--- title: "Ferroelectrics, Piezoelectrics, and Structural Transitions" module: Dielectrics and Ferroelectrics moduleNumber: 8 lessonNumber: 2 order: 802 summary: > Some crystals carry a polarization with no applied field and switch it under a reversing field, tracing a hysteresis loop. This lesson develops the ferroelectric transition through the perovskite BaTiO3 displacive instability and its soft transverse-optical mode, builds the Landau free-energy theory of first- and second-order polar transitions, derives the Curie-Weiss divergence of the dielectric constant, and closes with piezoelectricity and pyroelectricity and their devices. topics: [Dielectrics and Ferroelectrics] sources: - book: Kittel ref: "Ch. 16 — Dielectrics and Ferroelectrics" - book: Ashcroft & Mermin ref: "Ch. 27 — Dielectric Properties of Insulators" - book: Hook & Hall ref: "Ch. 9 — Dielectrics" --- The [previous lesson](/condensed-matter/dielectrics-and-ferroelectrics/dielectrics-and-polarization) ended at the polarization catastrophe: when the local field of the dipoles themselves is strong enough to sustain the polarization, the dielectric constant diverges and the crystal polarizes with no applied field. A **ferroelectric** is a crystal in which this happens below a transition temperature, the **Curie temperature** $T_c$. It carries a **spontaneous polarization** $P_s$ that can be reversed by an external field, and near $T_c$ its dielectric constant runs up to enormous values. This lesson works through the archetype BaTiO$_3$, the soft-mode mechanism, the Landau theory that classifies the transition, and the piezoelectric and pyroelectric effects that make polar crystals technologically central. ## Spontaneous polarization and the hysteresis loop A ferroelectric below $T_c$ has two or more symmetry-equivalent orientations of $P_s$, and a field can switch between them. Plotting polarization against applied field traces a **hysteresis loop**, the defining experimental signature. Starting from an unpoled crystal (zero net polarization, because domains of opposite $P_s$ cancel), a rising field aligns the domains until the polarization **saturates** at $P_s$. Reducing the field to zero leaves a **remanent polarization** $P_r$: the crystal stays poled. Reversing the field, the polarization drops to zero at the **coercive field** $E_c$ and then saturates in the opposite direction. The loop encloses an area equal to the energy dissipated per unit volume per cycle. $$ % caption: The ferroelectric hysteresis loop. As the field cycles, the % polarization saturates at plus or minus P_s, retains a remanent value P_r at % zero field, and reverses only once the field exceeds the coercive value E_c. % The enclosed area is the energy dissipated per cycle. The dashed curve is the % initial poling of a virgin crystal from the unpoled state. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (-4.0,0) -- (4.0,0) node[below right] {$E$}; \draw[->, black] (0,-3.0) -- (0,3.0) node[left] {$P$}; % upper branch (field increasing then decreasing): loop right side \draw[acc, very thick] (-3.4,-2.3) .. controls (-1.2,-2.35) and (-1.4,2.05) .. (0.9,2.25) .. controls (2.4,2.38) and (3.0,2.45) .. (3.6,2.5); % lower branch \draw[acc, very thick] (3.4,2.3) .. controls (1.2,2.35) and (1.4,-2.05) .. (-0.9,-2.25) .. controls (-2.4,-2.38) and (-3.0,-2.45) .. (-3.6,-2.5); % initial poling curve (virgin) \draw[black, dashed] (0,0) .. controls (0.6,0.9) and (1.2,2.0) .. (2.6,2.42); % markers \draw[black, dashed] (0,2.25) -- (2.6,2.25); \fill[acc] (0,1.55) circle (1.8pt); \node[black, anchor=east, font=\scriptsize] at (-0.1,1.55) {$P_r$}; \node[black, anchor=south west, font=\scriptsize] at (2.6,2.28) {$P_s$}; \fill[acc] (-1.05,0) circle (1.8pt); \node[black, anchor=north east, font=\scriptsize] at (-0.15,-0.1) {$E_c$}; \end{tikzpicture} $$ > **Definition (Ferroelectric, spontaneous polarization, coercive field).** A > **ferroelectric** is a crystal possessing a **spontaneous polarization** $P_s$ > in the absence of an applied field, whose direction can be reversed by a field. > The **remanent polarization** $P_r$ is the polarization at zero field after > saturation; the **coercive field** $E_c$ is the reverse field that reduces the > polarization to zero. Above the **Curie temperature** $T_c$ the spontaneous > polarization vanishes and the crystal is a paraelectric. The reversible spontaneous polarization is a much stronger condition than merely being polar. Many crystals have a built-in polarization from their structure (they are **pyroelectric**), but only a ferroelectric can have that polarization reversed by an accessible field. Ferroelectricity therefore requires both a polar structure and a low enough energy barrier between the two polar states. ## The perovskite BaTiO$_3$ and displacive transitions Barium titanate is the prototype. Above $T_c \approx 393\ \text{K}$ ($120\,^\circ\text{C}$) it has the cubic **perovskite** structure ABO$_3$: Ba$^{2+}$ ions at the cube corners, O$^{2-}$ ions at the face centers forming an octahedron, and a small Ti$^{4+}$ ion at the body center of that oxygen octahedron. In the cubic phase the Ti sits exactly at the center, the structure has inversion symmetry, and there is no net dipole. Below $T_c$ the Ti$^{4+}$ ion displaces off-center along a cube axis by about $0.1\ \text{\AA}$, the surrounding oxygen octahedron shifts the opposite way, and the cell distorts from cubic to tetragonal. The displacement of positive against negative charge creates a dipole in every cell, and the parallel alignment of these cell dipoles is the spontaneous polarization, $P_s \approx 0.26\ \text{C/m}^2$. This is a **displacive** transition: the atoms shift slightly from their high-symmetry sites, in contrast with an **order–disorder** transition in which pre-existing dipoles (as in KH$_2$PO$_4$) merely align. $$ % caption: The perovskite BaTiO3 cell. Ba ions sit at the cube corners and the % oxygen octahedron at the face centers. Above T_c the Ti ion is centered and the % cell is nonpolar; below T_c it displaces along a cube axis (arrow), pulling the % positive charge off center and giving the cell a permanent dipole. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} % oblique cube \def\dx{1.1} \def\dy{0.7} % back face corners \coordinate (bfl) at (0.9,0.9); \coordinate (bfr) at (3.9,0.9); \coordinate (btl) at (0.9,3.9); \coordinate (btr) at (3.9,3.9); % front face corners \coordinate (ffl) at (0,0); \coordinate (ffr) at (3.0,0); \coordinate (ftl) at (0,3.0); \coordinate (ftr) at (3.0,3.0); % edges \draw[black] (ffl)--(ffr)--(ftr)--(ftl)--cycle; \draw[black] (bfl)--(bfr)--(btr)--(btl)--cycle; \draw[black] (ffl)--(bfl); \draw[black] (ffr)--(bfr); \draw[black] (ftl)--(btl); \draw[black] (ftr)--(btr); % Ba at corners \foreach \c in {ffl,ffr,ftl,ftr,bfl,bfr,btl,btr}{ \fill[acc] (\c) circle (3pt); } \node[acc, anchor=north east, font=\scriptsize] at (ffl) {Ba}; % O at face centers (front and a couple visible) \coordinate (ofront) at (1.5,1.5); \coordinate (otop) at (2.0,3.45); \coordinate (oright) at (3.45,1.95); \fill[black] (ofront) circle (3.4pt); \fill[black] (otop) circle (3.4pt); \fill[black] (oright) circle (3.4pt); \node[black, anchor=north west, font=\scriptsize] at (1.55,1.45) {O}; % Ti near center, displaced up \coordinate (ti) at (1.95,2.35); \fill[black] (ti) circle (2.6pt); \node[black, anchor=west, font=\scriptsize] at (2.05,2.3) {Ti}; % displacement arrow \draw[acc, ->, very thick] (1.95,1.95) -- (1.95,2.75); \node[acc, anchor=south, font=\scriptsize] at (1.95,2.82) {shift}; \end{tikzpicture} $$ The physical driver is a competition already visible in the Clausius–Mossotti denominator. The short-range repulsion of the closed ion cores holds Ti at the center; the long-range dipole field (the local-field term $P/3\varepsilon_0$) pushes it off-center, because an off-center Ti polarizes its neighbors, whose fields push it further. As temperature falls the balance tips, and below $T_c$ the off-center configuration wins. ## The soft mode The lattice-dynamical statement of this instability is the **soft mode**. A displacive ferroelectric transition is driven by a particular transverse-optical (TO) phonon at the zone center, the mode whose atomic pattern reproduces the Ti-against-oxygen displacement that produces $P_s$. The restoring force for this mode is the difference between the stiff short-range core repulsion and the destabilizing long-range dipole field. As $T \to T_c$ that difference shrinks, the effective spring constant softens, and the mode frequency falls toward zero: $$ \omega_{\text{TO}}^2 \propto (T - T_c). $$ At $T_c$ the frequency reaches zero, the restoring force for that displacement vanishes, and the atoms freeze into the displaced configuration — the mode "condenses" into the static distortion. Above $T_c$ the mode is a genuine oscillation that stiffens as the crystal is heated away from the transition; inelastic neutron scattering measures $\omega_{\text{TO}}(T)$ directly and confirms the linear $\omega_{\text{TO}}^2$ in $T$. $$ % caption: The soft mode. The squared frequency of the ferroelectric transverse- % optical phonon falls linearly toward zero as the temperature approaches T_c from % above. At T_c the restoring force vanishes and the displacement pattern freezes % into the static polar distortion. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (6.6,0) node[below] {$T$}; \draw[->, black] (0,0) -- (0,3.4) node[left] {freq squared}; % Tc marker at x=1.6 \draw[black, dashed] (1.6,0) -- (1.6,0.15); \node[black, anchor=north, font=\scriptsize] at (1.6,0) {$T_c$}; % linear line from Tc up to the right \draw[acc, very thick] (1.6,0) -- (6.1,3.05); \fill[acc] (1.6,0) circle (1.8pt); \node[acc, anchor=south east, font=\scriptsize] at (5.7,2.7) {linear in T}; \end{tikzpicture} $$ The soft mode also explains the huge dielectric constant. The static ionic contribution to $\varepsilon$ scales as $1/\omega_{\text{TO}}^2$: a soft restoring force means an enormous displacement per unit field. The **Lyddane–Sachs–Teller relation** makes this exact, $$ \frac{\varepsilon_s}{\varepsilon_\infty} = \frac{\omega_{\text{LO}}^2}{\omega_{\text{TO}}^2}, $$ so as $\omega_{\text{TO}} \to 0$ the static dielectric constant $\varepsilon_s$ diverges while the longitudinal-optical frequency and $\varepsilon_\infty$ stay finite. The Curie–Weiss divergence derived below and the soft-mode vanishing are two faces of the same instability. ## Landau theory of the transition The transition is classified with a **Landau free energy**: an expansion of the free energy in powers of the polarization, the order parameter that is zero above $T_c$ and nonzero below. Symmetry under $P \to -P$ (present because the two polar states are equivalent) forbids odd powers, so $$ F(P,T) = F_0 + \tfrac12 a\,P^2 + \tfrac14 b\,P^4 + \tfrac16 c\,P^6 - E\,P, $$ with the last term the coupling to an applied field. The temperature enters through the quadratic coefficient, which changes sign at the transition: $$ a = a_0\,(T - T_0), \qquad a_0 > 0. $$ The equilibrium polarization minimizes $F$, so $\partial F/\partial P = 0$: $$ a P + b P^3 + c P^5 = E. $$ The sign of $b$ decides the order of the transition. ### Second-order transitions If $b > 0$, the sixth-order term is not needed and can be dropped. In zero field $aP + bP^3 = 0$, so either $P = 0$ (stable for $a > 0$, i.e. $T > T_0$) or $$ P_s^2 = -\frac{a}{b} = \frac{a_0}{b}(T_0 - T) \quad (T < T_0). $$ The spontaneous polarization grows continuously from zero as $(T_0 - T)^{1/2}$ below the transition. Here $T_0 = T_c$, the free energy evolves smoothly from a single well into a double well, and there is no latent heat: the transition is **second order** (continuous). $$ % caption: The Landau free energy F(P) at three temperatures. Above T_c the % quadratic coefficient is positive and F has a single minimum at P = 0 % (paraelectric). At T_c the well flattens. Below T_c the coefficient turns % negative and F becomes a symmetric double well whose minima sit at plus or % minus P_s, the spontaneous polarization. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (-3.2,0) -- (3.2,0) node[right] {$P$}; \draw[->, black] (0,-1.2) -- (0,2.6) node[left] {$F$}; % above Tc: single well 0.55 x^2 + 0.06 x^4 \draw[black, very thick, domain=-2.7:2.7, samples=90, variable=\x] plot ({\x},{0.42*\x*\x + 0.05*\x*\x*\x*\x}); \node[black, anchor=west, font=\scriptsize] at (1.9,2.15) {$T > T_c$}; % at Tc: flat quartic 0.09 x^4 \draw[black, thick, domain=-2.7:2.7, samples=90, variable=\x] plot ({\x},{0.075*\x*\x*\x*\x}); \node[black, anchor=west, font=\scriptsize] at (2.15,1.35) {$T = T_c$}; % below Tc: double well -0.55 x^2 + 0.09 x^4 \draw[acc, very thick, domain=-2.7:2.7, samples=120, variable=\x] plot ({\x},{-0.6*\x*\x + 0.11*\x*\x*\x*\x}); \node[acc, anchor=east, font=\scriptsize] at (-1.6,1.4) {$T < T_c$}; % minima markers of double well: dF=0 -> x^2 = 0.6/(2*0.11)=2.72 -> x=1.65 \fill[acc] (1.65,-0.82) circle (1.8pt); \fill[acc] (-1.65,-0.82) circle (1.8pt); \draw[black, dashed] (1.65,-0.82) -- (1.65,0); \node[black, anchor=south, font=\scriptsize] at (1.9,0.02) {$P_s$}; \end{tikzpicture} $$ ### First-order transitions If $b < 0$, the quartic term is itself destabilizing and the sixth-order term ($c > 0$) is required to bound the free energy. Now a second minimum at finite $P$ appears while $P = 0$ is still locally stable, and the two become degenerate at a temperature $T_c$ slightly **above** $T_0$. At $T_c$ the polarization jumps discontinuously from zero to a finite value, and the transition absorbs a latent heat: it is **first order** (discontinuous). BaTiO$_3$ is weakly first order; the polarization drops abruptly at $T_c$ rather than tapering to zero. $$ % caption: The order parameter versus temperature. In a second-order transition % (solid) the spontaneous polarization rises continuously below T_c as the square % root of T_c minus T. In a first-order transition (dashed) it appears % discontinuously, jumping to a finite value at T_c. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (6.6,0) node[below] {$T$}; \draw[->, black] (0,0) -- (0,3.1) node[left] {$P_s$}; % Tc at x=4.2 \draw[black, dashed] (4.2,0) -- (4.2,2.9); \node[black, anchor=south west, font=\scriptsize] at (4.2,2.5) {$T_c$}; % second order: sqrt curve from Tc down to left, P ~ sqrt(Tc-T) \draw[acc, very thick] (4.2,0) .. controls (3.4,1.55) and (2.2,2.15) .. (0.2,2.65); \node[acc, anchor=south west, font=\scriptsize] at (0.3,2.55) {second order}; % first order: flat then jump. below Tc' higher value, dashed \draw[black, very thick, dashed] (4.2,1.55) .. controls (3.2,1.95) and (1.8,2.25) .. (0.2,2.55); \draw[black, very thick, dashed] (4.2,1.55) -- (4.2,0); \fill[black] (4.2,1.55) circle (1.8pt); \node[black, anchor=west, font=\scriptsize] at (4.3,1.2) {discontinuous}; \end{tikzpicture} $$ ## The Curie–Weiss dielectric anomaly The dielectric response above $T_c$ follows from the same free energy. For small field and small $P$ in the paraelectric phase ($T > T_c$, $P = 0$ equilibrium), keep only the quadratic term in the equilibrium condition: $$ aP = E \implies \chi = \frac{\partial P}{\partial E} = \frac{1}{a} = \frac{1}{a_0(T - T_0)}. $$ Since $\varepsilon_r = 1 + \chi/\varepsilon_0 \approx \chi/\varepsilon_0$ for the huge susceptibilities near a ferroelectric transition, the dielectric constant obeys the **Curie–Weiss law** $$ \varepsilon_r = \frac{C}{T - T_0}, \qquad C = \frac{1}{\varepsilon_0 a_0}, $$ with $C$ the **Curie constant** (of order $10^5\ \text{K}$ for BaTiO$_3$) and $T_0$ the **Curie–Weiss temperature** (equal to $T_c$ for a second-order transition, a few kelvin below it for a first-order one). The dielectric constant diverges as $T \to T_0$ from above, reaching values of $10^4$ or more near the transition. Below $T_c$ a parallel calculation about the shifted minimum gives a susceptibility half as steep, $$ \chi = \frac{1}{2a_0(T_c - T)}, $$ so a plot of $1/\varepsilon_r$ against $T$ is a straight line hitting zero at $T_0$ from above and rising twice as steeply below $T_c$. $$ % caption: The Curie-Weiss anomaly. The dielectric constant (left axis, solid) % diverges as the temperature approaches T_c, following C over T minus T_0 above % the transition. Plotting the reciprocal 1 over eps-r (right, dashed) linearizes % the data: it falls to zero at T_0 from above and rises with twice the slope % below T_c. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (7.4,0) node[below] {$T$}; \draw[->, black] (0,0) -- (0,3.7) node[left] {permittivity}; % Tc at x=3.6 \draw[black, dashed] (3.6,0) -- (3.6,3.6); \node[black, anchor=south, font=\scriptsize] at (3.6,3.5) {$T_c$}; % eps diverging: right branch (T>Tc), 1/(T-Tc) \draw[acc, very thick] (7.0,0.55) .. controls (5.2,0.75) and (4.2,1.5) .. (3.75,3.4); % left branch (T **Theorem (Curie–Weiss law).** Above a ferroelectric transition the dielectric > constant diverges as $\varepsilon_r = C/(T - T_0)$, with Curie constant > $C = 1/\varepsilon_0 a_0$ set by the Landau coefficient and $T_0$ the > Curie–Weiss temperature. Below $T_c$ the reciprocal susceptibility rises with > twice the slope of the paraelectric branch. ## Piezoelectricity and pyroelectricity Two related effects arise in polar crystals more generally. **Piezoelectricity** is the linear coupling between mechanical strain and electric polarization, present in any crystal lacking a center of inversion (20 of the 32 point groups). Applying a stress produces a polarization (the **direct effect**), and applying a field produces a strain (the **converse effect**): $$ P_i = d_{ijk}\,\sigma_{jk} \quad\text{(direct)}, \qquad \epsilon_{jk} = d_{ijk}\,E_i \quad\text{(converse)}, $$ with the same third-rank **piezoelectric tensor** $d_{ijk}$ governing both, a consequence of the thermodynamic symmetry of the coupling. In a centrosymmetric crystal every $d_{ijk}$ vanishes, because reversing all coordinates would reverse $P$ but leave the symmetric strain unchanged. Quartz, a nonferroelectric piezoelectric, provides the frequency standard of electronic oscillators through the converse effect driving a mechanical resonance; poled ferroelectric ceramics such as PZT (lead zirconate titanate) have piezoelectric coefficients orders of magnitude larger and serve as ultrasonic transducers, sonar projectors, accelerometers, and precision actuators. **Pyroelectricity** is the change of spontaneous polarization with temperature, $\Delta P = p\,\Delta T$, present in the 10 polar point groups. Every ferroelectric is pyroelectric (it has a spontaneous polarization), and the pyroelectric coefficient is largest just below $T_c$, where $P_s(T)$ is steepest. Pyroelectric detectors sense infrared radiation by the charge a small temperature rise liberates, giving room-temperature motion sensors and thermal imagers. | Property | Requirement | Coupling | Example / device | | --- | --- | --- | --- | | Piezoelectric | no inversion center (20 groups) | strain $\leftrightarrow$ polarization | quartz oscillator, PZT transducer | | Pyroelectric | polar axis (10 groups) | temperature $\to$ polarization | infrared detector | | Ferroelectric | reversible spontaneous polarization | field switches $P_s$ | FeRAM, BaTiO$_3$ capacitor | The three properties nest: every ferroelectric is pyroelectric, and every pyroelectric is piezoelectric, but not conversely. The distinguishing feature of the ferroelectric — a switchable $P_s$ tracing the hysteresis loop — underlies nonvolatile ferroelectric memory (FeRAM), where the two remanent states store a bit, and the enormous near-$T_c$ permittivity makes BaTiO$_3$ and its relatives the dielectric of choice for multilayer ceramic capacitors. The soft-mode instability that produces all of this is the structural analog of the magnetic ordering transition taken up in the [magnetism module](/condensed-matter/magnetism/exchange-and-ferromagnetism), where an exchange interaction plays the role of the destabilizing local field and the magnetization plays the role of the order parameter.