MINISTRY OF EDUCATION AND TRAINING
HANOI NATIONAL UNIVERSITY OF EDUCATION (cid:63) (cid:63) (cid:63) (cid:63) (cid:63)
Le Thu Lam
DIFFUSIONAL AND ELECTRICAL PROPERTIES
OF OXIDE MATERIALS WITH FLUORITE STRUCTURE
Speciality: Theoretical and Mathematical Physics
Classification: 9.44.01.03
SUMMARY OF PhD THESIS
Ha Nội, 2020
The work was completed at Hanoi National University of Education
Science supervisor: Assoc.Prof.Dr. NGUYEN THANH HAI
Assoc.Prof.Dr. BUI DUC TINH
The first reviewer: Assoc.Prof.Dr. Nguyen Hong Quang
Institute of Physics
The second reviewer: Assoc.Prof.Dr. Nguyen Nhu Dat
Duy Tan University
The third reviewer: Assoc.Prof.Dr. Nguyen Thi Hoa
University of Communications and Transport
The thesis will be defended at Hanoi National University of Education on . . . . . . . . . .
The thesis is available at:
1. Library . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Library . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
INTRODUCTION
1. Rationale
Nowadays, solid oxide fuel cell (SOFC) can be seen as a re-
newable energy source which has the most potential. To improve
performance and commercialize of SOFC, diffusion and electrical
properties of fluorite structure materials need to be investigated.
Most of the previous theoretical studies concerning the diffusion
and electrical properties of the bulk material with fluorite structure
are based on the simple theory of thermal lattice vibrations. Dif-
fusion coefficient and ionic conductivity of undoped material with
the thermal oxygen vacancy were calculated by statistical moment
method (SMM). In the doped materials, the most oxygen vacancy
are generated by dopant and they requires a available calculation
models. Moreover, there is a lack of theoretical methods related the
oxide thin film with fluorite structure and the experimental meth-
ods give the conflicting results about the effect of thickness on the
conductivity ionic of the thin films.
For those reasons mentioned above, we decided to choose the
topic:“Diffusional and electrical properties of oxide materials with
fluorite structure”.
2. Purpose, object and scope of research
The purpose of thesis is to study the influence of temperature,
pressure and dopant on the diffusion coefficient and ionic conduc-
tivity of the bulk material as CeO2, c-ZrO2, YDC and YSZ. For YDC and YSZ thin films, we find the thickness dependence of the
lattice constant, diffusion coefficient and ionic conductivity. The
roles of grain boundary and substrate are canceled in our study.
2
3. Research method
In the thesis, the diffusion and electrical properties of the fluo-
rite structure materials are investigated using SMM including the
anharmonicity effects of thermal lattice.
4. Scientific and practical senses
The results related the vacancy migration paths, the vacancy-
dopant associations, the influences of temperature, pressure, dopant
and thickness on the diffusion coefficient and ionic conductivity
provide the important information about the diffusion and electrical
properties of the fluorite structure materials. The obtained results
of dopant and thickness dependences of the ionic conductivity could
be applied to make the electrolytes with high ionic conductivity
used in SOFC.
5. New contribution of thesis
The thesis has established a new theoretical model using SMM
to investigate the diffusional and electrical properties of fluorite
structure materials from simple to complex. In comparison with
the last model for CeO2, the advantage of this model is finding the preference migration path of oxygen vacancy. Some of calculated
results related bulk materials are in better agreement with the ex-
perimental results compared to the other theoretical results. The
calculated results of thin film supplement to the experimental re-
sults. The obtained results related the oxide thin film with fluorite
structure supplement the experimental investigation
6. Thesis outline
In addition to the introduction, the conclusions, the references,
the thesis content is divided into 4 main chapters.
3
CHAPTER 1
OVERVIEW OF DIFFUSIONAL AND ELECTRICAL
PROPERTIES OF FLUORITE STRUCTURE MATERIALS
1.1. Fluorite structure materials
Ceria (CeO2) and cubic phase of zirconia (c-ZrO2) with the open fluorite structure allow the oxygen vacancy to migrate through the
lattice with relative ease. However, the number of oxygen vacancy
is small due to the high vacancy formation energy. Hence, they are
not good ionic materials.
Yttria (Y2O3) doped with CeO2 (YDC) and c-ZrO2 (YSZ) sta- bilize the cubic phase of c-ZrO2 down to room temperatures and increase the vacancy concentration. The migration of oxygen va- cancy occurs via the site exchange with O2− ions at the opposite neighbour sites. However, the vacancy migration is blocked by the
vacancy-vacancy interactions and vacancy-dopant associations.
Thin films have a high surface-to-volume ration and large con-
centration of grain boundary. The vacancy formation and migration
at the surface and grain boundary effect strongly on the vacancy
formation and migration in the whole thin film.
Due to the high ionic conductivity, these materials are widely
applied as the useful electrolytes in solid oxide fuel cell (SOFC).
1.2. The main research methods and results
The diffusional and electrical properties of the fluorite structure
materials are investigated by the theoretical methods (molecular
dynamics simulation, density functional theory, Monte-Carlo sim-
ulations, ...) and the experimental method (sputtering, chemical
vapor deposition, pulse laser deposition, ...).
4
For the bulk materials, the previous results showed that the
oxygen vacancy in defect clusters is located at the 1NN sites relative to Y3+ ion in YDC and at the 2NN sites relative to Y3+ ion in YSZ. The oxygen vacancy migrates dominantly along the <100>
direction and the presence of dopant in the cation barriers blocks
the vacancy migration. Remarkable, the ionic conductivity depends
linearly on the dopant concentration.
For CeO2 thin film, the charged particle is mainly electron. But the current in doped thin films (YDC and YSZ) is predominantly
carried by the oxygen vacancy. The measured results of ionic con-
ductivity depend strongly on the substrate, measurement and fab-
rication methods. The enhancement of ionic conductivity with the
decreasing thickness was observed for YDC thin film. For YSZ thin
film, the experiments showed the conflicting results about the ef-
fects of substrate and thickness on the ionic conductivity.
3. The statistical moment method
The statistical moment method (SMM) investigate the physical
properties of crystal including the anharmonicity effects of thermal
lattice vibrations. Based on statistical operator ˆρ, the authors es-
tablished the general formula of moments to determine high level
moments from low level moments. The expressions related mechan-
ical, thermal, electrical properties are derived in closed analytic
forms in terms of the power moments of the atomic displacements.
The calculated results are in good agreement with the other theo-
retical and experimental results.
5
CHAPTER 2
INVESTIGATE OF DIFFUSIONAL AND ELECTRICAL
PROPERITES OF CERIA AND ZIRCONIA
2.1. Anharmonicity vibration and Helmholtz free energy
2.1.1. Anharmonicity vibration
The expression of ionic interaction potential in RO2 system [96]
(cid:88) (cid:88)
U =
ϕR
ϕO
(2.1)
i0 (|(cid:126)ri + (cid:126)ui|) +
i0 (|(cid:126)ri + (cid:126)ui|) .
NR 2
NO 2
i
i
The displacements of ions in RO2 system are determined as
(cid:115)
(2.22)
AR,
yR 0 ≈
2γRθ2 3k3 R
(cid:115) (cid:18) (cid:19)
+
1 +
AO −
yO 0 ≈
βO 3γO
1 KO
−
. (2.28)
(xOcothxO − 1) −
Oθ2 6γ2 K4 O 2β2 O 27γOkO
(cid:20) 1 3 (cid:21) βOkO γO
2γOθ2 3K3 O 2γOθ 3k2 O
2.1.2. Helmholtz free energy
The Helmholtz free energy of ions in RO2 system are given by
[96,98]
(cid:21)
+
ΨR ≈ U R
0 + ΨR
0 + 3NR
(cid:20) 2 (XR)2 − γR
aR 1
2γR 1 3
(cid:26) θ2 k2 R
(γR
(cid:21)(cid:27) ,
+
2 )2XR − 2(cid:0)(γR
1 )2 + 2γR
1 γR 2
(cid:1)(1 + XR) (cid:20) 4 3
2θ3aR 1 k4 R
(2.39)
(cid:21)
+
ΨO ≈ U O
0 + 3NO
O −
aO 1
0 + ΨO
(cid:20) 2 X 2 γO
2γO 1 3
(cid:26) θ2 k2 O (cid:21)(cid:27)
+
(γO
+
2 )2XO − 2(cid:0)(γO
1 )2 + 2γO
1 γO 2
(cid:1)(1 + XO) (cid:20) 4 3
2θ3aO 1 (kO)4
6
2
(cid:19) (cid:19) 1 (cid:26) θβ
−
− 1
+
+ 3NO
6KOγO
(cid:18) kO KO
θ2β KO
−
(cid:20)(cid:18) 2γOaO 1 3K3 O (cid:21)(cid:27) .
(2.42)
+
+
(XO − 1)
βO 6KOkO
βOaO 1 9K2 O
βOkOaO 1 9K3 O
Eqs. (2.39) and (2.42) enable us to calculate the Helmholtz free
energy of RO2 system
(2.46)
Ψ = CRΨR + COΨO − T SC.
2.1.3. Equation of states
At temperature T = 0 K, the equation of states can be written
as [97]
(cid:110) (cid:105)
P v = −a
+
+
(cid:105)(cid:111) .
CR
+ CO
(cid:104) 1 6
∂uR 0 ∂a
∂kR ∂a
(cid:104) 1 6
∂uO 0 ∂a
(cid:126)ωR 4kR (cid:126)ωO 4kO
∂kO ∂a (2.49)
2.2. Electrical and diffusion theories
2.2.1. Diffusion coefficient and ionic conductivity
The diffusion coefficient and ionic conductivity are given by
[2,3,92-94]
(cid:32) (cid:33) (cid:18) (cid:19)
−
,
(2.59)
D = r2
1n1f
exp exp
ωO 2π
Sf v kB
Ea kBT (cid:16)
1n1f ωO
8 a3 r2
(Ze)2 kBT
(cid:17) (cid:17) exp (cid:18) (cid:19) (cid:16) Sf v kB
− gf v kBT
σ =
−
.
2π exp T
exp
Ea kBT
(2.64)
with Ea is the vacancy activation energy
(2.60)
Ea = Ef + Em.
7
in which Ef and Em are the vacancy formation and migration en- ergies, respectively.
2.2.2. Vacancy activation energy
2.2.2.1. Vacancy formation energy
The vacancy formation energy can be written as
(cid:104) (cid:105)
−
∆Ψ =
CRNRψva
R + CO(NO − 1)ψva O (cid:105)
(cid:104)
−
,
(2.68)
CRNRψlt
CO
O
− COψlt
Ef ≈ ∆Ψ +
O + T Sf
v + P ∆V,
R + CO(NO − 1)ψlt O (cid:1) (cid:0)ψ∗min O + ψ∗max 2
(2.72)
with the free energies are determined via the average interaction
potentials of an ion in RO2 system having an oxygen vacancy.
(cid:88) (cid:88)
+
,
(2.80)
uva O =
bO−O i
ϕ∗O−O i0
bO−R i
ϕ∗O−R i0
NO − 2 NO − 1
i (cid:88)
i (cid:88)
+
.
(2.81)
uva R =
bO−R i
ϕ∗O−R i0
bR−R i
ϕ∗R−R i0
NO − 1 NR
i
i 2.2.2.2. The vacancy migration energy
The vacancy migration energy Em is given by
(2.82)
Em = Ψva−Ψyn
va +P ∆V, va is determined by the average interaction potentials uyn R O of an ion in RO2 system that has an ion O2− hopping to
R−O
with Ψyn and uyn the saddle point
,
(2.86)
R−R + uva
R−O +
uyn R = uva
+ NRuva
R−O + ϕB
R−O − ∆u1
O−O + ∆u2
O−O
U O−O O
.
uyn O =
R−O − ϕA ϕB NR R−O − ϕA NO − 1
(2.98)
8
2.3. Results and discussion
The interaction potential between ions in RO2 system is the Buckingham – Coulomb potential. The damping parameter α and
the cutoff radius Rc to turn the Coulomb interaction effectively into spherically symmetric potentials with relatively short-ranges.
The results in Table 2.2 show that the oxygen vacancy in CeO2 and c-ZrO2 hops dominantly along the <100> direction and the migrations along the <110> and <111> directions hardly happen.
The next calculations will performed with this preferred migration
Table 2.2. The vacancy migration energies along the <100>, <110> and
<111> directions in CeO2 and c-ZrO2.
of oxygen vacancy.
Migration direction m (eV) (eV)
(cid:104)100(cid:105) 1,1878 2,3337
(cid:104)110(cid:105) 3,9120 5.3918
(cid:104)111(cid:105) 4,4423 6,0328
ECeO2 Ec−ZrO2
m
≈
a
The obtained activation energies have quite large values (ECeO2
≈ 5,8 eV) and they are the increase function of
a
2,6 eV, Ec−ZrO2 temperature. Because the diffusion coefficient and ionic conductiv-
ity are inversely proportional to the activation energy and they pro-
portional to temperature via exponential function then their values
is quite small and increase rapidly with temperature (Fig. 2.7, Fig.
(2.9)).
Under high pressure, crystal lattice is shrinked. This deforma-
tion prevents the vacancy formation and migration processes in the
lattice space. For this reason, the diffusion coefficients and ionic
conductivities decrease rapidly with the increase of pressure.
In summary, in RO2 system, due to the oxygen vacancy with the high activation energy so the diffusion coefficient and ionic conduc-
9
Figure 2.7. The temperature dependence of diffusion coefficient and ionic
conductivity of CeO2.
Figure 2.9. The pressure dependence of the ionic conductivities of CeO2 and c-ZrO2.
tivity have relatively low values. Consequently, the ionic conduc-
tivities of CeO2 and c-ZrO2 systems are less studied. In fact, these materials are appropriately doped to enhance the diffusion coeffi-
cient and ionic conductivity.
10
CHAPTER 3
INVESTIGATE OF DIFFUSIONAL AND ELECTRICAL
PROPERTIES OF YTTRIA-DOPED CERIA AND
YTTRIA-STABILIZED ZIRCONIA
The general chemical formula including the concentration of ions in YDC and YSZ is R1−xYxO2−x/2, in which x is the dopant con- centration.
3.1. Anharmonicity vibration and Helmholtz free energy
3.1.1. Anharmonicity vibration The displacements of R4+, Y3+ và O2− ions in R1−xYxO2−x/2
system are given by
(cid:115)
≈
(3.10)
AR,Y ,
yR,Y 0
2γR,Y θ2 3k3
,Y R
(cid:115) (cid:18) (cid:19)
+
1 +
AO −
yO 0 ≈
βO 3γO
1 KO
−
. (3.11)
(xOcothxO − 1) −
(cid:20) 1 3
6γ2 Oθ2 K4 O 2β2 O 27γOkO
(cid:21) βOkO γO
2γOθ2 3K3 O 2γOθ 3k2 O
3.1.2. Helmholtz free energy
The expression determines the free energy of R1−xYxO2−x/2 sys-
tem [117]
(3.21)
Ψ = CRΨR + COΨO + ΨY − NY uR
0 − T S∗∗ c ,
with ΨR, ΨO are the free energies of R4+, O2− ions in RO2−x/2 system and ΨY is the free energy of Y3+ ions in R1−xYxO2−x/2 system
(cid:21)
+
ΨY ≈ U Y
0 + ΨY
0 + 3NY
(cid:20) 2 (XY )2 − γY
aY 1
2γY 1 3
(cid:26) θ2 k2 Y
11
(cid:21)(cid:27)
+
(γY
(cid:1)aY
. (3.22)
2 )2aY
1 XY − 2(cid:0)(γY
1 )2 + 2γY
1 γY 2
1 (1 + XY )
(cid:20) 4 3
2θ3aY 1 k4 Y
In Eq. (3.22), the free energies of R4+, Y3+ and O2− ions are
determined by the average interaction potentials of an ion [118]
(cid:18) (cid:88) (cid:19) (cid:88)
+
1 −
, (3.36)
uO =
ϕ∗O−R i0
ϕ∗O−O i0
bO−R i
bO−O i
Nva 2N − 1
i
i (cid:88)
(cid:18) (cid:19) (cid:88)
+
1 −
, (3.37)
uR =
bR−R i
ϕ∗R−R i0
bR−O i
ϕ∗R−O i0
Nva 2N − 1
i (cid:88)
(cid:88)
+
. (3.38)
uY =
bY −R i
ϕ∗Y −R i0
bY −O i
ϕ∗Y −O i0
i NR N − 1
NY − 1 N − 1
i
i
3.2. Diffusion coefficient and ionic conductivity
The diffusion coefficient and ionic conductivity are given by
(cid:19) (cid:19) (cid:18)
D = r2
,
(3.41)
−
1n1f
exp exp (cid:18) Sass v kB
Ea kBT (cid:16)
1n1f ωO
(cid:17) (cid:17) exp (cid:18) (cid:19)
ωO 2π 8 a3 r2
(Ze)2 kBT
(cid:16) Sass v kB
− gass v kBT
σ =
−
.
2π exp T
exp
Ea kBT
(3.42)
with [38,39,51]
(3.40)
Ea = Eass + Em.
in which Eass is the vacancy-dopant association energy. 3.2.1. Vacancy-dopant association energy
The expression of vacancy-dopant association energy Eass is
(cid:1) − given by Eass = (cid:0)ΨRNR YNY ONO
+ ΨRNR−2YNY +2ONO −1
v + P ∆V,
(cid:1) + T Sass
− (cid:0)ΨRNR−1YNY +1ONO
+ ΨRNR−1YNY +1ONO −1
(3.46)
12
with the free energies is determined by Eq. (3.21) with Eqs. (3.36) –
(3.38), ∆V is the volume change of system between the associated
state and isolated state of defect pair.
3.2.2. Vacancy migration energy
The vacancy migration energy Em can be written as [49-51,54,55]
(3.71)
Em = Ψyn − Ψ0 + P ∆V,
with Ψyn is determined by the average interaction potentials of an ion R4+, Y3+ and O2−
(3.72)
Y = uY + ∆uyn Y ,
,
(3.89)
R = uR + ∆uyn uB O = uO + ∆uO−R uB
O
O + ∆uO−O
R , uB O + ∆uO−Y
O
O
Y , ∆uO−R
R , ∆uyn
depend strongly on the and ∆uO−Y
in which ∆uyn cation configurations around site A and saddle point B.
3.3. Results and discussion
3.3.2. Vacancy activation energy
Table 3.2. Vacancy - dopant association energies at the 1NN and 2NN sites
in YDC and YSZ.
a. Vacancy - dopant association energy
Eass(eV)
YDC
YSZ
Method SMM DFT [38] SMM MD [51] MD [121] DFT [49]
1NN -0.2971 -0.086 -0.2080 -0.28 0.18 -0.2988
2NN 0.48352 0.1055 -0.2798 -0.45 -0.26 -0.3531
The results of vacancy-dopant association energy at 1NN and
2NN sites enable us to evaluate the vacancy distribution around
13
dopant (Table 3.2). For YDC, the dopant traps the oxygen vacancy
at the 1NN site and repels it from the 2NN site but for YSZ, the oxy-
Fig. 3.4. The dopant concentration dependence of the vacancy-dopant
association energy for YDC (a) and YSZ (b) at various temperatures.
gen vacancy locates mainly at the 2NN site relative to the dopant.
Figure 3.4 shows that the vacancy-dopant association energy
decrease with the increasing of dopant concentration. The reducing
of association energy lead to the consequence as the number of
vacancy increases quickly with an increase in dopant concentration.
b. Vacancy migration energy
The calculated results of vacancy migration energy through three cation barriers R4+ - R4+, R4+ - Y3+ and Y3+ - Y3+ show that the presence of Y3+ ions in the cation barrier blocks the vacancy migration due to the formation of R4+ - Y3+ and Y3+ - Y3+ cation barriers with high barrier (Table 3.4). The vacancy diffusion pro- cess occurs mainly pass the R4+ - R4+barrier and contributes dom- inantly on the vacancy diffusion process in the crystal lattice.
Figure 3.5 shows that the migration energy increases with the
increasing dopant concentration. This dependence arises from an
increase of the dopant concentration and leads to a greater fraction
14
Table 3.4. The vacancy migration energy pass three cation barriers.
Em(eV)
YDC
YSZ
Method SMM DFT [38] DFT+MC [122] SMM DFT+MC [40] DFT [49]
R4+ - R4+ R4+ - Y3+ Y3+ - Y3+ 0,7295 0,533 0,57 1,0528 1,29 1,19
0,2334 0,48 0,52 0,3625 0,58 0,2
1,0521 0,8 0,82 1,5091 1,86 1,23
Figure 3.5. The dependence of the vacancy migration energies Em on the dopant concentration in YDC (a) và YSZ (b) at various temperatures.
of cation barriers R4+ - Y3+ and Y3+ - Y3+ with high migration energies.
c. Vacancy activation energy
At low dopant concentration, the number of high energy barriers R4+ - R4+, R4+ - Y3+ is small and the activation energy is nearly equal that for the vacancy migration across the R4+ - R4+ barrier. As dopant concentration increases, the oxygen-vacancy exchange through the R4+ - Y3+, Y3+ - Y3+ edges rather than R4+ - R4+ barrier can be expected to increase. Therefore, the calculated ac-
tivation energy increases with the increasing dopant concentration
(Fig. 3.6).
15
Figure 3.6. The dopant concentration dependence of the vacancy activation
energy Ea for YDC (a) at 773 K and YSZ (b) at 1000 K.
3.3.3. Diffusion coefficient and ionic conductivity
a. Vacancy diffusion coefficient
Fig. 3.7 shows that the diffusion coefficient increases with the
increasing temperature and it decreases with an increase in dopant
concentration. The dopant concentration dependence generates from
the influence of cation barrier on the vacancy-oxygen ion exchange.
The increase of dopant concentration promotes the number of the
cation barriers with high energy and that prevents the vacancy mi-
Figure 3.7. The diffusion coefficient D is as a function of temperature
inverse (1/T ) at the various dopant concentrations in YDC (a) và YSZ (b).
gration process.
The diffusion coefficient decreases strongly with an increase in
16
Figure 3.8. The pressure dependence of diffusion coefficient D in YDC (a)
và YSZ (b) at various dopant concentrations.
pressure (Fig. 3.8). The shrinked lattice crystal blocks the vacancy
migration and increases the vacancy-dopant association energy.
b. Ionic conductivity
The ionic conductivity increases with the dopant concentration
but after reaching to the maximum value, it reduces fast with fur-
ther increase in the dopant concentration (Fig. 3.9). The vacancy-
dopant associations and the cation barriers with high energy are
the reasons for the nonlinear dependence of the ionic conductivity
Figure 3.9. The dependence of ionic conductivity on the dopant
concentration in YDC (a) at 1073 K and YSZ (b) at 873 K and 973 K.
on the dopant concentration.
The mobility of oxygen vacancy is enhanced due to temperature
17
and the diffusion process occurs more easier. Consequently, the ionic
Figure 3.10. The dependence of ionic conductivity vs 1/T at the various
dopant concentrations for YDC (a) and YSZ (b).
conductivity increases with the temperature (Fig. 3.10).
Figure 3.11 presents the dependence of ionic conductivities of
YDC and YSZ on pressure P at various dopant concentrations x =
0,1; x = 0,2; x = 0,3. The shrinked crystal lattice at high pressure
blocks the diffusion processes and decreases the ionic conductivities.
We predict that the pressure dependence of ionic conductivity in
Figure 3.11. The dependence of ionic conductivity on pressure P in YDC
(a) and YSZ (b) at dopant concentrations as x = 0,1; x = 0,2; x = 0,3.
YSZ is larger than that in YDC.
18
CHAPTER 4
INVESTIGATE OF DIFFUSIONAL AND ELECTRICAL
PROPERTIES OF YTTRIA-DOPED CERIA AND
YTTRIA-STABILIZED ZIRCONIA THIN FILMS
One divides thin film into n crystal layers that consist of two
external layers and (n-2) internal layers. We assume that the in-
teraction potentials between ions in the internal layers is similar to
those in the bulk material. The next calculation will be performed
for these types of layers.
4.1. Anharmonicity vibration and Helmholtz free energy
4.1.1. Anharmonicity vibration
a. The internal layers The displacement expressions of R4+, Y3+ and O2− ions have
the similar forms to those of bulk materials.
b. The external layers The displacements of R4+, Y3+ and O2− ions are determined by
the expression as
(cid:17)2 (cid:16)
6
θ2
+
yR,Y 0ng =
R,Y −
1 +
2γng (cid:16)
(cid:17)3 Ang (cid:17)4
γng R,Y (cid:16)
(cid:118) (cid:117) (cid:117) (cid:117) (cid:116)
βng R,Y 3γng R,Y
1 Kng R,Y
3
R,Y θ2 Kng R,Y
Kng R,Y
R,Y
(cid:16) (cid:17)2 (cid:16)
−
−
,
R,Y cothxng xng
(cid:17) R,Y − 1
2γng (cid:16)
(cid:17)2
1 3
βng 2 R,Y R,Y kng 27γng
R,Y
R,Y kng βng γng R,Y
3
R,Y θ kng R,Y
(4.32)
19
(cid:32) (cid:33) (cid:115)
1 +
+
yO 0ng =
O −
(cid:1)3 Ang (cid:1)2 θ2 (cid:1)4
βng O 3γng O
1 Kng O
6 (cid:0)γng O (cid:0)Kng O (cid:35)
(cid:34)
−
(cid:0)xng
.(4.33)
O cothxng
O − 1(cid:1) −
1 3
2β2 O 27γOkO
O kng βng O γng O
2γng O θ2 3 (cid:0)Kng O 2γng O θ (cid:1)2 3 (cid:0)kng O
4.1.2. Helmholtz free energy
a. The internal layers
The expression of the Helmholtz free energy of the internal layers
has the similar forms to that of bulk materials.
b. The external layers
The Helmholtz free energy of the internal layers is given by
− N ng
(4.41)
Ψng = Ψng
0ng + Ψng
Y uR
Y − T S∗ng C ,
RO2−x/2
with the Helmholtz free energies of R4+, Y3+ and O2− are deter- mined by the general formula as
(cid:21) (cid:26) θ2
+
= U0ng + Ψ0ng + 3N ng
a1ng
(cid:20) γ2ngX 2 −
ΨR,Y,O ng
2γ1ng 3
(kng)2
(cid:21)(cid:27)
+
+
(γ2ng)2Xng − 2(cid:0)(γ1ng)2 + 2γ1ngγ2ng
(cid:1)(1 + Xng)
2θ3a1ng (kng)4
2
(cid:19) (cid:19) 1 (cid:20) 4 3 (cid:26) θβng
− 1
+
−
+ 3Nng
6Kngγng
θ2βng Kng
(cid:20)(cid:18) 2γngaR,O 1ng 3 (Kng)3 (cid:21)(cid:27) .
(4.47)
−
(Xng − 1)
βng 6Kngkng
βngaR,O 9 (Kng)2 +
(cid:18) kng Kng βngknga1ng 9 (Kng)3 + 4.2. Lattice constant, diffusion coefficient and ionic con-
ductivity
a. Lattice constant
The expression of lattice constant can be written as
,
(4.56)
amm =
(n − 2)atr + 2ang n
20
with the value of atr is equal to that of lattice constant in the bulk material.
b. Diffusion coefficient and ionic conductivity
The diffusion coefficient and ionic conductivity are given by
,
(4.57)
Dmm =
,
(4.58)
σmm =
(n − 2)Dtr + 2Dng n (n − 2)σtr + 2σng n
with the values of Dtr and σtr are equal those of the diffusion coefficient and ionic conductivity in the bulk materials, Dng and σng are determined by the expression (3.41) and (3.42). The method to calculate Dng and σng is similar to the method used for the bulk materials YDC and YSZ in chapter 3.
4.3. Results and discussion
Figure 4.2. The dependence of lattice constant on the number of layers (a)
and dopant concentration (b) for YDC thin film.
4.3.1. Lattice constant
A discontinuity in periodic crystalline structure at the exter-
nal layers changes the interaction potential of ions and the anhar-
monicity effect in these layers. These changes increase distances
between ions and the anharmonicity vibrations of crystal lattice.
21
Consequently, the calculated lattice constant of the external layers
is larger than that of the internal layers. The Figs. 4.2 and 4.3 show
that the lattice constants of YDC and YSZ thin films decrease with
an increase in the thickness and the dopant concentration depen-
Figure 4.3. The dependence of lattice constant on the thickness (a) at T = 650 0C and dopant concentration (b) at T = 773 K for YSZ thin film. The experimental results [141,149] are presented for comparison.
dence is similar with that in bulk materials.
4.3.2. Diffusion coefficient and ionic conductivity
The calculated results show that the diffusion coefficient and
ionic conductivity of the external layers are enhanced approximately
three order of magnitude than those of the internal layers. In com-
parison to the internal layers, the vacancy diffusion in the external
layers become much easier and the concentration of mobile vacancy
is enhanced. At the small thickness, the diffusion coefficient and
ionic conductivity of the external layers influence mainly on the
whole thin film. The role of external layers decreases with the in-
creasing thickness. Consequently, the diffusion coefficient and ionic
conductivity of YDC thin film decrease with an increase of thick-
ness (Fig. 4.4). Our results show that the diffusion coefficient and
ionic conductivity of YDC thin films approach the values of bulk
22
Figure 4.4. The thickness dependence of the diffusion coefficient (a) and
ionic conductivity (b) of YDC thin film.
Figure 4.5. The dependence of ionic conductivity on the thickness (a) and
temperature (b) for YSZ thin film.
when the thickness is about some µm.
The calculated ionic conductivity of YSZ thin film is enhanced
approximately three order of magnitude compared that of bulk ma-
terial and it decreases quickly with the increasing thickness (Fig.
4.5). This dependence of the ionic conductivity on the thickness
are in good agreement with the most of experimental results [86-
88,141,150-153]. Our study for YDC and YSZ thin films shows that
the interaction potential and anharmonicity effect in the external
layers are the reasons for the enhanced ionic conductivity of thin
23
films compared to that of the bulk materials. However, the calcu-
lated results are significant differences with the experimental results
[72,151]. There are two reasons for this feature: (i) The roles of sub-
strate and grain boundary are ignored in the calculations. (ii) The
experimental results depend strongly on the substrate type [72-75],
the thin film preparation techniques [154] and the measurement
methods [155,156].
24
CONCLUSION
The thesis has achieved the main results as follow:
1. Based on the expressions of Helmholtz free energy, the
thesis have performed the model to calculate the vacancy activation
energy in the materials with fluorite oxide structure. The diffusion
coefficient and ionic conductivity can be derived.
2. The thesis has performed the expressions for the anhar-
monicity effect and Helmholtz free energy of ions in the external
layer of YDC and YSZ thin films. The diffusion coefficient and ionic
conductivity in these layers can be derived.
3. The calculated results related the bulk materials show the
preferred migration path of oxygen vacancy, the distribution of oxy-
gen vacancy and the influence of dopant on the oxygen vacancy
migration. The dependences of diffusion coefficient and ionic con-
ductivity on the temperature, dopant concentration and pressure
are evaluated in detail.
4. The calculated results for thin films YDC and YSZ eval-
uated the important role of external layers on the diffusional and
electrical properties of those thin films. The lattice constants, diffu-
sion coefficients and ionic conductivities increase with the decreas-
ing in thickness of thin films.
5. The obtained results include the anharmonicity effects of
thermal lattice vibrations and the role of the vacancy distribution
around dopant on the diffusional and electrical properties of these
materials.
6. This calculation model could be extended to study those
properties of CeO2 and c-ZrO2 with others dopants and mineral materials with perovskite structure.
25
LIST OF THE PUBLISHED WORKS
RELATED TO THE THESIS
1. V.V. Hung, L.T. Lam (2018), Investigate the vacancy
migration energy in ZrO2 by statistical moment method, HNUE
Journal of Science 63 (3), 56.
2. V.V. Hung, L.T. Lam (2018), Investigate the vacancy
diffusion in ZrO2 by statistical moment method, HNUE Journal of
Science 63 (3) 34.
3. L.T. Lam, V.V. Hung (2019), Investigation of oxy-
gen vacancy migration energy in yttrium doped cerium, IOP Conf.
Series 1274 (012004), 1.
4. L.T. Lam, V.V. Hung (2019), Effects of temperature
and dopant concentration on oxygen vacancy diffusion coefficient
of yttria-stabilized zirconia, IOP Conf. Series 1274 (012005), 1.
5. L.T. Lam, V.V. Hung, B.D. Tinh (2019), Investi-
gation of electrical properties of Yttria- doped Ceria and Yttria-
Stabilized Zirconia by statistical moment method, Journal of the
Korean Physical Society 75 (4), 293.
6. L.T. Lam, V.V. Hung, N.T. Hai (2019), Effect of tem-
perature on electrical properties of Yttria-doped Ceria and Yttria-
stabilized Zirconia, HNUE Journal of Science, HNUE Journal of
Science 64 (6), 68.
7. L.T. Lam, V.V. Hung, N.T. Hai (2019), Study of oxy-
gen vacancy diffusion in Yttria-doped Ceria and Yttria-stabilized
Zirconia by statistical moment method, Communications in Physics
29 (3), 263.
