1.3: Periodic Boundary Conditions

The definition of a crystalline lattice is an infinite set of points, vectors, or unit cells, but all real crystals are finite in size. The best quality single crystals consist of grains that contain 10¹⁹-10²¹ unit cells, so the lattice is, at best, a quasi-infinite set. Such dimensions, however, make computations of electronic structures and other properties intractable, so an approximation is needed to lower the order of the lattice set while keeping important characteristics of the crystal. Periodic (Born-von Karman) boundary conditions are designed to make physical problems for crystalline structures reasonably solvable and involve selecting a large, finite subset of lattice points for any crystalline lattice.

Consider the following 2-d lattice in which any lattice vector is \({\boldsymbol T}_{mn} = m{\boldsymbol a}_{1} + n{\boldsymbol a}_{2}\) for all integers m and n. The lattice vectors a₁ and a₂ outline the basic primitive cell for this lattice, outlined in red. The finite region for which periodic boundary conditions apply is designated by the set {\({\boldsymbol T}_{n_{1}n_{2}}\): 0 ≤ n₁ < N₁; 0 ≤ n₂ < N₂} for large integers N₁ and N₂ and is emphasized by green:

Figure 1.24: Expansive 2-d lattice showing one primitive unit cell and a larger region corresponding to periodic boundary conditions.

Then, for all functions f(r) of the crystalline structure, application of periodic boundary conditions means that f (r + \({\boldsymbol T}_{N_{1}N_{2}}\)) = f (r). In other words, the lattice vector \({\boldsymbol T}_{N_{1}N_{2}}\) behaves like the identity T₀₀. As a result, the subset of the quasi-infinite lattice remains an abelian group. Applied to a 1-d lattice, the periodic boundary condition converts a quasi-infinite chain of lattice points into a ring of N points:

Figure 1.25: Periodic boundary conditions applied to a 1-d chain of lattice points is similar to a ring of N points.

Likewise, the 2-d plane is mathematically transformed into a torus, and so on. Now, the electron density ρ(r) of a crystal has the full periodicity of the lattice, \(\rho\left({\boldsymbol r} + {\boldsymbol T}_{n_{1}n_{2}} \right) = \rho({\boldsymbol r})\) for all integers n₁ and n₂. On the other hand, the electronic wavefunctions (crystal orbitals) do not, but they must obey the periodic boundary conditions, i.e.,

\[\psi\left({\boldsymbol r} + {\boldsymbol T}_{n_{1}n_{2}} \right) = e^{i\omega T_{n_{1}n_{2}}}\psi({\boldsymbol r}) \nonumber \]

but

\[\psi\left({\boldsymbol r} + {\boldsymbol T}_{N_{1}N_{2}} \right) = e^{i\omega T_{N_{1}N_{2}}}\psi({\boldsymbol r}) = \psi({\boldsymbol r}). \nonumber \]

The resulting finite lattice set allows for effective computations of electronic states.