Hyperfine interactions in silicon quantum dots

A fundamental interaction for electrons is their hyperfine interaction (HFI) with nuclear spins. HFI is well characterized in free atoms and molecules, and is crucial for purposes from chemical identification of atoms to trapped ion quantum computing. However, electron wave functions near atomic sites, therefore HFI, are often not accurately known in solids. Here we perform an all-electron calculation for conduction electrons in silicon and obtain reliable information on HFI. We verify the outstanding quantum spin coherence in Si, which is critical for fault-tolerant solid state quantum computing.

Figure 1

Electron spin density in a (100) plane for bulk Si with an extra electron in the conduction band minimum at $k_z$. The vertical axis is $z$ and the color scheme runs from red (high density) to violet (low density), following the rainbow sequence. The circular high density spots are the Si atomic sites.

Figure 2

Squares give the calculated hyperfine parameters versus $1/N$, where $N$ is the number of Si atoms in the supercell. The asterisks give the interpolated values expected for natural Si ($~$$5\%$ of ${}^{29}\mathrm{Si}$), with $a\approx 1.9$ mT and $b\approx 0.06$ mT. Solid straight lines are $a_N=(37/N)$ mT and $b_N=(1.1/N)$ mT, and give good fits to the data points. These results are the basis of a local spin density approximation used in our quantum dot calculations, similar to LDA in electronic calculations.

Figure 3

Calculated $\eta$ for different supercell sizes (red squares). The average value and standard deviation are given by the lowest (red) asterisk and respective error bar. Experimental results are given by the upper (blue) horizontal dashed lines, following the average values of Refs. 23 and 24, labeled S&W (1956) and Wilson (1964), respectively. The error bar for Wilson was obtained as an upper limit of the error estimated from Ref. 24, where ${\eta }_{\hspace{0.5em}\mathrm{Si}}$ is given in an expression related to ${\eta }_{\hspace{0.5em}\mathrm{Ge}}$. The horizontal position of the asterisks is arbitrary, as they represent average values. The estimated value of $\eta \gtrsim 300$ in Ref. 25 is off the scale here.