Ab initio calculation of valley splitting in monolayer δ-doped phosphorus in silicon

Abstract

The differences in energy between electronic bands due to valley splitting are of paramount importance in interpreting transport spectroscopy experiments on state-of-the-art quantum devices defined by scanning tunnelling microscope lithography. Using vasp, we develop a plane-wave density functional theory description of systems which is size limited due to computational tractability. Nonetheless, we provide valuable data for the benchmarking of empirical modelling techniques more capable of extending this discussion to confined disordered systems or actual devices. We then develop a less resource-intensive alternative via localised basis functions in siesta, retaining the physics of the plane-wave description, and extend this model beyond the capability of plane-wave methods to determine the ab initio valley splitting of well-isolated δ-layers. In obtaining an agreement between plane-wave and localised methods, we show that valley splitting has been overestimated in previous ab initio calculations by more than 50%.

Electronic properties ofδ-doped Si:P and Ge:P layers in the high-density limit using a Thomas-Fermi method

“…The rate of decay of the gradient of the potential in the z direction and the depth of the potential are sensitive to ML doping density (see Table IV), donor atom configuration 28,33 and basis set size. 36 We note that including XC effects in TB1D and DFT1D (PWO) 24 and short-range effects in DFT1D (PWO) 24 does not affect the magnitude of the valley splitting. And, that when a large basis set is used effects of basis set size become relatively minor.…”

“…The path Γ to Y is equivalent to the path Γ to X by the four-fold symmetry of the Si(001) doping plane and therefore not shown. The path Γ to Z is affected by zone-folding 36,49 because of the height of the tP supercell in the z direction and therefore also not shown. The change from an FCC unit cell to a tP supercell projects the conduction valleys in the ±z directions to Γ.…”

“…10,31 Complimentary DFT models of Si:P δ-layers and quantum wires have been proposed using the siesta and vasp packages, with localised atomic orbital (LAO) bases [32][33][34][35] and a planewave basis. 36 However, the applicability of these models to realistic device architectures is restricted by the N 3 scaling in calculation time associated with DFT. Historically, the effective mass theory (EMT) has also been used to study n-type δ-doped layers in Si 37 and was recently put on a firm theoretical footing for Si:P δ-layers.…”

Electronic structure of phosphorus and arsenicδ-doped germanium

“…Density functional theory (DFT) calculations on δ-doped germanium are conducted by adapting the general approach previously applied to Si:P by ourselves [16][17][18] and others [19][20][21] to the specific requirements of Ge:P and Ge:As. All calculations are performed using the SIESTA software.…”

Large-scale atomistic density functional theory calculations of phosphorus-doped silicon quantum bits

“…Although some of this disagreement can be attributed to geometrical effects of dopant placement which we will not explore in detail (see instead Refs. [33] and [16]), the accuracy of the description of dopant electronic structure contributes to these discrepancies. Additionally, electrically detected magnetic resonance (EDMR) [34] has called into question a theoretical picture of the scattering of electrons in a two-dimensional electron gas (2DEG) from dopants in silicon [35,36].…”