3 −MgO system, and the orderinginduced superlattices were found in the samples with low Mgconcentration [51]. According to these cases, the superlattice domains were likely to be universally formed where once the complex sites of either the ordered complex perovskites or the ordered rock-salt ceramics were substituted by a small amount of
non-equivalent ions, and the ordering-induced superlattices were
expected to be highly related to the low dielectric loss of the materials.
η=
β
2tanθ
(22)
where η represents the internal strain/fluctuation of d-spacing, θ
is Bragg angle, and β is the full width at half maximum (FWHM)
of X-ray diffraction peaks. The results of the η and β as a function of the substitution amount (x) are shown in Fig. 17(b), and
the variation trend of the η is similar to that of the β and is
14
X. Zhang, Z. Fang, H. Yang et al.
Acta Materialia 206 (2021) 116636
Fig. 18. (a) Dielectric permittivity (ε ’), (b) dielectric loss (tan δ ), and (c) Q × f values as a function of frequency in the range 0.5 − 1 THz for the x = 0.1, 0.2, 0.5, 0.8, and
0.9 samples sintered at 1250 #x00B0;C for 4 h. The Q × f values measured by the Hakki-Coleman method in the microwave range for the corresponding samples are as well
given in the (c) figure. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
roughly inversely correlated with the changing trend of the Q × f
values. However, the very large internal strain of the d-spacing of
the x = 0.25 and 0.28 samples did not lower the Q × f values
much, hence, the ordering continually showed a great influence on
the low dielectric loss in these strained samples. Besides, the small
amount of the substitution (x = 0.08−0.20) interestingly resulted
in lower internal strains of d-spacing compared to that of the unsubstituted sample, which as well led to the increase of the Q × f
values. Notably, in the x = 0.2 sample, because the CPI formed with
a good “match”, as we have mentioned above, very tiny unfavorable elastic strains would produce on the interface. The reduced
internal strain may be ascribed to that a certain amount of internal strain was relieved when a small amount of phase transformed
from the ordered to the disordered structure, as that the disordering was more favorable for the stabilization of the strained structure.
Furthermore, the Terahertz time-domain spectroscopy was conducted to specifically evaluate the magnitude of the intrinsic dielectric loss that was associated with phonon oscillation. Fig. 18
shows the dielectric properties of the representative samples
(x = 0.1, 0.2, 0.5, 0.8, and 0.9) measured in the range 0.5 − 1.0
THz. The dielectric permittivities are almost following the values
that are measured in the microwave frequencies, as seen in Fig. 16,
indicating that the polarizability and hence the relative permittivity of the materials is insensitive to the extrinsic factors except for
porosity. The loss tangent (tan δ =ε ’’/ε ’ =1/Q) of the x = 0.2 sam-
ple is small and the Q × f values of the x = 0.2 sample in the
THz regimes that are calculated based on the gigahertz unit are
much higher than other ones, indicating a low anharmonicity of
lattice vibrations in the x = 0.2 sample [28,75]. Therefore, accordingly, the formation of the reconstructed superlattices might result
in a low degree of lattice anharmonicity, which was favorable for
the decrease of the intrinsic dielectric loss.
The temperature coefficient of resonant frequency (τ f ) of the
Li3+ x Mg2–2 x Nb1- x Ti2 x O6 (0 ≤ x ≤ 1) ceramics firstly decreases and
then increases as a function of the substitution amount (x), as
shown in Fig. 19. It is well known that the τ f decreases with
the increase of the amplitude of the oxygen octahedral tilting in
perovskites [76]. The recent literature has proved that this relation was also applicable in the rock-salt dielectrics [18,77]. Hence,
the lower τ f values of the cubic phase compared to that of the
orthorhombic and monoclinic phases should be attributed to a
greater degree of octahedral tilting that was owed to the larger
amounts of cation-anion bond length mismatches in the completely disordered cubic structures. Besides, the variations of the τ f
values in the phase transition stages are dependent on the mixing
rule, which can be described by the Lichtenecker equation [78]:
τ f = V1 τ f 1 + V2 τ f 2
(23)
where V1 and V2 are the volume fractions and τ f 1 and τ f 2 are the
τ f value of each phase. The near-zero τ f value was obtained in the
15
X. Zhang, Z. Fang, H. Yang et al.
Acta Materialia 206 (2021) 116636
sample with ε r =20.4, Q × f = 90,300 GHz (f = 7.9 GHz), and
τ f =2.9 ppm/ °C.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to
influence the work reported in this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant No. 51672038).
Supplementary materials
Supplementary material associated with this article can be
found, in the online version, at doi:10.1016/j.actamat.2021.116636.
Fig. 19. Variation of the temperature coefficient of resonant frequency (τ f ) as a
function of the substitution amount (x) in the Li3+ x Mg2–2 x Nb1- x Ti2 x O6 (0 ≤ x ≤ 1)
ceramics sintered at 1250 °C for 4 h. (For interpretation of the references to colour
in this figure legend, the reader is referred to the web version of this article.)
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4. Conclusions
The Li3+x Mg2–2x Nb1-x Ti2x O6 (0 ≤ x ≤ 1) solid-solution ceramics were designed by means of the Li2 TiO3 –Li3 NbO4 −MgO pseudo
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