Ferroelectric Response
Creator: Stefan Bringuier
Schematic representation of ferroelectric response for BaTiO3 depicting the double-well free energy potential and polarization plots. The figure illustrates the movement of the Titanium atom within the crystal lattice as it shifts between two stable positions, corresponding to minima in the free energy curve. This Titanium displacement results in a large change in the material's polarization.
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LaTeX
ferroelectric-response.tex (149 lines)
\documentclass{standalone}
\usepackage{tikz}
\usepackage{pgfplots}
\pgfplotsset{compat=1.18}
\usepgfplotslibrary{groupplots}
\usetikzlibrary{fadings,shadings,calc}
% Shape rendering specs
\pgfdeclareradialshading{atomshade}{\pgfpoint{0cm}{0cm}}{%
color(0cm)=(pgftransparent!0);
color(0.2cm)=(pgftransparent!20);
color(0.5cm)=(pgftransparent!50);
color(0.7cm)=(pgftransparent!70);
color(1cm)=(pgftransparent!100)%
}
\tikzset{
atom/.style={circle, shading=atomshade, minimum size=0.4cm},
bond/.style={
line width=0.5mm,
shading=axis,
color=black, %Change to #1 to specify bond color
shading angle=45
}
}
% Define BaTiO3 (#221) unit cell projection
\newcommand{\DrawUnitCell}[1]{%
\begin{tikzpicture}[scale=1.5]
% Draw the unit cell
\draw[thick] (0,0,0) -- (1,0,0) -- (1,1,0) -- (0,1,0) -- cycle; % Bottom face
\draw[thick] (0,0,1) -- (1,0,1) -- (1,1,1) -- (0,1,1) -- cycle; % Top face
\draw[thick] (0,0,0) -- (0,0,1);
\draw[thick] (1,0,0) -- (1,0,1);
\draw[thick] (1,1,0) -- (1,1,1);
\draw[thick] (0,1,0) -- (0,1,1);
% Draw lines connect Oxygen to Titanium
\begin{scope}
\ifnum \pdfstrcmp{#1}{0.5} < 0
% Connect bottom 4 oxygens to Ti
\draw[bond=red] (0,0.5,0.5) -- ($(0,0.5,0.5)!0.5!(0.5,#1,0.5)$);
\draw[bond=gray] ($(0,0.5,0.5)!0.5!(0.5,#1,0.5)$) -- (0.5,#1,0.5);
\draw[bond=red] (1,0.5,0.5) -- ($(1,0.5,0.5)!0.5!(0.5,#1,0.5)$);
\draw[bond=gray] ($(1,0.5,0.5)!0.5!(0.5,#1,0.5)$) -- (0.5,#1,0.5);
\draw[bond=red] (0.5,0,0.5) -- ($(0.5,0,0.5)!0.5!(0.5,#1,0.5)$);
\draw[bond=gray] ($(0.5,0,0.5)!0.5!(0.5,#1,0.5)$) -- (0.5,#1,0.5);
\draw[bond=red] (0.5,0.5,0) -- ($(0.5,0.5,0)!0.5!(0.5,#1,0.5)$);
\draw[bond=gray] ($(0.5,0.5,0)!0.5!(0.5,#1,0.5)$) -- (0.5,#1,0.5);
\draw[bond=red] (0.5,0.5,1) -- ($(0.5,0.5,1)!0.5!(0.5,#1,0.5)$);
\draw[bond=gray] ($(0.5,0.5,1)!0.5!(0.5,#1,0.5)$) -- (0.5,#1,0.5);
\else
\ifnum \pdfstrcmp{#1}{0.5} > 0
% Connect top 4 oxygens to Ti
\draw[bond=red] (0,0.5,0.5) -- ($(0,0.5,0.5)!0.5!(0.5,#1,0.5)$);
\draw[bond=gray] ($(0,0.5,0.5)!0.5!(0.5,#1,0.5)$) -- (0.5,#1,0.5);
\draw[bond=red] (1,0.5,0.5) -- ($(1,0.5,0.5)!0.5!(0.5,#1,0.5)$);
\draw[bond=gray] ($(1,0.5,0.5)!0.5!(0.5,#1,0.5)$) -- (0.5,#1,0.5);
\draw[bond=red] (0.5,1,0.5) -- ($(0.5,1,0.5)!0.5!(0.5,#1,0.5)$);
\draw[bond=gray] ($(0.5,1,0.5)!0.5!(0.5,#1,0.5)$) -- (0.5,#1,0.5);
\draw[bond=red] (0.5,0.5,1) -- ($(0.5,0.5,1)!0.5!(0.5,#1,0.5)$);
\draw[bond=gray] ($(0.5,0.5,1)!0.5!(0.5,#1,0.5)$) -- (0.5,#1,0.5);
\else
% Connect 8 oxygens to form octahedron
\draw[bond=red] (0,0.5,0.5) -- (1,0.5,0.5);
\draw[bond=red] (0.5,0,0.5) -- (0.5,1,0.5);
\draw[bond=red] (0.5,0.5,0) -- (0.5,0.5,1);
\fi
\fi
\end{scope}
% Draw the Oxygen atoms (Red) (back perspective)
\node[atom, anchor=center,ball color=red] at (0,0.5,0.5) {};
\node[atom, anchor=center,ball color=red] at (0.5,0,0.5) {};
\node[atom, anchor=center,ball color=red] at (0.5,0.5,0.0) {};
% Draw the Barium atoms (Cyan)
\node[atom, anchor=center,ball color=cyan] at (0,0,0) {};
\node[atom, anchor=center,ball color=cyan] at (1,0,0) {};
\node[atom, anchor=center,ball color=cyan] at (0,1,0) {};
\node[atom, anchor=center,ball color=cyan] at (1,1,0) {};
\node[atom, anchor=center,ball color=cyan] at (0,0,1) {};
\node[atom, anchor=center,ball color=cyan] at (1,0,1) {};
\node[atom, anchor=center,ball color=cyan] at (0,1,1) {};
\node[atom, anchor=center,ball color=cyan] at (1,1,1) {};
% Titanium atom (Gray)
\node[atom, anchor=center,ball color=gray, minimum size=0.25cm] at (0.5,#1,0.5) {};
% Draw the Oxygen atoms (red) (front perspective)
\node[atom, anchor=center,ball color=red] at (0.5,1,0.5) {};
\node[atom, anchor=center,ball color=red] at (1,0.5,0.5) {};
\node[atom, anchor=center,ball color=red] at (0.5,0.5,1) {};
\end{tikzpicture}%
}
\begin{document}
\begin{tikzpicture}[scale=0.8]
% Define top and bottom plots
\begin{groupplot}[
group style={group size=1 by 2, vertical sep=0pt},
width=10cm,
height=6cm,
xtick=\empty,
ytick=\empty,
axis lines*=box,
enlargelimits=false,
xmin=-2.5, xmax=2.5,
axis line style={very thick},
]
% Double-well free energy potential
\nextgroupplot[
ylabel={Free Energy},
ylabel style={rotate=0},
ymin=0, ymax=3.5,
xlabel={}, % Remove x-axis label from top plot
tick style={draw=none},
]
\addplot[domain=-2.5:2.5, samples=100, very thick] {0.5*(\x^4-4*\x^2+4)};
% Polarization vs. displacement
\nextgroupplot[
ylabel={Polarization},
ylabel style={rotate=-90},
ymin=-2.5, ymax=2.5,
xlabel={Ti Displacement},
xlabel shift={-10pt},
xlabel near ticks,
ylabel near ticks,
xtick={-2.25, 2.25},
xticklabels={negative, positive},
tick style={draw=none},
]
% Additional axes through origin
\draw[black!50, thick] (axis cs:0,-2.475) -- (axis cs:0,2.475);
\draw[black!50, thick] (axis cs:-2.49,0) -- (axis cs:2.49,0);
% Polarization line
\addplot[domain=-2.5:2.5, samples=2, blue, very thick] {\x};
\end{groupplot}
% Overlay the unit cells manually
\node[anchor=center] at (1.75,3.5) {\scalebox{0.5}{\DrawUnitCell{0.3}}};
\node[anchor=center] at (4.25,3.5) {\scalebox{0.5}{\DrawUnitCell{0.5}}};
\node[anchor=center] at (6.75,3.5) {\scalebox{0.5}{\DrawUnitCell{0.7}}};
\end{tikzpicture}
\end{document}
Typst
ferroelectric-response.typ (243 lines)
#import "@preview/cetz:0.3.4": canvas, draw
#import draw: rect, line, circle, content, hobby, scale, set-origin
#set page(width: auto, height: auto, margin: 8pt)
#canvas({
let arrow-style = (mark: (end: "stealth", fill: black, scale: .75))
let plot-height = 4
let plot-width = 10
let y-offset = 4.5
// Helper to draw axes
let draw-axes(origin, width, height) = {
line(
(origin.at(0) - 0.3, origin.at(1)),
(origin.at(0) + width, origin.at(1)),
..arrow-style,
name: "x-axis",
)
line(
(origin.at(0), origin.at(1) - 0.3),
(origin.at(0), origin.at(1) + height),
..arrow-style,
name: "y-axis",
)
}
// Top plot: Double-well potential
let top-origin = (-5, y-offset)
draw-axes(top-origin, plot-width, plot-height)
// Draw double-well potential curve using hobby spline
hobby(
(top-origin.at(0) + .5, top-origin.at(1) + 4), // start high
(top-origin.at(0) + 1.7, top-origin.at(1) + 0.4), // left minimum
(top-origin.at(0) + 1.8, top-origin.at(1) + 0.3), // left minimum
(top-origin.at(0) + 5, top-origin.at(1) + 1.5), // up to middle peak
(top-origin.at(0) + 8.2, top-origin.at(1) + 0.3), // right minimum
(top-origin.at(0) + 8.3, top-origin.at(1) + 0.4), // right minimum
(top-origin.at(0) + 9.5, top-origin.at(1) + 4), // up high again
omega: 0,
name: "potential-curve",
stroke: 1.5pt,
)
// Add "Free Energy" label
content(
"y-axis.mid",
[Free Energy],
angle: 90deg,
anchor: "south",
padding: (0, 0, 2pt),
)
// Bottom plot: Polarization vs. displacement
let bottom-origin = (-5, 0)
draw-axes(bottom-origin, plot-width, plot-height)
// zero lines
line(
(bottom-origin.at(0), bottom-origin.at(1) + plot-height / 2),
(bottom-origin.at(0) + plot-width, bottom-origin.at(1) + plot-height / 2),
stroke: gray + 0.5pt,
)
line(
(bottom-origin.at(0) + plot-width / 2, bottom-origin.at(1)),
(bottom-origin.at(0) + plot-width / 2, bottom-origin.at(1) + plot-height),
stroke: gray + 0.5pt,
)
// Add x-axis labels
content(
(bottom-origin.at(0), bottom-origin.at(1)),
[negative],
anchor: "north-west",
padding: (4pt, 2pt, 0),
name: "neg-label",
)
content(
(bottom-origin.at(0) + 8.5, bottom-origin.at(1)),
[positive],
anchor: "north-west",
padding: (4pt, 2pt, 0),
name: "pos-label",
)
// Draw linear polarization line
line(
(bottom-origin.at(0), bottom-origin.at(1)),
(bottom-origin.at(0) + plot-width, bottom-origin.at(1) + plot-height),
stroke: blue + 1.5pt,
name: "polarization-line",
)
// Add "Polarization" and "Ti Displacement" labels
content(
"y-axis.mid",
[Polarization],
angle: 90deg,
anchor: "south",
padding: 4pt,
)
content(
"x-axis.mid",
[Ti Displacement],
anchor: "north",
padding: (10pt, 0, 0),
)
// Helper function to draw BaTiO3 unit cell
let atom(pos, radius: 0.20, fill: luma(50), ..args) = {
circle(
pos,
radius: radius,
stroke: none,
fill: fill,
..args
)
circle(
(),
radius: radius,
stroke: none,
fill: gradient.radial(
fill.lighten(75%), fill, fill.darken(15%),
focal-center: (30%, 25%),
focal-radius: 5%,
center: (35%, 30%),
),
)
}
let draw-unit-cell(center-x, center-y, ti-y, cell-name) = {
let (x, y) = (center-x, center-y)
let z-offset = -1.0 // Consistent offset for back face
let cube-style = (stroke: 0.7pt)
// Draw unit cell cube with consistent offsets
rect(
(x - 1, y - 1, 0),
(x + 1, y + 1, 0),
..cube-style,
name: cell-name + "-front",
) // Front face
line(
(x - 1, y - 1, 0),
(x - 1, y - 1, z-offset),
..cube-style,
name: cell-name + "-left",
) // Left edge
line(
(x + 1, y - 1, 0),
(x + 1, y - 1, z-offset),
..cube-style,
name: cell-name + "-right",
) // Right edge
line(
(x - 1, y - 1, z-offset),
(x + 1, y - 1, z-offset),
..cube-style,
name: cell-name + "-back",
) // Back edge
line(
(x - 1, y + 1, z-offset),
(x + 1, y + 1, z-offset),
..cube-style,
name: cell-name + "-top-back",
) // Top back edge
line(
(x - 1, y - 1, z-offset),
(x - 1, y + 1, z-offset),
..cube-style,
name: cell-name + "-left-back",
) // Left back edge
line(
(x + 1, y - 1, z-offset),
(x + 1, y + 1, z-offset),
..cube-style,
name: cell-name + "-right-back",
) // Right back edge
line(
(x + 1, y - -1),
(x + 1, y - -1, z-offset),
..cube-style,
name: cell-name + "top-right",
) // top right edge
line(
(x + -1, y - -1),
(x + -1, y - -1, z-offset),
..cube-style,
name: cell-name + "top-left",
) // top right edge
// Define helper functions for each atom style
let Ba-atom(pos, name) = atom(pos, fill: rgb("#00ffff"), name: cell-name + "-ba-" + name)
let O-atom(pos, name) = atom( pos, fill: red, name: cell-name + "-o-" + name)
let Ti-atom(pos) = atom( pos, fill: gray, name: cell-name + "-ti")
let Ti-O-bond(end-pos, name) = line(
((x, y + ti-y, z-offset/2), 15%, end-pos),
((x, y + ti-y, z-offset/2), 85%, end-pos),
stroke: 1pt,
name: cell-name + "-bond-" + name,
)
// --- Back Plane (z = z-offset) ---
Ba-atom((x - 1, y - 1, z-offset), "back-bl")
Ba-atom((x + 1, y - 1, z-offset), "back-br")
Ba-atom((x - 1, y + 1, z-offset), "back-tl")
Ba-atom((x + 1, y + 1, z-offset), "back-tr")
O-atom((x, y, z-offset), "back")
Ti-O-bond((x, y, z-offset), "back")
// --- Middle Plane (z = z-offset/2) ---
if ti-y >= 0 {
Ti-O-bond((x, y + 1, z-offset/2), "top")
}
if ti-y <= 0 {
Ti-O-bond((x, y - 1, z-offset/2), "bottom")
}
Ti-O-bond((x - 1, y, z-offset/2), "left")
Ti-O-bond((x + 1, y, z-offset/2), "right")
O-atom((x, y + 1, z-offset/2), "top")
O-atom((x, y - 1, z-offset/2), "bottom")
O-atom((x - 1, y, z-offset/2), "left")
O-atom((x + 1, y, z-offset/2), "right")
Ti-atom((x, y + ti-y, z-offset/2))
// --- Front Plane (z = 0) ---
Ti-O-bond((x, y, 0), "front")
Ba-atom((x - 1, y - 1, 0), "front-bl")
Ba-atom((x + 1, y - 1, 0), "front-br")
Ba-atom((x - 1, y + 1, 0), "front-tl")
Ba-atom((x + 1, y + 1, 0), "front-tr")
O-atom((x, y, 0), "front")
}
// Draw three unit cells with different Ti displacements
scale(0.64)
set-origin("potential-curve.mid")
draw-unit-cell(-4.5, 2, -0.2, "cell1")
draw-unit-cell(-0.5, 2, 0, "cell2")
draw-unit-cell(3.5, 2, 0.2, "cell3")
})