Fast molecular-dynamics simulation for ferroelectric thin-film capacitors using a first-principles effective Hamiltonian

A newly developed fast molecular dynamics method is applied to ${\text{BaTiO}}_3$ ferroelectric thin-film capacitors with short-circuited electrodes or under applied voltage. The molecular dynamics simulations based on a first-principles effective Hamiltonian clarify that dead layers (or passive layers) between ferroelectrics and electrodes markedly affect the properties of capacitors, and predict that the system is unable to hop between a uniformly polarized ferroelectric structure and a striped ferroelectric domain structure at low temperatures. Simulations of hysteresis loops of thin-film capacitors are also performed, and their dependence on film thickness, epitaxial constraints, and electrodes are discussed.

Figure 1

Simplified flow chart for calculating forces on $u_{\alpha }\left(\mathit{R}\right)$. FFT and IFFT enable rapid calculation of long-range dipole-dipole interactions.

Figure 2

(Color online) Schematic illustrations of ferroelectric thin films of thickness $l$ unit cells (here $l=2$). (a) Isolated thin film in vacuum. (b) Thin film sandwiched between short-circuited perfect electrodes. Doubly periodic boundary conditions for simulations of films sandwiched between perfect and imperfect short-circuited electrodes are depicted in (c) and (d), respectively. Horizontal thick lines marked with “E” represent the electrostatic mirrors used to model electrodes. They are a distance $d/2$ away from the ferroelectric film surface [$d=0$ in (c), $d=1$ in (d)]. Each thin arrow represents a local dipole within a unit cell $\left(a^3=3.94{\text{Å}}^3\right)$ of the ${\text{BaTiO}}_3$ crystal. Thick dashed lines enclose the periodic cell used for simulations.

Figure 3

(Color online) Average homogeneous strains $e_{xx}$, $e_{yy}$, and $e_{zz}$ as a function of temperature in heating-up ($+5\text{K}$/simulation, solid lines) and cooling-down ($-5\text{K}$/simulation, dashed lines) simulations for a $16\times 16\times 16$ supercell. Strains are measured relative to the LDA minimum-energy cubic structure with lattice constant $3.948\text{Å}$.

Figure 4

(Color online) $⟨u_z⟩$ of heating-up (solid lines) and cooling-down (dashed lines) molecular dynamics simulations of ${\text{BaTiO}}_3$ thin-film capacitors with short-circuited electrodes under 1% in-plane biaxial compressive strain for (a) thickness $l=15$ layer with dead layer $d=1$, (b) $l=31$ with $d=1$, (c) $l=127$ with $d=1$, (d) $l=255$ $-z$ with $d=1$, and (e) $l=32$ without dead layer $\left(d=0\right)$. $\sqrt{⟨u_z^2⟩}$ are also plotted in (a)–(e). In (c)–(e), heating-up $\sqrt{⟨u_z^2⟩}$ and cooling-down $\sqrt{⟨u_z^2⟩}$ are almost identical. $\sqrt{⟨u_x^2⟩}$ is plotted only in (e), because the behaviors of $\sqrt{⟨u_x^2⟩}$ and $\sqrt{⟨u_y^2⟩}$ are essentially identical in cases (a)–(e) (for both heating-up and cooling-down). Supercells are of size $32\times 32\times 2\left(l+d\right)$. Animations of these cooling-down and heating-up simulations are also available in the EPAPS (Ref. 27).

Figure 5

(Color online) Snapshots at $\text{T}=100\text{K}$ in heating-up [(a)] and cooling-down [(b) and (c)] simulations of ferroelectric thin-film capacitors of $l=15$ with $d=1$. (a) and (b) are horizontal slices. (c) is a vertical cross section. Points of snapshots are indicated with “X” marks in Fig. 4a. In horizontal slices, the $+z$-polarized and $-z$-polarized sites are denoted by $◻$ and $◼$, respectively. In vertical cross sections, the dipole moments of each site are projected onto the $xz$-plane and indicated with arrows. Layers which do not have arrows are dead layers.

Figure 6

(Color online) Horizontal slices of snapshots at 100 K in cooling-down simulations of ferroelectric thin-film capacitors with single dead layer $\left(d=1\right)$ of various thicknesses $l=7$, 15, 31, 63, 127, and 255. The $+z$-polarized and $-z$-polarized sites are denoted by $◻$ and $◼$, respectively.

Figure 7

(Color online) Calculated thickness $l$ dependence of wavelength $\lambda$ of striped domain structures in thin film ${\text{BaTiO}}_3$ capacitors with a dead layer $\left(d=1\right)$. $+$ marks are from $32\times 32\times 2\left(l+d\right)$ supercell calculations and $\times$ are those of $40\times 40\times 2\left(l+d\right)$. Data of $l<=127$ are fitted with $\lambda =c\sqrt{l}$ (dotted line). $l=255$ data are omitted, because of their large supercell-size dependence.

Figure 8

(Color online) Effective potential surfaces of ${\text{BaTiO}}_3$ thin-film capacitors with short-circuited electrodes: (a)–(e), under 1% in-plane biaxial compressive strain arising from epitaxial constraints; (f)–(j), without epitaxial constraints (i.e., for “free” films). The thicknesses of ferroelectric films and dead layers are indicated in each panel with $l$ and $d$, respectively. Total energies as functions of $u_z$ are compared among striped domain structures with wave vectors $\mathit{k}$ parallel to (110). $\mathit{k}=\left(000\right)$ corresponds to the uniformly polarized structure. The zero of the energy scale is placed at the total energy of the nonpolarized $u_z=0$ structure. A negative pressure $p=-5\text{GPa}$ is applied to correct the underestimation in $T_{\mathrm{C}}$.

Figure 9

(Color online) Schematic comparison between the epitaxially constrained film and the “free” film. In the epitaxially constrained film, switching may have to climb over a potential barrier, but, in the “free” film, dipoles can be easily rotated and switching can go around a valley of the potential.

Figure 10

(Color online) Schematic illustrations of triangle-wave electric field used to measure ferroelectric hysteresis loops experimentally (inset) and in the present simulations.

Figure 11

(Color online) Calculated hysteresis loops for capacitors with (a) epitaxially constrained films, and (b) “free” films of various thickness $l$ and with dead layer $d$. Supercell sizes were $16\times 16\times 2\left(l+d\right)$. $\Delta t=2\text{fs}$ and $n_{\text{steps}}=50,000$. For epitaxially constrained films, ${\mathcal{E}}_0=4,000\text{kV}/\text{cm}$ and $\Delta \mathcal{E}=100\text{kV}/\text{cm}$ are employed. For “free” films, ${\mathcal{E}}_0=400\text{kV}/\text{cm}$ and $\Delta \mathcal{E}=10\text{kV}/\text{cm}$ are employed.

Figure 12

(Color online) Vertical cross section of a simulated ferroelectric “free” film capacitor with a single dead layer; $16\times 16\times \left(l=63,d=1\right)$. The snapshot was taken at the point marked “x” in Fig. 11b. The projections of the dipole moments onto the $xz$-plane are indicated with arrows.