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Domain evolution of BaTiO3 ultrathin films under electric field: a first-principles study

A first-principles-derived method is used to study the morphology and electric-field-induced evolution of stripe nanodomains in (001) BaTiO3 (BTO) ultrathin films, and to compare them with those in (001) Pb(Zr,Ti)O3 (PZT) ultrathin films. The BaTiO3 systems exhibit 180o periodic stripe domains at null electric field, as in PZT ultrathin films. However, the stripes alternate along [1-10] in BTO systems versus [010] in PZT systems, and no in-plane surface dipoles occur in BTO ultrathin films (unlike in PZT materials). Moreover, the evolution of the 180o stripe domains in the BaTiO3 systems, when applying and increasing an electric field along [001], involves four regions: Region I for which the magnitude of the down dipoles (i.e., those that are antiparallel to the electric field) is reduced, while the domain walls do not move; Region II in which some local down dipoles adjacent to domain walls switch their direction, resulting in zigzagged domain walls - with the overall stripe periodicity being unchanged; Region III in which nanobubbles are created, then contract along [110] and finally collapse; and Region IV which is associated with a single monodomain. Such evolution differs from that of PZT ultrathin films for which neither Region I nor zigzagged domain walls exist, and for which the bubbles contract along [100]. Discussion about such differences is provided.


I. INTRODUCTION
Ferroelectric materials are the heart of many applications such as non-volatile memories, communication devices, and microactuators. [1] The continuing miniaturization of these devices has stimulated considerable research attention on ferroelectric thin films (see, e.g., Refs. 2 and 3 and references therein). Such systems can exhibit striking phenomena because of some finite-size effect. For instance, they can adopt periodic stripe domains with exceptionally small periods (i.e., in the order of a few nanometers), when experiencing some specific mechanical and electrical boundary conditions. [2][3][4][5][6][7][8][9][10][11][12] Interestingly, the morphology of such nanostripe patterns seem to dramatically depend on the materials, as suggested by the facts that 180 o nanostripes are known to alternate along the [010] direction in (001) Pb(Zr,Ti)O 3 (PZT) and PbTiO 3 thin films [5,6,[9][10][11] while Ref. 4 predicts that such stripes alternate along another direction (namely, along [1][2][3][4][5][6][7][8][9][10]) in (001) BaTiO 3 (BTO) ultrathin films. The exact reason behind such difference, that is reminiscent of distinct behaviors found in Pb-versus Ba-based alloys, [13,14] remains to be addressed. One may also wonder if other differences exist for the morphology of the stripe domains in PZT and BTO films, and, if so, why. For instance, do dipoles at the surfaces of BTO films also lie in-plane (to close the flux as in PZT films [6,9] and in the Landau-Lifschitz model of domains [15]), or rather prefer to all lie along the growth direction (as in the Kittel model of domains [16]), or even adopt an intermediate case (e.g., form an angle of 45 degree between the in-plane and out-ofplane directions)? Another fundamental question to be addressed regarding stripe domains in BTO thin films is their precise atomistic evolution under the application of the factor that lies at the heart of many ferroelectric devices, namely an external electric field. More specifically, how do the differences between the initial stripe domains in BTO and PZT films affect the (very unusual) stripe ferroelectric nanobubble monodomain transition sequence that has been recently predicted and documented in details for PZT ultrathin films under applied fields [9]? In particular, it would be worthwhile to have a deep microscopic insight about the mechanisms leading to the disappearance of the nanostripes in favor of ferroelectric nanobubbles (that bear resemblance with ferromagnetic bubbles in ferromagnetic films [17]), and about the evolution of the morphology of these ferroelectric nanobubbles, under an electric field in BTO films -especially, when realizing that BaTiO 3 bulk is the most extensively studied system among all ferroelectrics [18] while little is still known about the domains [4] and their electric-field-induced domain evolution [9] in epitaxial BaTiO 3 ultrathin films.
The aims of this article are to use first-principles-based techniques to i) provide atomistic details of domains in BTO ultrathin films; ii) indicate how such domains evolve as a function of applied electric field; iii) compare the properties of BTO ultrathin films with those of PZT ultrathin films, as well as, to discuss and understand their similarities and differences.
The remainder of this paper is organized as follows. In Section II, the theoretical method is described. The stripe domains in 20Å-thick BTO thin films, and their energetic origins, are presented in Section III.A, with a special emphasis on explaining the difference in morphology between periodic nanostripes in BTO and PZT thin films.
Section III.B depicts the electric-field-induced domain evolution in the BTO ultrathin films, and compares it with the corresponding domain evolution in PZT ultrathin films.
Finally, a conclusion is provided in Section IV.

II. THEORETICAL METHOD
The presently studied BTO and PZT ultrathin films are assumed to be grown where u i is the local soft mode in the unit cell i of the film -whose product with the effective charge Z* yields the local electrical dipole in this cell; while η and {v i } are the homogeneous strain tensor and inhomogeneous strain-related variables in unit cell i, , and denote its magnitude as E z . Note also that we mimic epitaxially-grown (001) films by freezing of some components of the η homogeneous strain tensor, that are (in Voigt notation) η 6 =0 and η 1 =η 2 =δ -with δ being the value forcing the film to adopt the in-plane lattice constant of the substrate. [6,25,26]

A. Ferroelectric Nanodomains in Thin Films
We focus here on a 20 Å-thick (m=5) BTO film, as mimicked by a 24×24×5 supercell, under a compressive strain of -2.2% and having a realistic electrical boundary condition corresponding to β=0.8. [27] As consistent with Refs. 5, 6 and 9, the combination of a significant compressive epitaxial strain (that favors dipoles along the z-direction) with such electrical boundary condition (that will lead to a large enough residual depolarizing field if all the dipoles would parallely point along the z-axis) generates stripe domains at low temperature, when no external electric field is applied. Figure 1a displays such domains at 10K (that were obtained by slowly cooling down the system) and indicates that the stripe domains in the BTO thin films consist of periodically alternating "up" and   Fig. 1a) and that we will denote by BTO-010-4; and 3) the stripe domains also alternating along the [010] direction but now with a period of 8 lattice constants (i.e., as in the PZT film having similar thickness) and that we will refer to as BTO-010-8. Note that these two last domains were first created from scratch and then allowed to relax during a 80,000 sweeps of MC procedure at 10K. Interestingly, they were found to be thermodynamically (meta)stable at low temperature since the periodicity and directions of these stripe domains remained the same during the whole MC procedure at 10K.  Table I indicates that ε dpl is lowered more than ε short is raised, while we numerically found (not shown here) that the opposite occurs for PZT films because of the different parameters inherent to that latter material -which explains the difference in morphology of stripe domains in these two films.
As it can be seen in Figure 2, that shows the real-space distribution of the local dipoles in the ground-states, two other main differences exists between the morphology of the stripe domains in BTO versus PZT films. They are: (1) the dipoles of a given stripe domain are nearly constant in direction (i.e., parallel or antiparallel to the z-axis) and magnitude inside the BTO films, while such dipoles continuously rotate across the stripe inside the PZT films; (2) the dipoles at the surfaces can "only" deviate from the z-axis by up to 45 degrees in the BTO film, while the existence of in-plane surface dipoles were previously reported in PZT films in order to close the flux (and thus minimize depolarizing effects). [6,9,11] To better understand such differences, we decided to  Table 1) reveals that such disliking takes a major part of its origin in the strain-soft mode interaction energy, ε int . In other words, the local soft modes in BTO prefer to follow the easy polarization (z-) axis introduced by the elastic compressive strain. In PZT, we numerically found (not shown here) that ε short becomes the predominant contributing factor for the PZT domain patterns to exhibit rotating dipoles across stripes because dipoles can more easily rotate.

B. Electric-Field-Induced Evolution of Ferroelectric Nanodomains
We now turn our attention to the evolution of the BTO-110-4.3 ground-state domain when subject to an external electric field − for a chosen temperature of 10 K and with the boundary conditions being the same as those indicated above (i.e., a compressive strain of −2.2% and β=0. 8). Consequently, Figure 3 shows the supercell average of the z-Cartesian component of the local modes (<u z >) and of the magnitude of the local modes Region I: Figs. 4 tell us that, when the electric field is applied from zero to 33×10 7 V/m (Region I), the local dipoles in the "down" domains do not switch their direction to point along the +z direction (i.e., parallel to the electrical field) but rather "simply" significantly decrease their magnitude -while the dipoles in the "up" domains more modestly increase in magnitude. Such behaviors lead to <u z > and <u M > increasing and slightly decreasing with E z , respectively, (see Fig. 3) and to domain walls not moving with respect to the E z =0 case (as shown in Fig. 1b). Interestingly, Region I (unlike the Regions II, III and IV that will be discussed below) was not found in PZT films subject to an electric field, [9] because of the easiness in rotating dipoles in these latter systems.

Region II:
A further increase of E z from 33×10 7 to 43×10 7 V/m (Region II) induces a switch, from the -z to +z direction, of some dipoles in the down domains − as indicated by Figs. 4a and 4c. Such switch thus results in an increase of <u z > when increasing E z (see Fig. 3) and in an expansion of the majority domains at the expense of minority domains while maintaining the same overall stripe periodicity -as in the corresponding region for PZT films. [9] However, unlike in PZT [9] and as seen in Fig.   1c, this expansion of majority domains is not synchronous throughout the domain walls.
Rather, scattered regions of switched dipoles (with five lattice constants along the z-axis and one lattice constant along both the x-and y-axes) that are bounded by (100) Fig. 2) and is accomplished by local dipoles rotating away from the z-axis while decreasing their magnitude at the same time. The latter applies to the dipoles that were initially aligned along the −z-direction (see Fig. 2) is preceded by a decreasing of the dipoles' magnitude and leads to a significant change of this magnitude after the flipping (see Figs. 4b and 4d). In contrast, the dipoles inside the films switch predominantly through flipping (see Figs. 1a-1c), with the direction being reversed and the magnitude of dipoles being only very slightly increased after such flipping (see Figs. 4b and 4d). In other words, the slight increase of <u M > when increasing E z in Region II (see Fig. 3) mostly originates from the surface dipoles.

Region III:
As E z keeps increasing, the minority (down) stripe domains become pinched along [110]. Nanobubbles now form (see Fig. 1d), as in PZT thin films under electric field. Note that the stripe to bubble domain transition is numerically found for E z in between 41×10 7 and 45×10 7 V/m (because of the zigzagged domain walls, it is difficult to exactly pin point the onset of stripe to bubble domain transition, unlike in PZT thin films [9]). Once the nanobubbles are formed, they dramatically contract along [110] while keeping their width along [1-10] more or less the same under further applied field − as seen in Figs. 1d and 1e. In other words, as in PZT films, [9] the bubbles contract parallely 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., [1][2][3][4][5][6][7][8][9][10] in BTO versus [010] in PZT ultrathin films). Interestingly, Fig. 4d reveals that, between 43×10 7 and 51×10 7 V/m, the average magnitude of the surface dipoles having a positive component along the direction of the field (i.e., with u z >0) decreases with E z . [This is due to the facts that the number of switched surface dipoles (i.e., having now u z >0 while initially exhibiting u z <0) increases rather rapidly with E z (see Fig. 4c) while such dipoles had a rather small magnitude before this switching (see Fig. 4b).] Figs. 4a further indicates that the critical field for which all the surface dipoles belonging to the initial down stripes have completely switched is around E z =61×10 7 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. 4a). In other words, the part of Region III located above E z =61×10 7 V/m possesses nanobubbles that are encapsulated (i.e., that do not touch the surfaces, as in PZT films). With further increase of E z (see Fig. 1e), 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 z . Note that we numerically found 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 z =104×10 7 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. 1f and as consistent with the fact that Fig. 3 indicates that <u z > and <u M > are now identical. In this monodomain state, all the dipoles increase in magnitude with E z , with the surface dipoles being larger in size.
(Dipoles can deviate from the [001] direction by "only" up to 45 degrees at the surfaces of the BTO films). Furthermore, the stripe bubble monodomain transition sequence recently predicted in PZT ultrathin films [9] also occurs in epitaxial BTO ultrathin films, but with some noteworthy differences. For instance, for BTO films, (1) the dipoles in the minority domains films first decrease in magnitude before switching their directions; (2) zigzagged domains walls exist in the stripe regime; (3) bubbles contract along [110], rather than [100], before "popping out". We hope that our predictions will be confirmed soon, and lead to a deeper knowledge of ferroelectric thin films.    I  II  III  IV   I  II  III  IV   I  II  III  IV   I  II  III  IV