Zuo-Guang. Ye Advanced Dielectric Piezoelectric and Ferroelectric Materials: Synthesis, Characterisation and Applications

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Handbook of dielectric, piezoelectric and ferroelectric materials1010

temperature calculated from the structural data determined by neutron

diffraction study. The V

Ti2

is 4.2 in the paraelectric state at 700°C, and

therefore Ti2 is overbonded in the octahedra. This indicates that the Ti2–O

bonds are too short and that the Ti2O

octahedra are under compressive

stress. The ferroelectric phase transition results in the V

Ti2

of approximately

4.0. This value is in good agreement with the nominal charge of Ti

, indicating

that Ti2 is positioned in the crystallographically moderate environment in

the Ti2O

octahedra in the ferroelectric state. In contrast, the ferroelectric

transition leads to an increase in V

Ti1

from 4.0 (700°C) to 4.3 (25°C). Although

the V

Ti1

of 4.3 (25°C) is slightly overbonded, the difference from the nominal

charge (+4) is smaller than that of Bi.

The ferroelectric phase transition is found to result in a marked change in

Bi1

and V

Bi2

. In the paraelectric state, Bi1 and Bi2 are characterized by

strong underbonding. In particular, the underbonding of Bi1 is remarkable,

since V

Bi1

is only 2.3 at 700°C. This is much lower than the nominal charge

of Bi

. This bond valence analysis exhibits that Bi1 in the paraelectric state

requires tighter bonding with oxide ions at the perovskite A site to satisfy

bonding requirement. The ferroelectric transition leads to an increase in V

Bi1

and V

Bi2

. The values of V

Bi1

and V

Bi2

are 3.0 at 25°C, which agrees well with

the nominal charge of Bi

. While Bi1 cannot get a tight bonding with oxide

ions in the paraelectric state due to the restricted movement only along the

c-axis in the I4/mmm structure, the descending symmetry due to the ferroelectric

transition provides a three-dimensional degree of freedom for the displacements

of Bi1 and Bi2 to fulfill bonding requirement. The satisfactory bonding of

Bi1 and Bi2 stabilizes the ferroelectric distortion in the crystal. It is strongly

suggested that the loose bonding of Bi in the paraelectric state, especially the

Temperature /°C

6004002000

4.5

4.0

3.5

3.0

2.5

2.0

Bond valence sum,

Bi1

Bi2

Ti2

Ti1

33.3

Temperature dependence of bond valence sum (

) estimated

from the structural data.

represents static charge of ions in

crystals.

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Crystal structure and defect control 1011

underbonded Bi1 at the A site acts as the structural trigger for the ferroelectric

transition in the BiT system.

Figure 33.4 depicts the detailed bond length around Bi1 and Bi2 in the

ferroelectric state at 25°C. In the paraelectric state, Bi is located at the high-

symmetry 4e site. The ferroelectric transition leads to several short bonds as

shown in Fig. 33.4(b), and the satisfactory bonding of Bi1 is established. The

shortest bond is seen in bond length (l)

between Bi1 and O3 of 0.2284 along

the a-axis, which is 0.0458 nm (17%) smaller than that in the paraelectric

state. The strong bonding of Bi1–O3 plays an essential role in the ferroelectric

polarization as well as in the appropriate bond valence of the Bi1 at the A

site. The shorter bond is also found in l

Bi1-O3

= 0.2418 nm (12% shorter)

along the b-axis and l

Bi1-O1

= 0.2527 nm (17% shorter), which are responsible

for the large TiO

octahedral rotation around the a-axis, as clearly seen in

Fig. 33.1(b).

It is interesting to note that a marked change in bond length is also observed

for Bi2–O. The smallest two bonds are observed in l

Bi1-O4

= 0.2556nm (12%

shorter) along the b-axis and l

Bi1-O4

= 0.2614nm (10% shorter) along the

a-axis. These bond lengths are relatively longer compared with l

Bi1-O3

. Although

the Bi2–O4 bonding contributes to the satisfactory bonding of Bi2 and to the

stabilization of the alternate stacking of the Bi

layers and the perovskite

layers, these bonds are suggested to play a minor role in the ferroelectricity

in the BiT system.

33.3 Electronic band structure and density of

states (DOS)

Figure 33.5 shows the electronic band structures for (a) the paraelectric

(700°C) and (b) the ferroelectric (25°C) phases of BiT calculated from the

structural data determined by the neutron diffraction study. The band dispersion

for the paraelectric phase is larger than that for the ferroelectric phase, which

is attributed to the ferroelectric displacements induced by the descending

symmetry. In the paraelectric state, the fundamental band gap (E

) is indicated

to be indirect (valence band maximum at P and conduction band minimum

at Brillouin-zone center Γ) [34], which is consistent with the calculation

performed by Shimakawa et al. [34]. In contrast, our calculations suggest

that the ferroelectric BiT has a direct band gap, with the valence band maximum

and conduction band minimum lying at Γ. As is typical in the calculations using

density functional theory (DFT) within generalized gradient approximation,

g(DFT)

is underestimated in our calculations. The experimental E

of 3.3eV

[35] at 25 °C is higher than the calculated E

g(DFT)

of 2.4eV. The calculated

g(DFT)

of the paraelectric phase is 1.1eV. The values of E

g(DFT)

agree

qualitatively with the results of the optical measurements of BiT crystals [35].

The electronic structure of the Nd-substituted BiT is discussed in Section 33.8.

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Handbook of dielectric, piezoelectric and ferroelectric materials1012

(a)

(b)

(c)

Ti2

Ti1

Ti2

Ti1

Bi1

0.242

0.228

0.240

0.232

0.253

Bi2

Ti2

0.261

0.256

0.221

0.227

33.4

Detailed bond length in the ferroelectric state at 25 °C: (a) the crystal structure projected along the b-axis, the

enlarged views of (b) around Bi1 at the perovskite A site and (c) around Bi

in the Bi

layers.

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Crystal structure and defect control 1013

Figure 33.6 presents the total electronic density of states (DOS) between

–25 and 7.5eV, where the Fermi level (E

) is set at 0eV. For both the

paraelectric and ferroelectric phases, the Bi 5d states are localized at a deep

level (–22 to –24eV). The O 2s and Bi 6s stats appear at –17 to –20eV and

–7 to –12eV, respectively. The valence band consists mainly of the O2p

states, while the Ti 3d states form the conduction band. The Bi 6p states

overlap with the Ti 3d states in the conduction band.

Figure 33.7 depicts the total DOS and partial (ion decomposed and sphere

projected) DOS (PDOS) of the constituent ions of the (a) paraelectric and (b)

ferroelectric phases of BiT. The ionic model suggests that the Ti 4d states are

formed only in the conduction band, while significant PDOS of the Ti 3d

states is seen in the valence band in both the paraelectric and ferroelectric

states. The presence of the Ti 3d states in the valence band indicates that the

orbital hybridization between the Ti 3d and the O 2p occurs in the BiT

system. Although the hybridization of the Ti 3d–O 2p stabilizes the TiO

octahedra, this orbital interaction does not play an important role in the

ferroelectric phase transition, because the value of the Ti-3d PDOS remains

unchanged regardless the phase transition.

The PDOS analysis shows a marked change in the orbital interaction

between Bi and O induced by the transition. The primary band of the Bi 6s

appears at around –10eV, while that of the Bi 6p lies in the conduction band,

which is consistent with the ionic model. The PDOS of the O 2p is observed

to some degree in the Bi 6s band due to the hybridization of the Bi 6s–O 2p.

Furthermore, the PDOS of the O 2p is seen also in the Bi 6p band caused by

the Bi 6p–O 2p bonding and antibonding interactions. Thus, the PDOS of the

O 2p in these Bi bands can be regarded as a qualitative indicator of the

degree of the orbital hybridization between Bi–O.

–1

Energy (eV)

(a) Bi

(Paraelectric,

|4|mmm

)

(b) Bi

(Ferroelectric,

B2cb

)

SYTZ

33.5

Band structures of Bi

calculated from the structure data

determined by the Rietveld analysis: (a) the paraelectric phase and

(b) the ferroelectric phase.

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Handbook of dielectric, piezoelectric and ferroelectric materials1014

In the Bi

layers, the Bi2 6s is strongly hybridized with the O2 2p in

both the paraelectric and ferroelectric states, and a large PDOS of the O2 2p

is seen at the same energy level in the Bi2 6s main band. The Bi1 6s–O 2p

hybridization is also recognized in both states. These results suggest that the

traditional model [36, 38, 44] based on the stereoactive lone-pair electrons

of Bi

is not sufficient to explain the structural distortion in the ferroelectric

state. The phase transition leads to a marked increase in the PDOS of the O3

2p in the Bi1 6s and 6p main bands, which is consistent with the short bond

length between Bi1 and O3 in the ferroelectric state. These electronic and

structural analyses demonstrate that the hybridization of the Bi1 6s(p)–O3

2p acts as a trigger of the ferroelectric transition. This hybridization results

in a satisfactory chemical bonding of Bi1 at the A site, and then the ferroelectric

distortion is stabilized in the perovskite layers.

DOS (1/eV)

Energy (eV)

–25 –20 –15 –10 –5 0 0

(Ferroelectric,

B2cb

)

(b) Bi

(Ferroelectric,

B2cb

)

(a) Bi

(Paraelectric,

|4|mmm

)

Bi 5

Bi 6

O 2

Bi 5

O 2

Bi 6

O 2

Ti 3

O 2

Bi 6

O 2

Bi 5

Nd 4

33.6

Total electronic density of states (DOS) of (a) the paraelectric

phase of Bi

, (b) the ferroelectric phase of Bi

, and (c) the

ferroelectric phase of Bi

. In the calculations, the perovskite

A site is assumed to be occupied only by Nd.

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Crystal structure and defect control 1015

33.4 Defect structure

33.4.1 Neutron diffraction study

The information of oxygen vacancies in BiT has been obtained by the Rietveld

analysis of the time-of-flight neutron powder diffraction data measured at

700°C in vacuum [19]. The refined crystal structure with the thermal ellipsoids

(probability 80%) is depicted in Fig. 33.8. It is interesting to note that the

thermal ellipsoid of O1 is large compared with those of other oxide ions and

that the nuclear density of O1 spreads in the a–a plane. This result strongly

indicates that self-diffusion of oxide ions occurs through the O1 site at which

a large number of oxygen vacancies

(V )

••

are present. Takahashi et al. [17]

have demonstrated, through the high-temperature impedance spectroscopy

study, that oxide ion is the dominant carrier for high electrical conduction

along the a-axis in BiT crystals [19], which is consistent with the results of

this neutron structural study.

33.4.2

Ab initio

band structure calculations

Vacancy formation energies of Bi and O in the paraelectric state in the BiT

system obtained by ab initio band structure calculations have been reported

Energy (eV)

–12 –8 –4 0 4

–12 –8 –4 0 4 –12 –8 –4 0 4

DOS (1/eV)

2eV

0.5 eV

20eV

Total

ferroelectric phase

(

B2cb

)

(b) Bi

ferroelectric phase

(

B2cb

)

(a) Bi

paraelectric phase

(

|4|mmm

)

O6 2

O5 2

O4 2

O3 2

O2 2

O1 2

Ti2 3

Ti1 3

Bi2

Bi1

Bi2

Bi1

O6 2

O5 2

O4 2

O3 2

O2 2

O1 2

Ti2 3

Ti1 3

O5 2

O4 2

O3 2

O2 2

O1 2

Ti2 3

Ti1 3

33.7

Partial (ion decomposed and sphere projected) DOS (PDOS) of

(a) the paraelectric phase of Bi

, (b) the ferroelectric phase of

, and (c) the ferroelectric phase of Bi

. In the calculations,

the perovskite A site is assumed to be occupied only by Nd.

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Handbook of dielectric, piezoelectric and ferroelectric materials1016

[19]. The supercell method is adapted for estimating vacancy formation

energy (E

vacancy

) to reduce vacancy–vacancy interaction. Figure 33.9 indicates

the relative E

vacancy

of Bi and O. Since Bi1 and O2 show the lowest E

vacancy

in Bi and O, these values are set to be the standard for comparison. The

calculations for BiT show that the E

vacancy

of Bi1 is lower by about 2eV than

that of Bi2. The much lower E

vacancy

of Bi1 indicates that Bi vacancies are

created preferentially at the Bi1 site in the perovskite layers rather than at the

Bi2 site in the Bi

layers. For oxygen vacancies, O1, O2 and O3 show

similar E

vacancy

within 0.2 eV, which is lower by about 0.4–0.6 eV than those

of O4 and O5. It is concluded that Bi1 vacancies and O1 vacancies are

primary ionic defects in BiT at high temperatures.

It is interesting to note that the presence of La at the perovskite A site

increases E

vacancy

of the adjacent oxygen atoms, O1, O3 and O5, by about

1eV, as shown in Fig. 33.9. These results suggest that La substitution is

effective for improving the chemical stability of oxygen in the perovskite

layers. The total/partial DOS analysis of BLT (x = 2.0) suggests that the La

5d is hybridized with the O 2p of the adjacent oxygen. The covalent bonding

between La and O in the perovskite layers creates the DOS of the La 5d

states in the valence band near the Fermi level (–5~0eV), which is a possible

origin of the high chemical stability of oxygen in BLT. The chemical

(b)

(a)

TiO

octahedron

Ti1

Bi1

Ti2

layer

Bi2

33.8

Refined crystal structure with thermal ellipsoids (probability

80%) of BiT at 700°C by neutron diffraction Rietveld analysis.

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Crystal structure and defect control 1017

stability of oxide ions seen in BLT is confirmed also in BNT by ab initio

calculations.

33.4.3 Electrical conductivity investigations by ac

impedance spectroscopy

Figure 33.10 shows the temperature dependence of electrical conductivity

(σ). The σ along the a-axis of BiT was about two orders of magnitude higher

than that along the c-axis. Takahashi et al. have revealed that the high σ

along the a axis is attributed to oxide ion conduction in the perovskite layers

[17, 18]. In contrast, the Bi

layers act as a barrier to oxide ion migration

along the c-axis; hence this yields the low σ along the c-axis (h

•

conduction

is dominant). The oxide ion migration is achieved through O1 vacancies, and

oxide ions are suggested to move in the middle of the perovskite layers in the

a–a (a–b) plane.

The σ along the c-axis of BLT (x = 0.46) crystals was the same as that of

BiT, while the La substitution decreases σ along the a-axis. In BLT (x =

0.46), most La occupies the perovskite A site [33]. The same σ along the

c-axis for BiT and BLT suggests that the Bi

layers play the same role for

BiT and BLT in the electrical conduction along the c-axis. The lower σ along

the a-axis of BLT strongly indicates that the La substitution is effective for

decreasing

(V )

••

, which is essentially consistent with the results of the

vacancy

calculations. The relatively high oxide-ion conductivity observed for

BLT (x = 0.46) indicates that a certain amount of

(V )

••

is present even in the

lattice.

Relative

vacancy

(eV)

2.0

1.5

1.0

0.5

0.0

2.0

1.5

1.0

0.5

0.0

Bi1 Bi2

O1 O2 O3 O4 O5

33.9

Relative difference in vacancy formation energy (

vacancy

) in

I4/mmm

tetragonal symmetry calculated by ab initio band structure

calculations. In the calculations for BLT, the perovskite A site is

assumed to be occupied by La.

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Handbook of dielectric, piezoelectric and ferroelectric materials1018

33.4.4 Defect-formation mechanism

Here, a mechanism of vacancy formation in BiT at high temperatures is

discussed. It has been shown that the Bi vacancy at the Bi1 site

(V )

Bi1

′′′

more easily generated than

′′′

Bi2

, where ′ denotes one negative charge (Fig.

33.9). Furthermore, the nuclear-density analysis has revealed that oxygen

vacancies preferentially reside at the O1 site

(V )

••

, where

•

indicates one

positive charge (Fig. 33.8). The vacancy formation of

′′′

Bi1

is likely to be the

trigger for generating

••

. At high temperatures,

′′′

Bi1

is formed accompanied

with the nearest-neighbor

••

, which is the origin of the defect formation in

the BiT system. The high oxide-ion conduction observed in the BiT crystals

is ascribed to the large number of

••

at the perovskite layers. The high

oxide-ion conductivity also indicates that the electrical neutrality during the

defect formation is satisfied through the generation of

Bi1

′′′

accompanied

with

••

as follows:

2Bi + 3O 2V + 3V + Bi O (g)

Bi1

O1 2 3

→

′′′

••

33.2

and the electron hole (h

•

) plays a minor role in the high-temperature defect

formation.

As a result of the thermogravimetric analysis of BiT powder under different

oxygen partial pressures

()

[19], the weight loss due to the vacancy

formation was promoted in low-

(reducing) atmosphere. The weight loss

–1

(10

–1

)

1.31.21.11.0

log (σ/S cm

–1

)

–2

–1

–3

–4

–5

–6

In air

BiT

-axis

BLT

(

= 0.46)

-axis

33.10

Electrical conductivity in air as a function of temperature

observed for the crystals of BiT and Bi

4–

(BLT,

= 0.46).

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Crystal structure and defect control 1019

due to the vacancy formation was observed above 1000°C in air [19]. Annealing

in air at 1000°C for 5 h led to a 0.25% decrease in weight, while annealing

in O

( = 0.1 MPa)

at 1000 °C did not result in weight loss. The O

annealing at 1050°C for 5 h resulted in a weight loss of 0.5%, which was

much smaller than the annealing at 1050°C for 5h in air (1.2% loss). The

dependence of weight loss shows that the limiting factor of the vacancy

formation is the surface reaction on BiT powder, and this result gives valuable

information that the vacancy formation can be suppressed by heat treatment

in high-

atmosphere [19].

Thermogravimetric analysis allows us to propose the mechanism of reaction

on the BiT surface that leads to the vacancy formation from the three possible

models. In model 1, two lattice Bi1

(Bi )

Bil

and three lattice O1

(O )

migrate

to the surface and react with each other to form Bi

. The resultant Bi

evaporates into the atmosphere, i.e. Bi

(g). This reaction, however, should

be independent of

and thus does not explain the experimental results. In

model 2, only

Bil

moves from the lattice to the surface. Then, the adsorbed

Bi (Bi

) reacts with O

gas in the atmosphere, O

(g), and evaporates as

(g). Although this reaction yields

′′′

Bi1

and electron holes (h

•

), an

increase in

should enhance the weight loss, i.e.

′′′

Bi1

formation, which

is not consistent with the experiments. In model 3, two

Bil

and three

migrate to the surface, but do not react with each other because of the low

possibility of Bi

meeting O

on the surface (due to low vacancy

concentrations), as schematically depicted in Fig. 33.11. Bi

reacts with

(g) and evaporates as Bi

(g) at a relatively higher

, while Bi

detaches from the surface as Bi(g) or Bi clusters in reducing atmosphere.

Rate-limiting

reaction

Bi1

″′

. .

Bi1

Bi (g)

(g)

BiT surface

33.11

Mechanism of vacancy formation in BiT at high temperatures.

Bi1

is Bi at the Bi1 site with no charge and

denotes O at the

O1 site with no charge.

Bi1

and

migrate to the surface and

then become adsorbed Bi

and O

. The limiting factor of the

vacancy formation is the surface reaction related to O

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