Quantum ferroelectricity in ${\mathrm{K}}_{1-x}{\mathrm{Na}}_x\mathrm{Ta}{\mathrm{O}}_3$ and $\mathrm{K}{\mathrm{Ta}}_{1-y}{\mathrm{Nb}}_y{\mathrm{O}}_3$

The effect known as ferroelectricity arises when forces between polarizable ions in a solid produce a spontaneous displacement of these ions which results in a lattice polarization below some characteristic (Curie) temperature. Fluctuations in this polarization may be thermally induced as in the case of classical ferroelectrics, or if the Curie temperature is near O K, the fluctuations can be due to quantum-mechanical zero-point motion. The term "quantum ferroelectric" is applied to those systems where fluctuations in the polarization result from the zero-point motion. Experimental determinations of variations in the dielectric constant, spontaneous polarization, and elastic compliance as a function of temperature and impurity concentration are reported for ${\mathrm{K}}_{1-x}{\mathrm{Na}}_x\mathrm{Ta}{\mathrm{O}}_3$ and $\mathrm{K}{\mathrm{Ta}}_{1-y}{\mathrm{Nb}}_y{\mathrm{O}}_3$, and these results show that the physical properties of quantum ferroelectrics differ from those of classical ferroelectrics in the following ways: First, for a quantum ferroelectric, the transition temperature depends on impurity concentration (i.e., on an effective order parameter) as $T_c\propto {(x-x_c)}^{\frac{1}{2}}$, as opposed to $T_c\propto (x-x_c)$ for the classical case. Second, the inverse dielectric constant varies with temperature as ${\epsilon }^{-1\mathrm{}}\propto T^2$ for the quantum-mechanical case, instead of ${\epsilon }^{-1\mathrm{}}\propto T$. Finally, the distribution of transition temperatures in a given macroscopic sample with a Gaussian impurity concentration distribution is $p\left(T_c\right)\propto T_c\exp (-\alpha T_c^4)$ for the quantum ferroelectric, as opposed to a Gaussian for the classical situation. These results are in agreement with previous theoretical predictions of some of the distinguishing characteristics of quantum ferroelectricity.