Trapping Layers Prevent Dopant Segregation and Enable Remote Doping of Templated Self-Assembled InGaAs Nanowires

Selective area epitaxy is a promising approach to define nanowire networks for topological quantum computing. However, it is challenging to concurrently engineer nanowire morphology, for carrier confinement, and precision doping, to tune carrier density. We report a strategy to promote Si dopant incorporation and suppress dopant diffusion in remote doped InGaAs nanowires templated by GaAs nanomembrane networks. Growth of a dilute AlGaAs layer following doping of the GaAs nanomembrane induces incorporation of Si that otherwise segregates to the growth surface, enabling precise control of the spacing between the Si donors and the undoped InGaAs channel; a simple model captures the influence of Al on the Si incorporation rate. Finite element modeling confirms that a high electron density is produced in the channel.

dry etching employing CHF 3 /SF 6 chemistry. Finally, the samples were etched for 10 s in a highly dilute buffered HF solution to remove oxide residues inside the patterned features and smoothen the mask surface. Prior to the growth, samples were degassed at a manipulator set temperature of 400 • C for 2 h. They were then transferred to the growth chamber and annealed further at 630 • C for 10 min. Following the annealing, GaAs supports were grown at the same temperature at an equivalent 2D GaAs growth rate of 1 Å/s with an arsenic BEP of 4×10 −6 Torr. The Al flux used for the marker layer growth is much smaller than the Ga flux such that no perturbation of the growth rate is observed.
In both samples studied, the manipulator temperature was lowered to 480 • C for Si doping. Nominally undoped spacer layers of GaAs or AlGaAs/GaAs were grown at the same temperature. Samples were then heated to 540 • C for InAs nanowire growth, The In flux corresponded to an InAs equivalent growth rate of 0.3 Å/s, and the As BEP was increased to 7×10 −6 Torr. The nanowire growth time was 4 min, and then the sample was then cooled for GaAs shell capping, which was performed at 430 • C at a Ga growth rate of 1 Å/s and As pressure of ≤ 1 × 10 −5 Torr. A thin marker AlGaAs marker layer was introduced during the shell growth, and then the sample was cooled down under an As flux.
The AlGaAs markers and the InGaAs nanowire are shown by HAADF-STEM in Figure   S1. Fig. S1: HAADF-STEM (left) and EDS maps (mid, right) of a representative branch of Sample-1. Scale bars are 70 nm.

S2 Sample preparation and analysis conditions
Specimens for APT analysis were prepared using standard FIB lift-out and sharpening procedures 2,3 with the analysis direction perpendicular to (111) planes. The final specimens were sharp needle-shaped tips with diameters less than 100 nm. APT was performed using LEAP 5000XS (CAMECA, Madison, WI) with a 355 nm laser at 250 kHz pulse frequency, a 0.004-0.006 ions/pulse detection rate, a background temperature of 30 K, and laser pulse energies between 0.5 and 0.8 pJ. Reconstructions and further analysis were conducted using the commercial software package IVAS. The tip profile method was applied, in which the SEM images of the tips before analysis were imported to determine the reconstructed radius as a function of analyzed depth.

S3 Details of modeling and genetic algorithm
The model used here is similar to that used to model dopant incorporation in vapor-liquidsolid (VLS) nanowires. 6,7 The dependence of the effective solubility of Si on Al concentration is represented in an effective partition coefficient k, defined as the ratio between the bulk and surface solubility (k(θ Al ) ≡ c bulk Si /c surf Si ), which takes the form k(θ Al ) = c 0 + c 1 θ Al , where Si /c surf Si , and c 1 captures the change in the partition coefficient due to the presence of Al. We then solve for the partition coefficient for GaAs and the Al-induced relative enhancement k/c 0 that best describes the observed Si distribution (black line in Figure   3(b)).

S3.1 Description of empirical model
The dopant incorporation rate is given by 8 where θ surf is the Si concentration at the growth surface in cm −3 , J is the incident dopant flux, k out is the desorption rate coefficient, and I d is the net incorporation rate across the growth surface. In our simple model, I d is a product of an Al-concentration dependent partition coefficient g(θ Al ) and the surface Si concentration θ surf (x): which is valid when the surface coverage of Si is much less than a monolayer. Further assuming that desorption can be neglected, the dopant incorporation rate simplifies to: The time evolution of surface Si concentration will thus be reflected in the spatial distribution. By assuming that the nanowire growth rate v is constant, we can eliminate the time dependence of the differential equation and write where k(θ Al ) ≡ g(θ Al )/v, and θ Al is the bulk Al concentration measured by APT.
Given that changes in the relevant chemical potentials governing dopant incorporation can be linearized for sufficiently small perturbations in composition, our simple empirical model assumes the Si incorporation rate is proportional to the Si surface concentration (as noted above), and that the incorporation rate increases linearly with Al surface concentra-tion. We represent these dependencies in the partition coefficient k, defined as the ratio between the bulk and surface solubility (k = c bulk Si /c surf Si ), which takes the form where c 0 = c GaAs Si /c surf Si , and c 1 captures the change in the partition coefficient due to the presence of Al. We then solve for the partition coefficient of Si on pure GaAs (c 0 ), and the Al-induced relative enhancement k/c 0 that best describes the observed Si distribution, which can be defined as: The physical picture of the Al-enhancement in Si incorporation is presented in Figure   3(c) of the main text. In Figure 3(c)i), excess Si accumulates on the surface of the growing nanowire after the Si shutter is opened, leading to a gradual increase in Si incorporation (purple region in Figure 3(b)). This picture follows from from Eq. (4) -(6) in the absence of an Al flux, in which case k can be simplified to k(θ Al ) = c 0 . Substituting k into Eq. (4), we find that dθ surf dx (x) = J v − c 0 θ surf (x) ≈ J v , and substituting k into Eq. (6), we find that θ bulk = c 0 θ surf . Combining these equations, θ bulk = (c 0 J v ) x ≡ l x, where the slope l can be directly derived from the increase in Si concentration upon opening the shutter, again assuming that the coverage is much less than a monolayer. When the Al shutter is opened (blue region in Figure 3(b)), the Si begins to incorporate more rapidly, creating a spike in the Si concentration. As the excess surface Si becomes depleted, the incorporated Si concentration decreases rapidly, as observed in Figure 3(c)(ii) → (iii). Because the Si shutter closes when the Al shutter is opened, J v = 0 in this region. Therefore, change in surface Si concentration as a function of position can be written as: where x Si is the position that Si flux stops.

S3.2 Description of genetic algorithm
Since the master equation Eq. (7) is not explicitly differentiable by the fitting parameters, optimization methods based on gradient descent are not applied here. Instead, we used a genetic algorithm 9 to find the best fit of the measured profile of Si bulk concentration.
Briefly, a genetic algorithm is an algorithm in which the solution is found by sequentially evolving a population of potential solutions to minimize the loss function. Here, we define the loss as a weighted L1 norm: In each iteration, the 20% of the population of parameters c 0 , c 1 , x Si that gives the smallest loss are selected as "elite", while a "mutation" is applied to the remaining 80% of the population, by adding a vector from a Gaussian distribution to the "elites". The new population of parameters is filled back into the loop and evaluated with the same loss function. The iteration is terminated when the distribution of each parameter has converged and the loss does not decrease further.
The fit parameters in Figure 3(b) also describe well the Si profile on the right-hand facet as shown in Figure S7.

S4.1 Details of finite element simulation
While the simulation is based on the composition and morphology extracted from APT and TEM analyses, it differs from the structure of Samples 1 and 2 in two ways. First, the nanomembranes analyzed in by APT analysis were ∼50 nm in width so that the entire upper faceting supporting the InGaAs nanowire could be captured within the APT field of view.
The simulated nanowires are positioned on wider (∼80 nm) nanomembranes, which is necessary for this approach to achieve a sufficiently high In concentration and the desired electron localization. 10 The nanowire cross-section in Figure 4(d) is similar to that demonstrated in a previous study. 1 Second, the APT samples were grown with a high Si concentration to improve the signal-to-noise of the dopant mapping, whereas the simulations use a lower active Si concentration of 2×10 18 cm −3 . Unnecessarily high Si doping can result in dopant compensation, strain, clustering, and additional carrier scattering, 11-13 as well as carrier accumulation near the nanowire-nanomembrane interface (Supporting Information Figure S8).
The key experimental advance reported here is the localization of the Si dopants with an AlGaAs layer to create a dopant-free spacer layer, which remains feasible at lower doping levels. Finally, the 2 nm AlGaAs layer and the thicker GaAs layer in the simulation are representative of the experimentally measured structure.

S4.2 Choice of the GaAs spacer thickness
A GaAs spacer layer was used to separate dopants from the nanowire channel. The thin AlGaAs trapping layer and the GaAs spacer layer diminish electron transfer to the InGaAs wire, so their combined thicknesses must be chosen to enable sufficient electron accumulation in the nanowire. The simulations in Figure S10 show that a 2 nm trapping layer and 3 nm spacer enable electron accumulation in the nanowire.