Electronic states and valley-orbit coupling in linear and planar molecules formed by coupled P donors in silicon

We present a theoretical investigation of the single-electron electronic structure of polyatomic phosphorus donor molecular planar structures embedded in silicon. Using an effective mass theory and multivalley envelope function representation, the effect of the valley-orbit coupling is systematically analyzed in such systems. The valley composition of the single-electron states strongly depends on the geometry of the dopant molecule and on its orientation relative to the crystallographic axes. The electron binding energy of a triatomic linear molecule is larger than that of a diatomic one oriented along the same crystallographic axis, but the energy gap between the ground state and the first excited state is not significantly different for internuclear distances from 1.5 to 6.6 nm. Three donor atoms arranged in a triangle geometry have larger binding energies than a triatomic linear chain of dopants with the same internuclear distance. Planar donor molecules are characterized by a strong polarization in favor of the valleys oriented perpendicular to the molecular plane, an effect that increases with the number of atoms in the molecule and is not present in diatomics. As a result, the amplitude of the in-plane wave function oscillations caused by the valley interference decreases which reduces the sensitivity of the electronic states to random displacements of dopants. The effect of weak spatial disorder and of the molecular orientation relative to the crystallographic axes are studied in detail for a hexagonal structure. The valley properties that we characterize are fundamental for the implementation of robust multiqubits systems on many-body states of interacting dopants.

Figure 1

Electronic energy spectra for (a) and (d) three phosphorus donors arranged in a linear chain, (b) and (e) two phosphorus donors, and (c) and (f) three phosphorus donors arranged in a triangular structure. (a)–(c) Solutions of single-valley problems neglecting the valley-orbit coupling, while (d)–(f) are for results with the valley-orbit coupling taken into account. Both diatomic and triatomic linear molecules are oriented along the crystallographic axis [100]. Numbers above curves indicate the degrees of degeneracy of corresponding energy levels. The dotted line in (d) corresponds to the energy spectrum in (f) at $b=0$. The red filled triangle in (f) designates the ground state energy that corresponds to the wave function shown in Fig. 3.

Figure 2

(a) The ground state electron density for the linear triatomic structure and (b) corresponding single-valley envelope functions and valley compositions. The energy spectra of the linear chain is marked by the dashed line in Fig. 1.

Figure 3

(a) The ground state electron density for the triangular structure and (b) corresponding single-valley envelope functions and valley compositions. The energy spectra of the triangular molecule is marked by the red triangle in Fig. 1.

Figure 4

(a) Ionization energies and (b) energy gaps between the ground state and the first excited state for the triatomic linear structure of donors (red line with round markers) and the diatomic structure (black line with rectangular markers).

Figure 5

(a) The ground state electron density for the hexagonal structure formed by six donor atoms with an edge of the hexagon aligned long [100] crystallographic axis and (b) corresponding single-valley envelope functions and valley compositions.

Figure 6

(a) The ground state electron density for the hexagonal structure formed by six donor atoms with an edge of the hexagon aligned long [110] crystallographic axis and (b) corresponding single-valley envelope functions and valley compositions.

Figure 7

The energy spectrum six phosphorus donors arranged in the hexagonal structures.

Figure 8

(a) An example of a generated mesh used in the finite-element-method-based computations for six phosphorus donors arranged in a hexagonal structure, and (b) orientations of the bulk silicon isoenergetic ellipsoids relative to the molecular orientation.

Figure 9

Possible alignments of six donor atoms in a regular hexagon structure in the crystallographic plane (001) of silicon (the small gray spheres designate silicon atoms, the large color spheres represent phosphorus impurities). Edge lengths equal $2a$.