Extended Hubbard model for mesoscopic transport in donor arrays in silicon

Arrays of dopants in silicon are promising platforms for the quantum simulation of the Fermi-Hubbard model. We show that the simplest model with only on-site interaction is insufficient to describe the physics of an array of phosphorous donors in silicon due to the strong intersite interaction in the system. We also study the resonant tunneling transport in the array at low temperature as a mean of probing the features of the Hubbard physics, such as the Hubbard bands and the Mott gap. Two mechanisms of localization which suppresses transport in the array are investigated: The first arises from the electron-ion core attraction and is significant at low filling; the second is due to the sharp oscillation in the tunnel coupling caused by the intervalley interference of the donor electron's wave function. This disorder in the tunnel coupling leads to a steep exponential decay of conductance with channel length in one-dimensional arrays, but its effect is less prominent in two-dimensional ones. Hence, it is possible to observe resonant tunneling transport in a relatively large array in two dimensions.

Figure 1

A 2D donor array coupled to a source and a drain under a bias $V_{\text{SD}}$. The chemical potential of the leads can be varied by a gate voltage. Electrons from the source can tunnel through a many-body state of the array that is delocalized along a path that connects one side of the array to the other.

Figure 2

Density plot of the central cell-corrected multivalley ground state wave function in the (100) plane. This wave function is more sharply peaked at the donor's site compared with the Kohn-Luttinger wave function [19] due to the strong attraction of the central cell potential. The oscillation in density is due to the Bloch part of the wave function.

Figure 3

Tunnel coupling $t$ and long range interactions $V$ and $W$ for various donor separations along the silicon crystal axes [100], [110], and [111]. The long range interactions are considerably larger than the tunnel coupling for all donor separations.

Figure 4

A density plot of the single site energy, scaled by the binding energy of an isolated neutral donor, in a $10\times 10$ array on a square lattice ($i$ and $j$ are the row and column indices of the sites in the array, respectively). The nearest-neighbor separation is $4.6\mathrm{nm}$. There is a huge variation in the single site energy due to the long range attraction from all the ion cores in the array.

Figure 5

Addition energy spectrum for a $3\times 4$ donor array as (a) on-site interaction is increased to its true value of 43.86 meV while long range interaction is neglected, and (b) with on-site interaction at $43.86\mathrm{meV}$ and long-range interaction increased to its true value of 123 $\mathrm{meV}\times \mathrm{nm}$. The on-site interaction opens the Mott gap, and the intersite interaction leads to broadening of the lower and upper Hubbard bands.

Figure 6

Electron number distribution in the array at quarter filling with (a) no long range interactions and (b) full long range interactions.

Figure 7

(a) Conductance spectrum [scaled by $g_T=e^2\mathrm{\Gamma }/\left(\hslash kT\right)]$ at 4 K for a $3\times 4$ array when (a) long range interactions are neglected and (b) long range interactions are included. The positions of the peaks match the addition energy spectrum in Fig. 5, where elastic scattering of the electrons from the leads to the arrays is allowed. The width of the peaks is approximately $kT$.

Figure 8

(a) The scaling with the channel length in 1D of the conductance peaks at various filling: single electron (diamond), quarter filling (circle), and half filling (square); $T=100$ mK. (b) Conductance spectrum of a 1D chain with 10 donors, which reveals how the peaks decrease when moving away from half filling.

Figure 9

(a) Two cubic unit cells of silicon whose centers are separated by 4.6 nm along the [110] direction. (b) Probability distribution of the tunnel coupling between two donors each occupying a substitutional site within its cubic unit cell. The donors are randomly placed at the sites according to a uniform distribution.

Figure 10

Probability distribution of the conductance peak at half filling, scaled by its corresponding value in the perfectly ordered case, for a (a) $1\times 4$ chain and (b) $4\times 4$ array. The arrays are oriented along the [110] axis with an average nearest neighbor separation of 4.6 nm. Each donor is randomly distributed within a cubic unit cell according to a uniform distribution. These results are obtained by exact diagonalization.

Figure 11

(a) The scaling of the median value of the conductance distribution (at half filling) with system size for (a) a 1D chain obtained by the noninteracting approximation (diamond), restricted Hartree-Fock mean field approximation (triangle), hole-Hubbard approximation (square), and exact diagonalization (circle), and (b) a 2D $N\times N$ array obtained by the noninteracting approximation (diamond), restricted Hartree-Fock mean field approximation (triangle), and hole-Hubbard approximation (square). The exact result for 1D (circle) is included in the bottom figure for a comparison with the mean field result for 2D.

Figure 12

A density plot of the single site energy, scaled by the binding energy of a neutral donor, in a $10\times 10$ array on a square lattice when (a) the contribution from the image charge is excluded and (b) included ($d=4.6$ nm). The image charges lead to a large shift in the magnitude of the single site energy (note the difference in the two colorbar scales) but there is little change in the energy variation.

Figure 13

A density plot of the single site energy in a $1\times 10$ array when (a) the contribution from the image charge is excluded and (b) included ($d=4.6$ nm).