Impact of Incorporation Kinetics on Device Fabrication with Atomic Precision

Scanning tunneling microscope lithography can be used to create nanoelectronic devices in which dopant atoms are precisely positioned in a $\mathrm{Si}$ lattice within approximately $1$ nm of a target position. This exquisite precision is promising for realizing various quantum technologies. However, a potentially impactful form of disorder is due to incorporation kinetics, in which the number of P atoms that incorporate into a single lithographic window is manifestly uncertain. We present experimental results indicating that the likelihood of incorporating into an ideally written three-dimer single-donor window is $63\pm 10\mathrm{\%}$ for room-temperature dosing, and corroborate these results with a model for the incorporation kinetics. Nevertheless, further analysis of this model suggests conditions that might raise the incorporation rate to near-deterministic levels. We simulate bias spectroscopy on a chain of comparable dimensions to the array in our yield study, indicating that such an experiment may help confirm the inferred incorporation rate.

Figure 1

Example STM lithography array. (a) STM image of 25 patterned windows, where the two marked windows correspond to (c),(e), respectively. (b) STM image of the same windows after ${\mathrm{PH}}_3$ dosing and incorporation. Each window is colored to indicate the number of P incorporation features in each window, with red corresponding to 0, green corresponding to 1, and blue corresponding to 2. The corresponding windows after incorporation for (c),(e) are shown in (d) and (f), respectively. (c) Example $w=3$ window after hydrogen depassivation lithography corresponding to the middle window in (a). (d) Illustration of the window from (c) after incorporation where a $\mathrm{Si}$:P heterodimer remains. It is likely that the corresponding $\mathrm{Si}$ adatom has diffused away. (e) Another example $w=3$ window after lithography, corresponding to the window in the lower left in (a). (f) Illustration of the window from (e) after incorporation where an ejected $\mathrm{Si}$ adatom is observed. This adatom feature is artificially broadened by a scan artifact and is likely masking any corresponding $\mathrm{Si}$:P heterodimer feature.

Figure 2

Statistics for our 50-window sample. (a) The occurrence of different depassivation feature widths in the predose images for a target width of $w=3$. These data are used to assign $w$ for a particular window. (b) The occurrence of distinct feature categories according to their height. These data are used to assign $n$ for a particular window. (c) The relative frequencies, $P_I(n|w)$ for $n=0,1,2$ and $w=2,3$. The number of windows realizing $w=2,3$ are indicated parenthetically.

Figure 3

Comparison of our incorporation model with our experimental results. (a) $P_I(n|w)$ calculated using KMC versus experimentally measured data. (b) P likely incorporates via two different pathways: a one-step process, or a two-step process with a thermodynamically unfavorable middle step. We indicate the lower end of the dimer row by underlining the $\mathrm{Si}$ label in red. Labels here refer to the notation used by Warschkow et al. [32]. (c) Incorporated P calculated using an instance of our KMC model in which the barrier of the B3 to C1 reaction is lowered by 0.1 eV. While there are minor changes in incorporation rates, our KMC model is largely robust to typical DFT errors in reaction barriers.

Figure 4

The simulated probability of incorporation as a function of both dose time and pressure generated using the kinetic model outlined in Sec. 2b. Diagonal lines represent lines of constant exposure, indicated in langmuirs (L). Even within constant exposure, moving to low-pressure and long-time regimes can dramatically increase incorporation. Near-deterministic incorporation is predicted in the extreme low-pressure, long-time limit at room temperature. Incorporation can also be improved by raising the temperature during dosing as demonstrated by the heatmaps at (a) $25{}^{\circ }\mathrm{C}$, (b) $75{}^{\circ }\mathrm{C}$, and (c) $150{}^{\circ }\mathrm{C}$.

Figure 5

Plots of the differential conductance (DC), $dI/d\mu$, as a function of the source-drain bias across the chain, $\mu$, and a uniform shift in the orbital energies, $ϵ$, caused by a nearby gate voltage. The geometry of exemplary devices are illustrated on the right side of each plot, with the source (top) and drain (bottom) biased as to pass a current across a chain of single incorporated donors (dark) including patterned sites in which incorporation failed (light). (a) DC across an ideal chain of five sites with five donors. (b) DC across a chain of five sites in which every other donor is missing. (c) DC across a chain of five sites in which two consecutive donors are missing. In all cases, the faint transitions at higher occupancies (i.e., on the left of the plots) are diminished due to the increased impact of the intersite Coulomb interaction. We note that our calculations do not include current through scattering states to clearly demarcate the transport through the eigenstates of the chain and thus our Coulomb diamonds do not have signatures of transport “above” or “below” them, as would be expected in a real experiment.

Figure 6

Common lithographic outcomes.

Figure 7

Illustrative ${\mathrm{PH}}_3$ dosing outcomes for lithographic templates.

Figure 8

Donor incorporation statistics expanded to include half dimer windows.

Figure 9

Incorporation rates for a low-temperature dosing scheme, where the dimer windows are initial dosed at 100 K, and then annealed at $500{}^{\circ }\mathrm{C}$. (a) The incorporation rate for dimer windows of width $w=2$ as a function of both dosing pressure and temperature. (b) The incorporation rate for dimer windows of width $w=3$ as a function of both dosing pressure and temperature. (c) The relative frequencies of differing numbers of incorporations as a function of dimer width with a dose pressure of $3\times {10}^{-8}\mathrm{Torr}$ and a dose time of 600 s. Notably, a significant portion of three-dimer window exposures result in two incorporation events.

Figure 10

Exhaustive set of bias spectra for a five-donor chain in which $n\le 4$ sites are missing. Here the spacing between donors is chosen to be 2.72 nm along [100]. The $x$ axes are all identical corresponding to varying ${ϵ}_{j,\sigma }$ from $-0.5$ to 0.1 eV. The $y$ axes are all identical corresponding to varying the source-drain bias from $-0.07$ to 0.07 eV. The source-chain and drain-chain tunnel rates are fixed at 9.9 meV, comparable to the tunnel coupling between two sites, ten lattice spacings apart.

Figure 11

Exhaustive set of bias spectra for a five-donor chain in which $n\le 4$ sites are missing. Here the spacing between donors is chosen to be 5.43 nm along [100]. The $x$ axes are all identical corresponding to varying ${ϵ}_{j,\sigma }$ from $-0.5$ to 0.1 eV. The $y$ axes are all identical corresponding to varying the source-drain bias from $-0.07$ to 0.07 eV. The source-chain and drain-chain tunnel rates are fixed at 9.9 meV, comparable to the tunnel coupling between two sites, ten lattice spacings apart.

Figure 12

Exhaustive set of bias spectra for a five-donor chain in which $n\le 4$ sites are missing. Here the spacing between donors is chosen to be 2.69 nm along [110]. The $x$ axes are all identical corresponding to varying ${ϵ}_{j,\sigma }$ from $-0.5$ to 0.1 eV. The $y$ axes are all identical corresponding to varying the source-drain bias from $-0.07$ to 0.07 eV.

Figure 13

Exhaustive set of bias spectra for a five-donor chain in which $n\le 4$ sites are missing. Here the spacing between donors is chosen to be 5.38 nm along [110]. The $x$ axes are all identical corresponding to varying ${ϵ}_{j,\sigma }$ from $-0.5$ to 0.1 eV. The $y$ axes are all identical corresponding to varying the source-drain bias from $-0.07$ to 0.07 eV.