Dielectrics and Paraelectric-Ferroelectric Phase Transitions

Abstract

In the linear approximation the dielectric displacement $\mathbf {D}= \mathbf {\mathcal {E}} + 4\pi \mathbf {P}$ and the electric field $\mathbf {\mathcal {E}}$ are connected by a second order tensor $\varepsilon$, the dielectric function, invariant under the point-group symmetry operation of the crystal. We shall assume for simplicity scalar $\varepsilon$ and dielectric susceptibility $\chi$ (in ${\mathbf {P}/\mathbf {\mathcal {E}}} =\chi$ and $\varepsilon =1+ 4\pi \chi$).

Appendices

Appendix 16.1 Pseudo-Spin Dynamics for Order-Disorder Ferroelectrics

In the following the response of an assembly of interacting dipoles ${\varvec{\mu }_{\mathbf {e}}}$ to a small and time-dependent external field is described in the framework of the mean field approximation, by resorting to the pseudo-spin formalism.

In the light of Eq. (16.33) one starts from the Hamiltonian

$$\begin{aligned} \mathcal {H}= -2\varGamma \hbar \sum _i S_x^i- \sum _{i,j}\,'\, I_{ij}S_z^i S_z^j- 2\mu _e\sum _i H_i(t)S_z^i\mathrm { }, \end{aligned}$$

(A.16.1.1)

$H_i$ being the local field. The statistical average of the spin operators is time-dependent according to the equation

$$\begin{aligned} \frac{d<{\mathbf {S}}^i>(t)}{dt}= - \frac{i}{\hbar }[<{\mathbf {S}}^i>, \mathcal {H}]_t \,\,\,\,\, \end{aligned}$$

(A.16.1.2)

The MFA extended to time dependent phenomena (the so called RPA, random phase approximation) corresponds to evaluate the commutator by substituting the density matrix of the N-body system with the product of single-spin density matrices (note 4 in the present chapter). In turn, this is equivalent to substitute the products as $<S_{\alpha }^iS_{\beta }^j>$ (with $i\ne j$ and $\alpha$, $\beta = x,y,z$) with the products of the expectation values of the type $<S_{\alpha }^i> <S_{\beta }^j>$. Thus the equations of motions can be written in terms of single particle, becoming

$$\begin{aligned} \frac{d<{\mathbf {S}}^i>(t)}{dt}= <{\mathbf {S}}^i>(t)\times \mathbf {H_i}(t) \,\,\,\,\, \end{aligned}$$

(A.16.1.3)

where the average field is

$$ \mathbf {H_i}(t)= - \frac{\partial<\mathcal {H}>(t)}{\partial <{\mathbf {S}}^i>(t)}, $$

$<\mathcal {H}>(t)$ being the expectation value of the Hamiltonian in MFA.

In the light of the description given at Sect. 16.4, Eq. (A.16.1.3) corresponds to the single spin precession around an effective instantaneous, time-dependent mean field.

In the assumption of linear response to the external field, $<\mathbf {S}^i>(t)$ is the sum of the expectation value plus $\delta <\mathbf {S}^i> exp(i\omega t)$, corresponding to the deviation due to the field

$$ \mathbf {H}= \mathbf {H}_i + \delta \mathbf {H}_i e^{i\omega t} \,\,\,\,\, \mathrm { } \,\,\, $$

From Eq. (A.16.1.3), by taking into account only the terms linear in $\delta <\mathbf {S}^i>$ and in $\delta \mathbf {H_i}$, one has

$$\begin{aligned} i\omega \delta<\mathbf {S}^i>= \delta<\mathbf {S}^i>\times \, \mathbf {H_i} + <\mathbf {S}^i>\times \,\delta \mathbf {H_i} \,\,\,\,\, \end{aligned}$$

(A.16.1.4)

$<\mathbf {S}^i>\times \, \mathbf {H_i}$ being zero, the mean value of the spin operator being along $\mathbf {H_i}$. The excitations are the deviations from the time-independent values and the eigenfrequencies are obtained from Eq. (A.16.1.4) for zero external field.

By taking into account that the average molecular field is

$$ H_i^x= 2\varGamma \hbar , \quad H_i^y=0, \quad H_i^z= \sum _{j}\,'\, I_{ij} <S_z^j>_t, $$

the equation of motions for the spin deviations turn out

$$ i\omega \delta<{S_x^i}>= \biggl (\sum _{j} I_{ij}<S_z^j> \biggr ) \delta <S_y^i> $$

$$\begin{aligned} i\omega \delta<{S_y^i}>= 2\varGamma \hbar \delta<S_z^i> - \sum _{j} \biggl [I_{ij}<S_z^j> \delta<S_x^i>- I_{ij}<S_x^i> \delta <S_z^j>\biggr ] \,\,\,\,\, \end{aligned}$$

(A.16.1.5)

$$ i\omega \delta<{S_z^i}>= -2\varGamma \hbar \delta <S_y^i>, $$

namely 3N coupled linear equations. Then we turn to the collective components

$$<\delta \mathbf {S_q}>= \sum _i \delta <\mathbf {S}^i> exp[- i \mathbf {q}\cdot \mathbf {R_i}]. $$

In the paraelectric phase, being $<S_z>= 0$, for a given wave vector $\mathbf {q}$ one has

$$ i\omega \delta <{S_{\mathbf {q}}^x}>= 0 $$

$$\begin{aligned} i\omega \delta<{S_{\mathbf {q}}^y}>= 2\varGamma \hbar \delta<S_{\mathbf {q}}^z> - I_{\mathbf {q}}<S_{\mathbf {q}}^x> \delta <S_{\mathbf {q}}^z> \,\,\,\,\, \end{aligned}$$

(A.16.1.6)

$$ i\omega \delta<{S_{\mathbf {q}}^z}>= -2\varGamma \hbar \delta <S_{\mathbf {q}}^y> $$

with $I_{\mathbf {q}}= \sum _{i,j} I_{ij} exp[ -i \mathbf {q}\cdot (\mathbf {R_i}- \mathbf {R_j})]$.

As usual, the excitation frequencies are given by the secular equation

$$\begin{aligned} \left( \begin{array}{ccc} i\omega &{} 0 &{} 0 \\ 0 &{} i\omega &{} -2\varGamma \hbar + I_{\mathbf {q}}<S_{\mathbf {q}}^x> \\ 0 &{} 2\varGamma \hbar &{} i\omega \end{array}\right) = 0. \end{aligned}$$

(A.16.1.7)

One of the eigenfrequencies pertains to the longitudinal motion along the molecular field (along $S_x$) while two frequencies describe the precessional motion of the pseudo-spin around the mean field.

For temperature well above the transition temperature the only excitation involves the tunnelling frequency $\omega _{2,3}^2(\mathbf {q})= 4\varGamma ^2$. On decreasing temperature one has

$$\begin{aligned} \omega _{2,3}^2(\mathbf {q})= \left( \frac{I_0<S_z>}{\hbar }\right) ^2 + 2\varGamma \left( 2\varGamma - \frac{I_{\mathbf {q}} <S_x>}{\hbar }\right) , \end{aligned}$$

(A.16.1.8)

the first term being zero for $T> T_c$, while $<S_x>= tanh(\varGamma \hbar / k_BT)$ (see Eq. (16.35)).

Equation (A.16.1.8) evidences the slowing down of the frequencies and the instability limit for $q=0$ in correspondence to the maximum value of the Fourier transform of the interaction, the one for $q=0$. On approaching $T_c$, for $q=0$ one can expand the above equation in power of $(T- T_c)$, then writing

$$ \omega _{2,3}^2(q=0)= \left( \frac{\partial \omega _{2,3}^2(q=0) }{\partial T}\right) _{T=T_c} (T- T_c)= \frac{\varGamma ^2I_{q=0}}{k_BT_c^2 cosh^2\left( \frac{\varGamma \hbar }{k_BT_c}\right) } \left( \frac{T-T_c}{T_c}\right) \equiv $$

$$\begin{aligned} \equiv A_{q_c=0} \left( \frac{T-T_c}{T_c}\right) , \end{aligned}$$

(A.16.1.9)

indicating a dynamical critical exponent $\gamma = 1$, as expected.

According to Eq. (A.16.1.9), the pictorial behaviour (derived from Eq. (A.16.1.5) without assuming $<S_z>= 0$) of the frequencies are sketched below (see also Chap. 15)

As regards the response to the external field, by resorting to Sect. 15.4 and adding at hand a damping factor , the generalized susceptibility is written

$$\begin{aligned} \chi ''(\mathbf {q},\omega )\propto \frac{N\mu _e^2}{(\omega (\mathbf {q})^2- \omega ^2)^2 + \gamma (\mathbf {q})^2\omega ^2}, \end{aligned}$$

(A.16.1.10)

The modes have resonant character for small damping while turn to relaxational modes for strong damping.

Two comments are in order. The tunnelling integral being expected mass-dependent, one can realize why the deuteration in KDP increases the transition temperature from 123 K to 213 K.

For the cubic lattice of dipoles at distance a, from

$$ \mathbf {I_q}= 2I[cos(q_xa) + cos(q_ya) + cos(q_za)], $$

by expanding $\mathbf {I_q}$ and $\omega _{2,3}^2(\mathbf {q})$ around $q=0$ Eq. (A.16.1.9) becomes

$$\begin{aligned} \omega _{2,3}^2(\mathbf {q})= A_{q=0} \frac{T-T_c}{T_c}+ 4\varGamma ^2 \frac{a^2 q^2}{6}, \end{aligned}$$

(A.16.1.11)

a dispersion relation for the pseudo-spin excitations consistent with the general form of the q-dependence discussed at Sect. 15.4.

It is also reminded that when $\mathbf {I_q}$ is maximum in correspondence to a zone-boundary wave vector $\mathbf {Q_{BZ}}$ then the transition involves the crossover to an antiferroelectric phase, the order parameter is the sublattice polarization and the crystal cell doubles below $T_c$.

Finally we just mention that when the maximum of the Fourier transform of the interaction is neither at $q=0$ nor at $\mathbf {Q_{BZ}}$, the transition can involve an incommensurate phase, in which the order parameter of the critical variable (e.g. the expectation value of the pseudo-spin or the lattice displacement of an atom) is not commensurate with the underlying lattice.

A few words about the order-disorder ferroelectrics below $T_c$ can be added. In the ferroelectric phase both $<S_x>$ and $<S_z>$ are different from zero and the fictitious effective field is in the xz plane. From the equations for the spin deviations and the secular equations one can obtain

$$\begin{aligned} \omega _{2,3}^2(\mathbf {q})= \left( \frac{I_0 <S_z>}{\hbar }\right) ^2+ 4\varGamma ^2 \left( 1- \frac{\mathbf {I_q}}{I_0}\right) . \end{aligned}$$

(A.16.1.12)

The critical mode is at $q=0$ and $\omega _{2,3}(q = 0)\propto I_0<S_z>$. The temperature dependence of the polarization related to $<S_z>$ follows. In fact, the effective field $(H_x+ H_z)$ below $T_c$ implies the eigenvalues $\pm W= \pm [\varGamma ^2 + (I_0<S_z>)^2]^{1/2}$.

The partition function being $Z= 2 cosh(W/ k_BT)$, the polarization is

$$ P= \frac{1}{k_BT} \frac{\partial lnZ}{\partial H}= \frac{I_0<S_z>}{2W} tanh\left( \frac{W}{k_BT}\right) \propto $$

$$ \propto tanh\biggl [ \frac{[\varGamma ^2 + (I_0<S_z>)^2)]^{1/2}}{k_BT} \biggr ]\mathrm { }. $$

For exhaustive presentation of the issues related to Sect. 16.4 and particularly to this Appendix, the book by Lines and Glass or the one by Blinc and Zeks are suggested.

Appendix 16.2 Distribution of Correlation Times and Effects around the Transition

For a system of permanent dipoles with single-particle response of Debye character Eq. (16.13) holds and then

$$\begin{aligned} \varepsilon '(\omega )= \varepsilon (\infty ) + \frac{\varepsilon (0)- \varepsilon (\infty ) }{1+ \omega ^2\tau ^2} \end{aligned}$$

(A.16.2.1)

$$ \varepsilon ''(\omega )= \frac{(\varepsilon (0)- \varepsilon (\infty ))\omega \tau }{1+ \omega ^2\tau ^2} \,\,\,\,\, $$

where $\tau (T)= \tau _0 exp(\varDelta E/k_B T)$ is the relaxation time.

In mono-dispersive crystals, where a single correlation time occurs, these equations can be used for the MFA dielectric response measured at $q=0$, with (see Appendix 15.1)

$$\begin{aligned} \tau _p= \tau _{q=0}= \frac{\tau (T)}{1- I_{q=0}\chi ^0} \,\,\,\,\, \end{aligned}$$

(A.16.2.2)

Real dielectrics are hardly mono-dispersive and rather exhibit a distribution of $\tau 's$. that has to be taken into account, particularly around the transition from the disordered to the ordered phase. Empirical account of dielectric dispersion and absorption measurements in poly-dispersive systems can be given by using for the dielectric constant the following relations:

$$ \varepsilon '(\omega )= [\varepsilon '(0)- \varepsilon (\infty )] \frac{1+ bZ}{1 + 2bZ + Z^2} $$

$$\begin{aligned} \varepsilon ''(\omega )= \varepsilon '(\omega ) \frac{aZ}{1+bZ} \,\,\,\,\, \end{aligned}$$

(A.16.2.3)

where $a= sin(\pi B/2)$, $b= cos (\pi B/2)$ and $Z = (\omega \tau _p)^B$, while B measures the width of the distribution of the relaxation times, with $0< B\le 1$.

These equations are known as Cole-Cole relationships and result from the integration over $\tau$ of Eq. (A.16.2.1) with a distribution function of the form

$$\begin{aligned} y(\tau )= \frac{1}{\pi } \frac{sin(B\pi )}{x^B + x^{-B}+ 2 cos(B\pi ) } \,\,\,\,\, \end{aligned}$$

(A.16.2.4)

with $x= \tau /\tau _p$, $\tau _p$ being the correlation time measured with the homogeneous electric field (see Eq. (A.16.2.2)), with the critical behaviour

$$ \tau _p\propto \left( \frac{T- T_c}{T_c}\right) ^{-\varDelta }, $$

for $T\rightarrow T_c^+$. The critical exponent is $\varDelta = \gamma = 1$, in the MFA. For $B= 1$ in Eq. (A.16.2.3) one again obtains the Debye relations, consistent with the MFA susceptibility at $q=0$.

For $B\ne 1$, by plotting $\varepsilon ''(\omega )$ versus $\varepsilon '(\omega )$ and then $[\varepsilon ''/(\varepsilon ' a - \varepsilon '' b)]^{1/B}$, one can extract $\varepsilon (0)$, B and $\tau _p$. For illustration and for comparison with a 3D MFA system (return to Fig. 15.4), see the plots reported below for a 2D system with short-range interactions ($SnCl_2.2H_2O$) .

Illustrative plots of the data around the transition temperature $T_c= 219.45$ K in a ferroelectric crystal characterized by planar structure of dipoles (data from Mognaschi, Rigamonti and Menafra, Phys. Rev. B 14, 2005 (1976)). It is noted that the MFA value $\gamma = 1$ is clearly ruled out.

Problems

Problem 16.1

By applying Eq. (16.25) to permanent electric dipoles with Ising-like pseudo-spin variable v of dichotomic character, show that the Brillouin function for $<v>$ (in terms of $x= v\mathcal {E}/k_BT$) and the Curie-like law are obtained, in the assumption of no interactions.

Solution: For no interaction one rewrites

$$ <v>= \frac{v exp(v\mathcal {E}/k_BT) + (-v) exp(-v\mathcal {E}/k_BT)}{Z} $$

and from the expansion of the Brillouin function

$$ \chi ^0= \frac{<v>}{\mathcal {E}}= \frac{v^2}{k_BT}. $$

Problem 16.2

By starting from Eq. (16.2) and considering two frequency dependent in-phase and out-of phase dielectric constants, derive Eqs. (16.3) and (16.4).

Solution: By changing the variable from $\tau$ to $t'= t-\tau$ in Eq. (16.2), one writes

$$ \mathbf {D}(t)= \varepsilon (\infty ) \mathbf {\mathcal {E}_0 }cos(\omega t) + \mathbf {\mathcal {E}_0}\int _{-\infty }^t cos(\omega \tau ) g(t-\tau )d\tau = $$

$$ =\varepsilon (\infty )\mathbf {\mathcal {E}_0}cos(\omega t) + \mathbf {\mathcal {E}_0}\int _0^{\infty } cos(\omega (t-t'))g(t')dt'. $$

Thus

$$ \mathbf {D}(t)= \varepsilon (\infty ) \mathbf {\mathcal {E}_0} cos(\omega t) + \mathbf {\mathcal {E}_0}\int _0^{\infty } g(t')[cos(\omega t)cos(\omega t')+ sin(\omega t)sin(\omega t')]dt'= $$

$$ =\mathbf {\mathcal {E}_0} cos(\omega t)\biggl [\varepsilon (\infty ) + \int _0^{\infty } cos(\omega t')g(t') dt'\biggr ]+ \mathbf {\mathcal {E}_0}sin(\omega t)\int _0^{\infty } sin(\omega t') g(t') dt'. $$

Then, from

$$ \mathbf {D}(t)= \mathbf {\mathcal {E}_0}\biggl [\varepsilon '(\omega ) cos(\omega t)+ \varepsilon ''(\omega ) sin(\omega t)\biggr ] $$

Equations (16.3) and (16.4) follow.

Problem 16.3

Derive the relationship between $\varepsilon '(\omega )$ and $\varepsilon ''(\omega )$.

Solution: By Fourier transformation of Eqs. (16.3) and (16.4)

$$ g(x)= \frac{2}{\pi } \int [\varepsilon '(\omega )- \varepsilon (\infty )] cos(\omega 'x) d\omega ' $$

$$ g(x)= \frac{2}{\pi } \int \varepsilon ''(\omega ) sin(\omega 'x) d\omega ', $$

Equations (16.6) follow.

Problem 16.4

In the light of Eq. (16.9) show that without the Onsager reaction field one would predict the ferroelectric catastrophe in water.

Solution: One has $\alpha = \mu _e^2/3k_BT$ and from Eqs. (16.7) and (16.9) for $\mu _e^{H_2O}= 1.87$ Debye

$$ \varepsilon = 1+ \frac{4\pi N\mu _e^2}{3k_B(T-T_c)} $$

with $T_c= (4\pi N\mu _e^2/9k_B)\simeq 1200$ K.

Problem 16.5

From the equation of motion of a single ion in the lattice under short-range elastic constant (temperature dependent in the form $k_{sh}+ bT$) and a long range electrostatic force $k_{el} x$, derive the susceptibility and the temperature at which the frequency goes to zero, with lattice instability.

Solution: The equation of motion is

$$ m \ddot{x}+ \gamma \dot{ x}+ (k_{sh}- k_{el}+ bT) x= q E_0 e^{i\omega t}. $$

The polarization is Nqx. Then, by solving this equation

$$ \chi (\omega )= \frac{Nq}{m(\omega _0^2- \omega ^2+ i{\gamma \omega }) } $$

(see Eq. (15.47)) and

$$ \omega _0^2= \frac{b}{m} \left( T+ \frac{k_{sh}- k_{el}}{b}\right) . $$

Thus $\omega _0\rightarrow 0$ for $T\rightarrow T_c$, with $T_c= -(k_{sh} -k_{el})/b$ (return to Problem 15.3).

Problem 16.6

By referring to the vibrational motion of a diatomic chain and by considering anharmonic terms of the form $(V_4 /4!)\sum _i (x_{i+1}- x_i)^4$, show that the frequency of the $q=0$ optical mode goes to zero at the critical temperature $T_c= 2|k_1| k_2 /(V_4k_B)$, with $k_1$ and $k_2$ elastic constants for nearest neighbours and next nearest neighbours.

Solution: The equations of motion for optical and acoustic modes in the absence of anharmonic terms follow from

$$ m \ddot{x_i}= -2(k_1 +k_2)x_i+ k_1(x_{i+1}+ x_{i-1})+ k_2(x_{i+2}+ x_{i-2}), $$

yielding $\omega ^2_{ac}(q)$ and $\omega ^2_{op}(q)$ (see Eq. (14.15)).

With the anharmonic contribution, by writing the Hamiltonian in terms of the normal coordinates and averaging the acoustic ones, one derives

$$ \omega ^2_{op}(q)= \left[ \frac{V_4k_BT}{2m k_2}- \frac{4|k_1|}{m}\right] cos^2\left( \frac{qa}{2}\right) + \left( \frac{4k_2}{m}\right) sin^2(qa). $$

Thus, for $T= T_c= 2 |k_1|k_2/V_4k_B$ the lattice instability occurs.