THE TOLERANCE FACTOR

Perovskite and Related Structure Systems

3.3. THE TOLERANCE FACTOR

In the ideal perovskite structure, if rA' rB' ro are the radii of the A, B cations and

°

anion, respectively, the ratio of rA + ro to 21/2(rB + ro) should be unity ifthe spheres are in direct contact. In reality, the ratio is less than unity. Thus, a tolerance factor is defined to represent a general case (Goldschmidt, 1926):

(3.1) In general the perovskite structure occurs only within the range 0.75 <

ft

< 1.00, which is given solely based on geometric considerations. From the bonding point of view the A and B cations have to be stable in 12-fold (12, 8 + 4, or 6 + 6) and sixfold coordinated anion surroundings. This can give a lower limit for the cation radii. In oxides, for example, the limits are r A> 0.09 nm and rB> 0.051 nm (Goodenough and Longo, 1970).

If 0.75 <

ft

< 0.90, a cooperative buckling of the corner-sharing octahedra for optimizing the A-O bond length causes an expansion in the unit cell. If 0.90 <

ft

< 1, this buckling may not be found, but small distortions to rhombohedral symmetry will occur. This distortion of the octahedra should be distinguished from distortion as a result of electron spin ordering. The ideal perovskite structure can be found in high-temperature phases or in compounds with more ionic A--'O bonds.

3.4.

FUNCTIONAL MATERIALS WITH PEROVSKITE-LIKE STRUCTURES Perovskite-like structures can be sorted out by the valence combination of the A and B cations as follows (Galasso, 1969):

2. A2+B4+03 type, in which the A2+ cations are alkaline earth ions such as cadmium or lead, and the B4+ ions can be Ce, Fe, Pr, Pu, Sn, Th, Hf, Ti, Zr, Mo, and U. Typical examples are BaTi03 and PbTi03. These two compounds are well known due to their remarkable ferroelectic properties, as the result of opposite displacements of the Ti and oxygen atoms.

3. A3+B3+03 type, such as GdFe03, YAl03, PrV03, PrCr03, NdGa03, and YSC03.

4. A2+(B~.673+B~.336+)03 type, such as Ba(ScO.67W0.33)03 and Sr(Cro.67Reo.33)03.

5. A2+(B~.332+B~.675+)03 type, such as Ba(SrO.33Tao.67)03, and Pb(Mgo.33NbO.67 )03.

6. A2+(B~}+ B~.55+)03' A2+(B~.52+ B~.56+)03' A2+(B~.51+ B~.s 7+)03, and A3+(B~.s2+B~.54+)03 types, such as Ba(SrO.5WO.5)03, Pb(Sco.5Tao.s)03, and Pb(Sco.5Nbo.5)03. The compounds Pb(Mgo.33Nbo.67 )03, Pb(Sco.s Tao.5)03, and Pb(Sco.5Nbo.5)03 are very important ferroelectric materials and they are usually called "relaxors."

7. A2+(B~.251+ B~.7/+)03 type, such as Ba(Nao.25Tao.75)03 and Sr(Nao.25 Tao.75)03.

8. A2+(B~i+ B~.55+)02.75 and A2+(B~.53+ B~.5 4+)02.75 with anion deficiency, such as Sr(Sro.s Tao.s)02.75 and Ba(FeO.5MoO.5)02.75.

9. A2+(B~.53+ B~.52+)02.25 (Section 8.10).

It is possible, of course, to have others. A common feature is that an alternate stacking of the (A03)4- and B4+ layers is the basis of the entire perovskite structures.

The valences of the A and B cations are usually close to 2+ and 4+, respectively, but in some special cases their valences can be 3+ and 3+ if the B3+ cation has a six- coordination. The valence variation at the A cation position can cause distortion or displacement of the oxygen anion array, resulting in the buckling of the (A03)4- layers.

This buckling may induce distortion of the octahedra with B cations at the centers. The B cation must have the flexibility to tolerate this effect, and the transition metal elements are the candidates for filling the B cation position because of its multi valency or the special 3d and 4d electron configurations. This is the reason that transition metal oxides have perovskite-type structures, and they usually have special physical properties (Rao and Reveau, 1995).

The principles illustrated in this section can be applied to predict and interpret the structural characteristics of many oxide functional materials. It is impossible, however, to illustrate the structural evolution of each compound. We now select a few typical examples to illustrate the application of the theory described above.

3.4.1. FERROELECTRICITY AND FERROELECTRIC COMPOUNDS

Ferroeiectricity is an important property of oxide functional materials for its potential applications in ultrahigh-density information storage. We first use Fig. 3.11 to illustrate this effect. A linear molecular M02 is taken as an example, in which the oxygen valence is 2-, the metal ion has a valence of 4+, and the molecule does not have an electric dipole due to the annihilation of the positive and negative electric moments. If an external electric field E is applied, the oxygen anions and the cation will be displaced in opposite directions due to the electrostatic force, resulting in a dipole moment that is a function of the applied electric field. The total electric field should be the sum of the

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PEROVSKITE AND RELATED STRUCTURE SYSTEMS

112

CHAPTER 3

external field E and the field generated by the dipole, E'. If E is withdrawn, the molecular dipole may still have some polarization as determined by the environment of the molecule; thus, the local field is E~ rather than zero. If an external field is applied in an opposite direction and the strength of the field is controlled just to restore the linear shape of the buckling molecular, the strength of the supplied electric field is called a coercive field. If the applied field exceeds the coercive field, the molecule is polarized in an opposite direction, and the local field is the sum of E and E'. By repeating the above exercise and plotting the total field E' as a function of the applied field E, a hysteresis loop, called the ferroelectric hysteresis loop, is obtained, similar to the magnetic hysteresis. This is the ferroelectricity.

Based on this simple illustration, a compound may need to meet the following requirements to exhibit ferroelectricity: (a) it must have a basic structure that has the flexibility to change the relative atom positions; and (b) it should have a slightly distorted (in one direction) crystal structure in which the centers of the positive and negative charges should not coincide; e.g., the crystal has some polarization along one direction.

If we look at the perovskite unit cell with A cation at the origin as shown in Fig.

3.1c, we can say that it is a good candidate for ferroelectricity because the four oxygen anions on the side faces are easier to move than the oxygens on the top and bottom faces if an electric field is applied along the z axis. If the A and B cations can make little buckling of the octahedron (within the (A03)4- layers), it may have ferroelectricity. If the covalency of the A-O bonding increases, the ferroelectricity vanishes. If the covalency of the A -0 bonding is weakened, the ferroelectricity will be dominated by the displacement of the B cation. Transition metal cations having empty d orbitals can create

Electric field System dipole

(a) _2- 4+ 2- 0 0

E' =E=O

!

^E"= E + p'3 Eo'"

t

^P

4+ E'

E=O p>O

4+ E'o = pof3£o

(b)Ec

Ec _E<O

t

^{2- 4+ 2-}

t

^{p=O E'=O}

^o

r

^2-

^{E' l}

^{E' = E + p3 E 0 < 0}

^tp

Figure 3.11. Schematics of a linear molecular polarization process induced by an external electric field for producing the ferroelectric hysteresis loop (see text). E is an external electric field, Eo and Ec are the polarization and coercive fields, respectively, and P the polarization.

spontaneous ferroelectric distortion. Ti, Zr, Nb, and Ta are the favorite ions for fabricating ferroelectric oxides (Goodenough and Longo, 1970), and BaTi03 and PbTi03

compounds are typical examples.

Figure 3.12 shows the process of the octahedron distortion in BaTi03 and PbTi03

perovskite compounds. When the oxygen anions and B cation are displaced in an opposite direction, the cia ratio of the unit cell changes, and the structure is perovskite- like with tetragonal distortion. In these compounds, the titanium (e.g., the B cation) does not exactly locate at the center of the octahedron, as indicated in Fig. 3.12.

Ferroelectricity, in general, occurs only in a specific temperature range. BaTi03, for example, has the ideal cubic perovskite structure above 120^DC, which is called the transition temperature, but below this temperature the oxygen and titanium ions are displaced to new positions, forming a tetragonal structure with cia

=

1.01. The dipole moment at 120^DC is 18 x 1O-⁶C/cm2, but below 120°C it is 26 x 1O-⁶C/cm² (Merz, 1953). The dielectric constant of BaTi03 is 1600 (Hench and West, 1990). PbTi03 has a similar property as BaTi03 with a transition temperature of 495°C (Shirane and Hoshino, 1951). The dielectric constant of PbTi03 varies from 100 at room temperature to 1000 at 490°C.

As the dipole changes its orientation the crystal should vary its shape. This phenomenon is the piezoelectric effect or piezoelectricity, which is a twin effect with ferroelectricity. The piezoelectricity constant, a quantity for characterizing the piezo- electricity, is a tensor for an anisotropically structured unit cell and is related to the crystallographic directions. BaTi03 (Cherry and Adler, 1948) and PbTi03 (Tien and Carlson, 1962) both exhibit this effect.

The unit cell of a ferroelectric compound usually has a small dipole moment. If a group of these unit cells are aligned, a ferroelectric domain will be formed, but the orientation of dipoles in different domains can be along the

±x, ±y,

or

±z

axes in a single-crystalline material. Therefore, a macroscopic piece of BaTi03, for example, may not have polarization because of the cancelling effect of the dipole moments belonging to different domains. A polycrystalline material usually does not exhibit ferroelectricity.

This discussion leads to two points. Domain structure is a key in ferroelectric materials, and the domain boundaries are also important which determine the interaction between domains, such as sharpness, thickness, size, and the crystallographic planes of the domain boundaries. Observation of the domain walls using transmission electron microscopy and electron holography will be addressed in Sections 7.5.3 and 6.4.4.

c :%: a c= a c:%:a

Figure 3.12. The polarization processes in a perovskite structure under an external electric field E. The short arrowheads indicate the displacement directions of the oxygen anions and the B cations. The cubic perovskite unit cell is distorted to be tetragonal.

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PEROVSKITE AND RELATED STRUCTURE SYSTEMS

114

CHAPTER 3

Pb(ZrxTi1- x)03 (PZT) is an important oxide for functional materials. The single- crystalline Pb(ZrxTi1 - x)03 thin films have many potential applications in sensors, memories, smart systems, and microelectronics, because the lattice distortion does not introduce local stress leading to cracking.

Pb(Sco.s Tao.s)03, Pb(Sco.sNbo.s)03, and Pb(Mgo.33Nbo.67)03 are also ferroelectric compounds. These materials show a broad dielectric permittivity peak versus temperature and dielectric dispersion at low frequencies (Cross et al., 1980). The structures of Pb(Sco.sTao.s)03 and Pb(Sco.sNbo.s)03 have been given by Galasso (1969). A common feature of these compounds is that the structures of the (A03^)4- [e.g., (Pb03^)4-] layers are the same, but the structures of the B layers are different since there are two types of B cations: Sc3+ and Tas+; Sc3+ and Nbs+; and Mg2+ and Nbs+. Thus, there are two possibilities: disordered or ordered B layers. The probability for an ordered arrangement of these two types of B cations is determined by the differences between their ionic charges and ionic radii. The transition temperature between the ordered and the disordered structure is approximately related to the charge difference between the two B cations as well. If I1q = q(B_{1) -} q(B2) represents the charge difference between the two B cations, the transition temperature is Tord ~ (l1qi (Goodenough and Longo, 1970).

Based on the known structures of these types of compounds, (l1q)2 usually is 36 or 16 for ordered structures. Ba(Feo.s3+ Tao.S5+)03, for example, is disordered, while Ba(Mgo.s²⁺ WO.s6+)03 is ordered. Moreover, although (l1q)2 equals 4, the structure can be ordered if the difference in ionic sizes between the two is large; Ba(Lao.s3+ TaO}+)03 is such an example. For ordered compounds the minimum difference between ionic sizes is required to satisfy IrBl - rB211rBl ~ 0.09. Nb and Ta are good candidates because they have empty d orbitals, with the possibility of forming stable octahedra, and the ion sizes are suitable as well. For example the difference in percentage between the radii of Ta (5+ ) (0.078 nm) and Sc (3+) (0.0885 nm) is 13% (> 0.09), and the compound might be ordered as judged from this criterion, but the charge difference is not favorable for the ordered structure. This situation gives the compound a unique ordering process and characteristics: ordering occurs gradually in a wide temperature range, and the boundary between the ordered and disordered domains is smeary. A high-resolution transmission electron microscopy image of Pb(Sco.sTao.s)03 is given in Fig. 3.13. The ordered structure has a face-centered cubic unit cell whose dimension is twice that of the perovskite unit cell (Fig. 3.14), while the disordered phase has the perovskite structure.

The ordered structure has two kinds of B cation layers: Sc and Ta layers and they are inserted alternately between the two (Pb03^)4- layers, but in the disordered structure the B layer only has one type of (Sco.s Tao.s) layer.

It is, however, difficult to distinguish the ordered from disordered regions experimentally because the average compositions of the two phases are the same and the lattices are almost the same, while the only difference is the distribution of Sc and Ta cations. On the other hand, the difference in lattice parameters (a

=

0.814 om for the ordered fcc and a

=

00407 nm for the disordered cubic) produces lattice fringes with different spacing when viewed along (110). The image in Fig. 3.13 clearly demonstrate this difference and the ordered and disordered phases are separated. In this image the white dots correspond to the metal atom columns arranged in a small hexagonal pattern, and the smeared region is the disordered area. This HRTEM technique has been used to identify the ordered and disordered regions in a group of "relaxor" compounds (Kang et al., 1990; Boulesteix et al., 1994).

Figure 3.13. HRTEM image demonstrating the ordered and disordered domains in a "relaxor"

Pb(Sco.sTao.5)03 compound. "Os" indicates the disordered domain having perovskite structure. and "Or"

indicates the ordered domain with a unit cell having twice the size of the perovskite unit cell. The imaging condition was chosen to reveal the two-dimensional pattern of the large unit cell. but only one-dimensional fringes of the perovskite unit cell. by choosing the size of the objective aperture (Section 6.2) .

• ^~ _•

^Pbcation _Tacation ^{Sc cation}

0

A stacking layer of 0

•

B stacking layer of 0

ED

^{I •• ~':.~ C} Stacking layer of 0

Figure 3.14. Structure of an ordered phase of Pb(SCO.5 Tao.5)03 compound.

115

PEROVSKITE ANDRELATEO STRUCTURE SYSTEMS

116

CHAPTER 3

An ordered structure usually refers to a system in which the substitution of cations occurs periodically, forming a long-range superlattice. In contrast, a disordered system is produced if the substitution occurs randomly. Between the ordered and disordered is the short-range order. Analysis of short-range order by electron diffraction is given in Section 7.4.

3.4.2. FERROMAGNETISM AND FERROMAGNETIC COMPOUNDS

Ferromagnetism is a phenomenon similar to ferroelectricity. The difference is the creation of ionic (or atomic) magnetic moments instead of ionic dipoles. The basic character of ferromagnetic compounds is the magnetism hysteresis loop and the Curie temperature for ferromagnetic transition.

As discussed in Chapter 1, transition metal ions have different numbers of electrons in the d orbitals. If the 4s electrons are lost, the 3d electrons are exposed to the neighboring atoms. Thus, the d orbitals of transition metal ions could interact with each other or with electrons belonging to the nearest neighbors. The crystal structure is closely related to the interactions between atoms, and the crystal field effect is vitally important for transition metal ions. Figure 3.15 gives the 3d electron spin alignment, spin-induced magnetic moment, and possible coordination numbers of the first few transition element series. It shows that, due to Hund's rule, the first five d electrons have parallel spins to maximize the moment, with ~B

=

5 for Mn²+, Fe³+, and C0⁴+ ions. When the 3d electrons increase from 0 to 5, the spin moment reaches the maximum, but as the 3d

Transition metal ions 3d electrons Spin alignment Resulting moment Favorable coordination in d orbitals in flB

number

Sc3+ Ti4+ 3d^o 0 6,5,4

Ti3+ V4+ 3di ~ ^{1 6,5,4,}

Ti2+ V3+ cr4+ 3ct2 ~~ ^{2 6,5}

y2+ Cr3+ Mn4+ 3d3 ~~~ ^{3 6}

Cr2+ Mn3+ Fe4+ 3d4

~~~'"

^{4 6,5}

Mn2+ Fe3+ Co4+ 3d⁵

~~~~~

^{5 6,4}

Fe2+ c03+ Ni4+ 3d6 ~t~~~~ ^{4 6,4}

Co2+ Ni3+ 3d⁷ ~t~t~# ^{3 6,5,4}

Ni2+ 3dB

~t~t~t"''''

^{2 6,5,4}

Cu2+ 3d9

~t~t~t"'t~

^{1 6,4}

Cu+ Zn2+ 3dlO

~t~t~t"'t~t

^{0 6,4,3,2}

Figure 3.15. The 3d electron spin alignments, spin-induced magnetic moments, and possible coordination numbers of the first few transition elements, where IlB = ~ = 9.274 X 10-²³ J T-¹ is the Bohr magneton.

4nmoC

electrons continue to increase the spin moment will decrease to zero for Cu+ and Zn2+, which have 3d lO. Therefore, the spin-induced moments of transition metal ions can vary from 0 for 3do (Sc³+ and Ti4+) and 3dlO (Cu+, Zn2+) electron configuration to a maximum of 5 for Mn2+, Fe3+, and Co4+. This wide range of spin-induced moments and oxidation state variation makes transition metal ions able to tailor the net magnetic moment of the compound. This will be used in Chapter 5 for designing new materials.

The experimental analysis of the magnetic moment will be discussed in Section 6.4.5.

The net magnetic moment of an atom should include two components: the orbital moment and the spin-induced moment. These two components may interact with each other to change the total magnetic moment of the atom (Le., the spin-orbit coupling). The coordination number and the crystal structure are the dominant factors that determine the magnetic properties of a crystal.

In the perovskite structure the transition metal cations are at or around the center of the oxygen octahedron. This coordination structure will modify the energy state of the

(a)

(b)

d electron energy level transition metal cation of in phcrical cnviromcnl

(c)

d electron energy level transition metal cation of in octahedral enviromenl

Figure 3.16. (a) The eg and t2g d electron orbitals in an octahedral coordinated environment. d electron energy levels of a transition metal in (b) spherical environment and (c) the octahedral coordinated environment.

117

PEROVSKITE AND RELATED STRUCTURE SYSTEMS

118

CHAPTER 3

transition metal ions (Fig. 3.16). The five types of 3d orbitals having different energies for transition metal cations in the oxygen octahedral environment are given in Fig. 3.16a, where the d_z² and d:x2_y'l orbitals are called eg states and the dxy, dyz, and dzx orbitals are called t_2g states, which are frequently used in describing the electronic structures of transition metals. In Fig. 3.16b, the five energy levels of the 3d electrons in a free atom are shown. Figure 3 .16c gives the 3d electron energy levels for the octahedral coordinated transition metal cations, where the energy levels are grouped into eg (dz² and dx2_y'l) and t2g(dxy, dyz, and dzx) and are separated by a large gap A, having different values for different transition metals. The energy levels in crystalline environment are apparently different from those in a free atom. For free cations the 3d electrons should obey Hund's rule to remain in different energy levels with spin up or down as shown in Fig. 3.15, but in the octahedral environment the spin arrangement can have two situations: high spin and low spin as pointed out in Chapter 1. High spin has more unpaired electron spins than the low spin does, and it is natural that the magnetic moment of high spin is higher than that of the low spin. If the octahedron is distorted by an elongation along the z axis (Fig.

3.17a), the energy levels are changed in such a way that the gap between the two eg levels increases and the t_2g levels are split into two groups: one goes higher approaching the lower energy level of the eg states, while the other two go toward low energy. These splittings create three gaps as indicated with AI. A20 and A3 in Fig. 3.17a. Observations of atom ionization edges clearly indicate the existence of these energy levels (as will be seen in Fig. 8.22). Figures 3.17b, c, and d give the high-spin state of d⁴ (Cr²+, Mn³+, Fe⁴+), d⁹ (Cu²+) and low-spin states of d⁸ (NiH) energy level configurations, respectively. The structural distortion can decrease the total energy of the high-spin state of d⁴ (Cr²+, Mn³+, Fe⁴+) and low-spin states of d⁸ (NiH), but increase the energy of the high-spin

~

~I

t

e

g

~ _ru + ^t

^{I~ ~}

,

".

^{". I~}

+ +

~

t2g

}~

^ill

# *

t '.

(a) (b) (c)

Cd)

Figure 3.17. (a) Energy level splitting due to the octahedral distortion with an elongation alonf the z axis, and (b) the high-spin state of d⁴ (C?+,Mn3+,Fe⁴+), (c) d⁹ (Cu²+), and (d) low-spin states.of d (Ne+) energy level configurations.