respect to basis completeness, as well as the usual size effects and $k$-point mesh density.

We have also studied the bandstructures with all three methods, finding that the DZP correctly determines the $\Delta$-band minima away from the $\Gamma$ point, where the SZP method does not. We show that these minima occur in the $\Sigma$ direction for the type of cell considered, not the $\Delta$ direction as has been previously reported. Having established the DZP methodology as sufficient to describe the physics of these systems, we then calculated the electronic density of states, and the electronic width of the $\delta$-layer. We found that previous SZP descriptions of these layers underestimate the width of the layers by almost 15%.

We have shown that the properties of interest of $\delta$-doped Si:P are well-converged for 40-layer supercells using a DZP description of the electronic density. We recommend the use of this amount of surrounding silicon, and technique, in any future DFT studies of these and similar systems - especially if inter-layer interactions are to be minimized.

Acknowledgements {#acknowledgements .unnumbered}
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The authors acknowledge funding by the ARC Discovery grant DP0881525. This research was undertaken on the NCI National Facility in Canberra, Australia, which is 