Film thickness versus misfit strain phase diagrams for epitaxial ${\text{PbTiO}}_3$ ultrathin ferroelectric films

We present a full scale nonlinear thermodynamic model based on a Landau-Ginzburg-Devonshire formalism and the theory of dense polydomain structures in a multiparameter space to predict the phase stability of (001) oriented ${\text{PbTiO}}_3$ epitaxial thin films as a function of film thickness and epitaxial strain. The developed methodology, which accounts for electrostatic boundary conditions as well as the formation of misfit dislocations and polydomain structures, produces a thickness-strain phase stability diagram where it finds that the rotational phases (the so-called $r$ and $ac$ phases) in epitaxial ${\text{PbTiO}}_3$ are possible only in a very small window. We find that for experimentally used thickness or strains (or both) that often fall outside this window, the film is in either single phase tetragonal ($c$ phase) or in a $c/a/c/a$ polydomain state; this explains why rotational polar domains are rarely observed in epitaxial ferroelectric thin films.

Figure 1

(Color online) (a) Theoretical MB critical thickness for misfit dislocation formation in ${\text{PbTiO}}_3$ as a function of misfit strain at a growth temperature $T_G=873\text{K}$. The graph shows the respective MB critical thicknesses for popular substrates. (b) Computed contour map showing actual misfit strain as a function of thickness and the imposed numerical (pseudomorphic) misfit.

Figure 2

(Color) Computed free-energy profiles as a function of misfit strain for each of the phases for different thickness; (a) $L=4$, (b) 8, (c) 12, and (d) 20 nm.

Figure 3

(a) Film thickness-strain phase diagram for epitaxial [001] PTO as a function of numerical misfit strain. The $y$ axis is the film thickness. The $x$ axis is the numerical misfit strain calculated based on the paraelectric effective cubic lattice parameter of the ferroelectric and the substrate lattice parameter at growth temperature $\left(T_G=873\text{K}\right)$; the inset shows a contour map displaying the variation of the $c$-domain fraction ${\varphi }_c^0$ in the stability area of the $c/a/c/a$ polydomain pattern at RT in the same parameter space; (b) film thickness-strain phase diagram at 473, (c) 673, and (d) 873 K.

Figure 4

(Color online) Computed free-energy profiles as a function of thickness for three different substrates. (a) STO, (b) KTO, and (c) MgO. Key to figure: $G_{a1a2}$—black $+$, $G_a\left(G_{a^{\prime }}\right)$—red $●$, $G_{aa}$—green $▲$, $G_{ac}\left(G_{a^{\prime }c}\right)$—blue $▼$, $G_r$—dark cyan $◆$, $G_p$—wine $▶$, $G_c$—dark yellow $\Delta$, $G_{caca}$—magenta $○$, and $G_{c^{\prime }{a}^{\prime }}$—navy $◻$.

Figure 5

(Color online) Phase profiles in PTO as a function of thickness for (a) KTO and (b) MgO. For these cases the film was forced to remain pseudomorphic and in a monodomain state. Due to these limitations, one now finds that only the $aa$ phase to be the most stable. Symbols hold the same meaning as in Fig. 4.