What a Born effective charge actually measures

Ask what the charge on the copper atom in Cu₂ZnSnS₄ is and you will get a different answer from every charge-partitioning scheme, because there is no unique way to carve a continuous electron density into atoms. The Born effective charge sidesteps the question. Instead of asking how much charge sits on an ion, it asks how much the macroscopic polarisation P changes when that ion is displaced:

Z*_{κ,αβ} = (Ω / e) ∂P_α / ∂u_{κ,β}

Here Ω is the unit-cell volume, u is the displacement of ion κ, and the indices make it a tensor: pushing an atom along one axis can polarise the crystal along another. Equivalently, Z* is the force on the ion produced by a unit macroscopic electric field. The two definitions agree because both are second derivatives of the same energy, once with respect to a displacement and a field, once in the other order.

Why it is not just the nominal charge

In a purely ionic picture Z* would equal the formal charge: +1 for copper, −2 for sulphur. In real covalent or mixed-bonding crystals it can be far larger, because moving an atom drags the bonding charge along with it, and that moving charge counts too. Anomalously large Born charges are a fingerprint of the electronic structure, and they are what make a material’s infrared response strong.

One constraint always holds. Displacing the whole crystal rigidly cannot polarise it, so the Born charges of all the atoms in the cell must add up to zero, tensor by tensor. This is the acoustic sum rule, and it is the first thing to check in any calculation: a residual sum says how far the numbers can be trusted.

Where it shows up

Born charges enter directly into the splitting between longitudinal and transverse optical phonons at the zone centre. The long-range Coulomb field set up by a longitudinal mode costs extra energy, so the LO frequency sits above the TO frequency by an amount that grows with Z*² and shrinks with the electronic dielectric constant. Get Z* right and computed infrared and Raman spectra line up with experiment; get it wrong and every polar mode is off.

For photovoltaic absorbers like the kesterites this matters twice over: Z* controls how strongly the lattice screens charged defects, and how strongly carriers couple to polar phonons, which limits their mobility.

How it is computed

There are two standard routes from first principles. Density-functional perturbation theory obtains Z* as a by-product of the same linear-response calculation that gives the phonons. Alternatively one applies a small finite electric field, evaluates the polarisation with the Berry-phase method, and reads Z* off the forces or the polarisation change. Both start from the same ground state and should agree; when they do not, the culprit is usually k-point sampling, because polarisation converges more slowly than the energy.

With José Coutinho and Vítor Torres I computed the Born effective charge tensors of Cu₂ZnSnS₄ in Thin Solid Films 535, 311 (2013).