The shape effect and its consequences for polar surfaces and for heterogeneous catalysis.

In this paper we develop the shape effect, which is relevant for crystalline materials whose size is larger than that of the thermodynamic limit. According to this effect the electronic properties of one surface of a crystal depend upon all of its surfaces, on the overall shape. At first, qualitative mathematical arguments are presented for the existence of this effect based on the conditions for the stability of polar surfaces. Our treatment explains why such surfaces are observed even though earlier theory indicated that they should not exist. Then, models are developed from which it is found computationally that changing the shape of a polar crystal can substantially alter the magnitude of its surface charges. Apart from surface charges, it follows that the crystal shape will also significantly affect bulk properties, most notably polarization and piezoelectric responses. Additional model calculations show a strong shape effect on the activation energy for heterogeneous catalysis primarily through local surface charges rather than a non-local/long range electrostatic potential.