Revisiting the theory of crystal polarization: The downside of employing the periodic boundary conditions

Abstract

Periodic boundary condition (PBC) is a standard approximation for calculating crystalline materials properties. However, a PBC crystal is not the same as the real macroscopic crystal, therefore, if applied indiscriminately, it can lead to erroneous conclusions. For example, unlike other extensive observables such as total energy, the polarization of a macroscopic crystal cannot always be described by a PBC model, because polarization is inherently nonlocal and strongly dependent on surface terminations, irrespective of crystal size, and moreover, the symmetry of the macroscopic crystal can be altered when the PBC is applied to a macroscopic crystal. We demonstrate in this paper that the polarization of a macroscopic crystal receives contributions from both the repeating bulk units and the crystal surfaces, which must be treated on an equal footing. When the combined system of the bulk and its surfaces are taken into account, materials traditionally classified as nonpolar can, in fact, admit polar symmetry, thus explaining why experimentalists have observed polarization in some nominally ``nonpolar'' systems. Our study, thus, clarifies that polarization can only exist in polar group systems and that apparent violations of Neumann's principle reported in some recent works originate from misinterpreting bulk PBC crystal as intrinsic macroscopic crystal, ignoring the contribution from the surfaces. We demonstrate that when the full bulk-plus-surface system is considered, the crystal polarization and symmetry is fully consistent with Neumann's principle.

Figures (5)

• Figure 1: The difference between an periodic crystal (top) and a real infinitely large crystal (bottom), illustrated using a one-dimensional charged chain as an example, is that in terms of symmetry, the periodic crystal belongs to the nonpolar symmetry group $D_{\infty}$, whereas the actual real crystal effectively exhibits polar symmetry group $C_{\infty}$.
• Figure 2: One-dimensional charge chain. The box on the top and the box on the bottom represent different choices of neutral unit cells. (without losing generality, we choose charge $q=1$ )
• Figure 3: A one-dimensional chain with the black square representing the unit cell. When the anion moves by half a lattice vector along the black arrow, the chain changes from (a) to (b).
• Figure 4: (a) The primitive cell $L$ of the zinc-blende structure. The configuration on the left is denoted as $L_1$. By moving the anion along the (111) direction by $(1/2, 1/2, 1/2)$, we obtain the configuration on the right, denoted as $L_2$. (b) The conventional cell $H$ of the zinc-blende structure. The left configuration is denoted as $H_1$ and the right as $H_2$. In the figure, green spheres represent the positive charge centers, and pink spheres represent the negative charge centers.
• Figure 5: Two different phase transition paths from $H_1$ to $H_2$.The black arrows indicate the direction of anion movement. $C_i$ represents the macroscopic crystal corresponding to $H_1$. (a) The final state obtained from $C_i$ moving along $Path1$ is $C_f$; (b) the final state obtained from $C_i$ moving along $Path2$ is $C'_f$.