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

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Handbook of dielectric, piezoelectric and ferroelectric materials590

of the dipoles’ magnitude and leads to a significant change of this magnitude

after the flipping (see Fig. 19.8b and d). In contrast, the dipoles inside the

films switch predominantly through flipping (see Fig. 19.2a–c), with the

direction being reversed and the magnitude of dipoles being only very slightly

increased after such flipping (see Fig. 19.8b and d). In other words, the slight

increase of 〈u

〉 when increasing E

in Region II (see Fig. 19.7) mostly

originates from the surface dipoles.

Region III: As E

keeps increasing, the minority (down) stripe domains

become pinched along [110]. Nanobubbles now form (see Fig. 19.2d), as in

PZT thin films under electric field. Note that the stripe to bubble domain

transition is numerically found for E

in between 41 × 10

and 45 × 10

V/m

(because of the zigzagged domain walls, it is difficult to exactly pinpoint the

onset of stripe to bubble domain transition, unlike in PZT thin films (Lai et

al., 2006). Once the nanobubbles are formed, they dramatically contract

along [110] while keeping their width along

[110]

more or less the same

under further applied field – as seen in Fig. 19.2(d) and (e). In other words,

as in PZT films (Lai et al., 2006), the bubbles contract parallel to the direction

along which the initial stripe domains were running (i.e. [110] in BTO versus

[100] in PZT films) while keeping their width fixed parallel to the direction

along which the initial stripe domains were alternating (i.e.

[110]

in BTO

versus [010] in PZT ultra-thin films). Interestingly, Fig. 19.8(d) reveals that,

between 43 × 10

and 51 × 10

V/m, the average magnitude of the surface

dipoles having a positive component along the direction of the field (i.e.

with u

> 0) decreases with E

. This is because the number of switched

surface dipoles (i.e. having now u

> 0 while initially exhibiting u

< 0)

increases rather rapidly with E

(see Fig. 19.8c) while such dipoles had a

rather small magnitude before this switching (see Fig. 19.8b). Figure 19.8(a)

further indicates that the critical field for which all the surface dipoles belonging

to the initial down stripes have completely switched is around E

= 61 × 10

V/m (note that half of the dipoles inside the film that belonged to the initial

down domains are still not flipped under such field, according to Fig. 19.8a).

In other words, the part of Region III located above E

= 61 × 10

V/m

possesses nanobubbles that are encapsulated (i.e. that do not touch the surfaces,

as in PZT films). With a further increase of E

(see Fig. 19.2e), the bubbles

still continue to contract along [110]. Once they reach a critical size of two

lattice constants along both x- and y-axes and of three lattice constants along

the z-axis, some bubbles pop out and thus the number of nanobubbles decreases

with E

. Note that the collapse of BTO bubbles is via flipping dipoles inside

the films, while the collapse of PZT bubbles is via dipole rotation.

Region IV: At E

= 104 × 10

V/m, all the nanobubbles have disappeared

and a single monodomain, with all the dipoles pointing along the +z direction,

is thus formed − as seen in Fig. 19.2(f) and as consistent with the fact that

Fig. 19.7 indicates that 〈u

〉 and 〈u

〉 are now identical. In this monodomain

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Domains in ferroelectric nanostructures from first principles 591

state, all the dipoles increase in magnitude with E

, with the surface dipoles

being larger in size.

19.4 Domains in one-dimensional and zero-

dimensional ferroelectrics

In this section, we will discuss the results of Ponomareva et al. (2005b), in

general, and will focus on the predicted domains in ferroelectric nanodots

and nanowires of Pb(Zr

0.4

0.6

, in particular. Technically, the investigated

nanodot is cubic in shape with a 48Å lateral size, while the nanowire is

periodic along the z-direction (chosen along the pseudo-cubic [001] direction)

and has a cubic cross-section of 48 × 48Å

The results of the simulation for the spontaneous polarization are shown

in Figs 19.9(a) and (c) for the stress-free dot and wire, respectively. The left

and right insets of Figs 19.9(a) and (c) further display a cross-section of a

snapshot of the dipole pattern for ideal OC and SC conditions, respectively,

in these two systems. Under SC-like conditions (i.e. β larger than 88%), the

dot and nanowire exhibit a spontaneous polarization.

Below the critical values of β ≈ 88%, the magnitude of the residual

depolarization field inside the ferroelectric nanostructure becomes so large

than the dipoles have to rearrange themselves to efficiently suppress it. A

structural phase transition thus occurs. In the case of the dot, such transition

is characterized by the vanishing of the spontaneous polarization − as also

found in Stachiotti (2004) for some BaTiO

nanodots under OC conditions

− and the formation of a vortex dipole structure (see left inset of Fig. 19.9a)

that results in the appearance of a spontaneous toroid moment of polarization

(which involves the cross-product between local dipoles at the sites i and the

vectors locating these sites i) (Naumov et al., 2004) − as indicated by Fig.

19.9(b). Such vortex can also be thought as an original pattern made by four

90° domains. On the other hand, the wire energetically chooses a different

dipole pattern (see left inset of Fig. 19.9c) to annihilate the macroscopic

depolarization field, namely the dipoles now align along the infinite direction

of this wire (rather than perpendicular to it as for SC conditions). This

difference between the dipole pattern in the dot versus the wire can be

simply understood by realizing that, unlike the wire, the dot is confined

along any direction. As a result, any non-zero macroscopic polarization pointing

along any direction would lead to the activation of a depolarizing field in the

dot. Moreover, the infinite wire prefers to have a macroscopic polarization

rather than adopting a vortex-like dipole pattern below a critical value of β

because the formation of the latter pattern is more costly in term of short-

range and elastic energies.

Similarly, stress-free thin films under OC-like conditions have also been

found to have a macroscopic polarization pointing along a periodic direction

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Handbook of dielectric, piezoelectric and ferroelectric materials592

19.9

Ground-state spontaneous polarization in free-standing dot (part

(a)) and wire (part (c)) as a function of the screening coefficient β.

The β-dependency of the toroid moment of polarization in the

freestanding nanodot is shown in part (b). Right and left insets to

both parts (a) and (c) show the projection of the dipole patterns on a

chosen plane for perfect SC and OC conditions, respectively, in the

corresponding nanostructure. (Note that these dipoles nearly lie in

the plane shown in the insets, that is there is only a slight difference

between the dipole vectors and their displayed projections.) The

infinite direction of the wire is along the

-axis. The

- and

-axis

are chosen along the pseudo-cubic [100], [010] and [001] directions,

respectively. (Reprinted with permission from Ponomareva

et al.

2005b). The toroidal moment,

is given in

units, where

minus the electron charge and 1

= 1.60217653 × 10

(–39)

C/m

SI units.

Dot

(a)

(b)

Polarization (C/m

)

1.1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

(eÅ

)

(Kornev et al., 2004). Interestingly, Fig. 19.9(b) and (c) and Kornev et al.

2004) tell us that the magnitude of the order parameter − that is the spontaneous

polarization in wires and films versus spontaneous toroid moment of

polarization in the dot − in stress-free ferroelectric nanostructures is

independent of β below the critical values. In other words, a given ferroelectric

nanostructure adopts a specific dipole pattern, once the depolarizing field is

large enough.

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Domains in ferroelectric nanostructures from first principles 593

We next turn our attention to the same dot and wire, but under different

mechanical boundary conditions, namely those mimicking an epitaxial growth

in the (x, z)-planes. The infinite z-direction of the wire is thus chosen to lie

in such planes. The results for a tensile strain δ of +2.65% are shown in Fig.

19.10, and are in overall qualitatively close to the corresponding ones in the

freestanding systems. There are, however, three differences worth attention.

First of all, dipoles aligned along the z- or x-axis are now preferred over

dipoles parallel to the y-axis because of our chosen tensile strains. This

implies that the toroid moment of the dot under OC-like conditions can only

be along the y-axis as opposed to the case of stress-free dots for which this

moment has an equal probability of being along any of the three Cartesian

directions. Secondly, the magnitudes of the spontaneous polarization and

toroid moment are larger than those of the freestanding systems for small β.

Such a feature is due to the fact that the x and z-components of the dipoles

expand as a result of their coupling with the tensile strains (Cohen, 1992;

Zhong et al., 1994, 1995). Thirdly, we numerically found another configuration

(Fig. 19.11) for a nanowire under tensile strain, and with β smaller than 86%,

that is slightly larger in energy than the configuration shown in the left part

of the inset of Fig. 19.10(c). It consists of an infinite number of vortexes

assembled along the z-direction of the wire, with neighboring vortexes rotating

in opposite directions. The formation of this configuration is driven by the

desire of the wire to have non-zero components of the dipole in both x- and

z-directions, as dictated by the tensile strain, while minimizing the depolarizing

field inside the wire.

We now study the effect of a compressive strain δ of –2.65% − still

applied in the (x, z)-plane − on the dipole pattern in the nanodot and wire.

(c)

Wire

0.75 0.8 0.85 0.9 0.95

Polarization (C/m

)

1.1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

19.9

(Continued)

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Handbook of dielectric, piezoelectric and ferroelectric materials594

The results of our simulation are shown in Fig. 19.12. Such compressive

strain favors dipoles aligned along the y-axis via the strain–dipole coupling

(Cohen, 1992; Zhong et al., 1994, 1995), which explains why the spontaneous

polarization of the dot and wire lies along such direction for electrical boundary

conditions close to SC (see Figs 19.12a and b and their right insets). However,

for the smallest β values, any macroscopic polarization aligned along the y-

axis would generate a relatively large depolarizing field that should strongly

Dot

(a)

(b)

Polarization (C/m

)

1.1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

(eÅ

)

19.10

Ground-state spontaneous polarization in dot (a) and wire (c),

under a tensile strain +2.65% applied in the (

) plane, as a function

of the screening coefficient β. The β-dependency of the toroid

moment of polarization in the nanodot is shown in (b). Right and left

insets to both (a) and (c) show the projection of the dipole patterns

on a chosen plane for perfect SC and OC conditions, respectively, in

the corresponding nanostructure. (Note that these dipoles nearly lie

in the plane shown in the insets, that is there is only a slight

difference between the dipole vectors and their displayed

projections.) The infinite direction of the wire is along the

-axis. The

- and

-axes are chosen along the pseudo-cubic [100], [010] and

[001] directions, respectively. (Reprinted with permission from

Ponomareva

et al.

, 2005b). The toroidal moment,

is given in

units, where

is minus the electron charge and 1

= 1.60217653 ×

(–39)

C/m

in SI units.

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Domains in ferroelectric nanostructures from first principles 595

oppose such polarization. The resulting ground-state configurations must

thus arise from a compromise between these strain versus depolarizing field

factors. As a result, both nanosystems form 180° stripe domains with the

dipoles being mostly ‘up’ or ‘down’ along the y-direction, with two neighboring

and opposite stripes having a very small overall width (namely, 24 Å in the

present case). Interestingly, such nanodomain patterns occur in any low-

dimensional ferroelectrics under compressive strain and under residual

(c)

Wire

0.75 0.8 0.85 0.9 0.95

Polarization (C/m

)

1.1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

19.10

(Continued)

19.11

A cross-section of another possible dipole pattern for a

nanowire periodic along the

-direction, and under OC-like electrical

boundary conditions and a tensile strain applied in the (

)-plane.

The arrows show the projection of the dipole patterns on a chosen

plane. (Note that these dipoles nearly lie in the plane shown in the

insets, that is there is only a slight difference between the dipole

vectors and their displayed projections.) The

- and

-axis are

chosen along the pseudo-cubic [100], [010] and [001] directions,

respectively. (Reprinted with permission from Ponomareva

et al.

2005b.)

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Handbook of dielectric, piezoelectric and ferroelectric materials596

Dot

(a)

(b)

Wire

0.75 0.8 0.85 0.9 0.95

Polarization (C/m

)

1.1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Polarization (C/m

)

1.1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

19.12

Ground-state spontaneous polarization in dot (part (a)) and

wire (part (b)), under a compressive strain –2.65% in the (

) plane,

as a function of the screening coefficient β. Right and left insets to

both parts (a) and (b) show the projection of the dipole patterns on a

chosen plane for perfect SC and OC conditions, respectively, in the

corresponding nanostructure. (Note that these dipoles nearly lie in

the plane shown in the insets, that is there is only a slight difference

between the dipole vectors and their displayed projections.) The

infinite direction of the wire is along the

-axes. The

- and

-axis

are chosen along the pseudo-cubic [100], [010] and [001] directions,

respectively. (Reprinted with permission from Ponomareva

et al.

2005b.)

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Domains in ferroelectric nanostructures from first principles 597

depolarizing field since they have also been found in ferroelectric thin films

experiencing similar mechanical and electrical boundary conditions (see Section

19.3.1, Fong et al., 2004, and Kornev et al., 2004). Note, however, that the

wires and films distinguish themselves from the dot in the sense that the

former exhibit stripe domains that are (infinitely) repeated along a periodic

direction, while the 3D-confinement of the nanodot results in a finite number

of stripes. Furthermore, unlike for the stress-free and tensile mechanical

boundary conditions in the dot, the ground-state dipole pattern in the

compressively strained dot leads to a vanishing spontaneous (total) toroid

moment of polarization − in addition to a vanishing spontaneous polarization

− for small β.

19.5 Conclusions

Recent works have shown that it is now possible to study domain patterns

occurring in ferroelectric nanostructures from direct first principles and first-

principles-based techniques. In particular, we used the effective Hamiltonian

approach (1) to describe a wide variety of dipole patterns that can exist in

ultra-thin films, nanodots, and nanowires, depending on the interplay between

film orientation and boundary conditions and (2) to provide detailed atomistic

information of the peculiar domain evolution of epitaxial ultrathin films

under an applied electric field.

We are thus confident that the use of the first-principles approaches for

obtaining important quantitative information leads to a deeper understanding

of low-dimensional ferroelectrics, in general, and will help in interpreting

future experimental and phenomenological data, in particular.

19.6 Acknowledgments

This work was supported by ONR grants N 00014-01-1-0365 and N 00014-

04-1-0413, NSF grants DMR-0404335, and DOE grant DE-FG02-05ER46188.

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