Abstract
Atomically precise donorbased quantum devices are a promising candidate for solidstate quantum computing and analog quantum simulations. However, critical challenges in atomically precise fabrication have meant systematic, atomic scale control of the tunneling rates and tunnel coupling has not been demonstrated. Here using a room temperature grown locking layer and precise control over the entire fabrication process, we reduce unintentional dopant movement while achieving high quality epitaxy in scanning tunnelling microscope (STM)patterned devices. Using the Si(100)2 × 1 surface reconstruction as an atomicallyprecise ruler to characterize the tunnel gap in precisionpatterned single electron transistors, we demonstrate the exponential scaling of the tunneling resistance on the tunnel gap as it is varied from 7 dimer rows to 16 dimer rows. We demonstrate the capability to reproducibly pattern devices with atomic precision and a donorbased fabrication process where atomic scale changes in the patterned tunnel gap result in the expected changes in the tunneling rates.
Introduction
Atomically precise siliconphosphorus (Si:P) quantum systems are actively being pursued to realize universal quantum computation^{1} and analog quantum simulation^{2}. Atomically precise control of tunneling rates is critical to tunnelcoupled quantum dots and spinselective tunneling for initialization and readout in quantum computation^{3,4,5}, and also essential in tuning correlated states in Fermi–Hubbard simulators^{2}. Although scanning tunneling microscope (STM)patterned tunnel junctions lack the degree of tunability of topgate defined tunnel barriers in conventional semiconductor heterostructures^{6}, it was shown by Pok^{7} and Pascher et al.^{8} that engineering the dimensions of the STMpatterned nanogaps can affect the tunnel barriers and the tunnel rates in STMpatterned devices: even a ~1 nm difference in the tunnel gap separation can drastically change the tunnel barrier and transport properties in atomically precise Si:P devices^{9}. Although the exponential dependence of the resistance on the tunnel gap at the atomic scale is a well established physical phenomenon, critical challenges in fabrication have meant a systematic demonstration of the exponential dependence of the resistance on the tunnel gap separation has not been demonstrated in STM patterned devices. Here using a room temperature grown locking layer and precise control over the fabrication process, we demonstrate the expected control of the tunnel coupling in response to atomicscale changes in STMpatterned single electron transistors (SETs). In this study, we define “atomicscale control of tunneling” as achieving the predicted response in the tunneling resistance relative to a given atomicscale change in the tunneling gap. (For example, if the dimension of a tunnel gap is 11 dimer rows, and the gap is changed by 1 dimer row, there is an expected one order of magnitude change in tunneling resistance.) We mention here that reliable device metrology is possible at two stages, measuring the STM lithographic pattern dimensions on an atomically ordered surface and lowtemperature transport measurements of the resulting device. Using the naturally occurring surface lattice of the Si(100)2 × 1 surface reconstruction as an atomically precise ruler, we measure the tunnel junction gap separations based on the number of lattice counts in the surface reconstruction and demonstrate exponential scaling of the tunneling resistance where the gap is varied from 7 dimer rows to 16 dimer rows. Varying the tunnel gap separation by only ~5 dimer rows, we demonstrate a transition in SET operation from a linear conductance regime to a strong tunnel coupling regime to a weak tunnel coupling regime. We characterize the tunnel resistance asymmetry in a pair of nominally identical tunnel gaps and show a fourfold difference in the measured resistances that corresponds to half a dimer row difference in the effective tunnel gap—the intrinsic limit of hydrogenlithography precision on Si(100)2 × 1 surfaces.
In this study, we overcome previous challenges by uniquely combining hydrogen lithography that generates atomically abrupt device patterns^{10,11} with recent progress in lowtemperature epitaxial overgrowth using a lockinglayer technique^{12,13,14} and silicide electrical contact formation^{15} to substantially reduce unintentional dopant movement. These advances have allowed us to demonstrate the exponential scaling of the tunneling resistance on the tunnel gap separation in a systematic and reproducible manner. We suppress unintentional dopant movement at the atomic scale using an optimized, room temperature grown locking layer, which not only locks the dopant position within lithographically defined regions during encapsulation, but also improves reproducibility since the critical first few layers are always grown at room temperature^{12}. Furthermore, our recent development of a highyield, lowtemperature method for forming ohmic contact to burried atomic devices enables robust electrical characteriation of STMpatterned devices with minimum thermal impact on dopant confinement^{15}. With improved capabilities to define and maintain atomically abrupt dopant confinement in silicon, we fabricated a series of STMpatterned Si:P SETs, where we systematically vary the tunnel junction gap separation, and have used them to demonstrate and explore atomicscale control of the tunnel coupling. Instead of geometrically simpler single tunnel junctions, we chose SETs in this study because observation of the Coulomb blockade signature is a direct indication that conductance is through the STMpatterned tunnel junctions. We chose SET lead widths and island size to be large enough that we are in the metallic regime and avoid the complications introduced by quantization and confinement in smaller lead widths. Additionally, SETs are wellunderstood devices that are ideal for developing and validating atomicscale control of device designs and fabrication methods. They enable characterization of capacitive coupling between the various gates and device elements, the two junctions that make up an SET are fabricated in a nearly identical process and can be individually characterized, electron addition/charging energies can be measured and compared to design values, and current flowing through the SET island shows a strong exponential dependence on the junction dimensions at the atomic scale. Furthermore, SETs are exemplary structures because they are fundamental components in a number of quantum devices: they can function as DC charge sensors and are used in spin to charge conversion, qubit initialization, charge noise characterization, radiofrequency (RF)SET reflectometry, and charge pumps.
Results
Atomically precise patterning of tunnel gaps
We define the tunnel gaps with atomically abrupt edges using ultraclean hydrogen lithography while utilizing the surface lattice of the Si(100)2 × 1 surface reconstruction to quantify the tunnel gap separations with atomicscale accuracy. The Si(100)2 × 1 surface reconstruction features dimer rows of pitch 0.77 nm that can serve as a natural “atomic ruler” allowing us to define the critical dimensions with atomic precision. Figure 1 shows atomically precise STM lithography for three SET charge sensors fabricated with nominally identical source/island and drain/island tunneling gaps. In these devices we targeted an 11 dimer row tunnel gap for the source/island/drain tunnel coupling for all three devices. In this set of devices our fabrication control resulted in a mean gap of 11.0 dimer rows with a standard deviation of the mean of 0.2 dimer rows (1 sigma) (see Table 1).
Controlled variation of the tunnel gap in donorbased SETs
Figure 2a shows ebeam patterned electrical contacts overlaid on a composite of STM images from an SET hydrogenlithography pattern. Patterning of a device begins with the central region where atomic precision is required. An atomic resolution STM image is taken after patterning the central region to verify dimensions. Subsequently, the interconnect leads and contact pads are patterned and local STM images are taken after patterning each component section to verify the lithographic quality. As a part of our standard fabrication protocol, we do not take STM images of the entire completed device pattern to avoid potential atomic/molecular contamination, tip damage, additional vacuum exposure, and tipsurface interactions during lengthy large area STM imaging. Figure 2b shows an STM image of the atomically precise central region of a typical SET device after hydrogen lithography, but before phosphine dosing. P dopants only incorporate into the bright regions where the STM tip has removed H atoms from the hydrogenterminated surface and exposed chemically reactive Sidangling bonds (Fig. 2c). The planar source and drain, island (quantum dot), and gates are saturationdosed resulting in degenerate dopant densities over three orders of magnitude beyond the Mott metalinsulator transition^{16}. The island is capacitively coupled to the two inplane gates through an effective capacitance C_{G} and to the source (drain) electrodes through tunnel barriers represented by a tunneling resistance R_{S} (R_{D}) and a capacitance C_{S} (C_{D}), where each resistance is coupled in parallel with its respective capacitance (Fig. 2d). The gate voltages applied to both gates tune the local electrochemical potential of the island and modulate the source–drain current flowing through the central island. Single electrons tunnel sequentially through each barriers due to the electron addition energy (charging effect) on the island^{17} (Fig. 2e).
Figure 3 shows a series of STM images acquired following hydrogen lithography with the dimer rows of the surface reconstruction clearly visible. Although not all device drain/source electrodes are aligned to the [110] lattice direction, we observe improved edge uniformity by orienting the device in the [110] lattice direction and aligning the geometries of the critical device region (island and tunnel junctions) with the surface lattice of the reconstruction. For SETG and SETI whose tunnel gaps are in the [100] direction, we have corrected for the 45° angle relative to the [110] direction when counting the number of dimer rows in their junction gaps. While attempting to keep lead width and island size identical, we systematically increase the number of dimer row counts within the tunnel junction gap starting from a continuous wire with zero gaps up to SET tunnel gap separations of ~16.2 dimer rows, covering a large range of SET device operation characteristics. Because isolated single dangling bonds do not allow dopants to incorporate, we disregard them in quantifying the device geometry. The critical dimensions after STMimaging correction are summarized in Table 2 for all devices in this study (see Methods for details). In addition, our regular use of highresolution STM imaging over the STMpatterned device region allows us to identify atomicscale defects in the device region, such as stepedges^{18} and buried charge defects^{19}, which can potentially affect device performance.
Exponential scaling of tunneling for atomicscale changes
In Fig. 4a, the current–voltage (I–V) characteristics of WireA exhibit Ohmic behavior with a 4point resistance of 96.8 kΩ. Considering the actual STMpatterned wire geometry (approximately \(57 \pm 4\) squares between the ebeampatterned voltage contact probes, see Fig. 2a), this corresponds to a sheet resistance of 1.70 ± 0.15 kΩ in the STMpatterned electrodes, in excellent agreement with previous results on metallically doped Si:P delta layers^{20}. Given the ultrahigh carrier density and small Thomas Fermi screening length^{16} in this saturationdoped Si:P system and the relatively large island size^{21} of the SETs, we treat the energy spectra in the islands and source and drain leads as continuous (\(\Delta E \ll k_{\mathrm{B}}T\), where \(\Delta E\) is the energy level separation in the island and source and drain reservoirs) and adopt a metallic description of SET transport^{17}. The tunneling rates, \(\Gamma _{\mathrm{{S,D}}}\), and the tunneling resistances, \(R_{\mathrm{{S,D}}} = \hbar /\left( {2\pi e^2\left A \right^2D_{\mathrm{i}}D_{\mathrm{f}}} \right)\), across the source and drain tunnel barriers can be described using Fermi’s golden rule^{22}, where A is the tunneling matrix element, \(D_{\mathrm{{i,f}}}\) represents the initial and final density of states, \(\hbar\) is the reduced Plank’s constant, and e is the charge of an electron.
In the interest of clarity, we define our use of the terms tunnel coupling and tunneling rates. In the context of the work presented here, Eq. (1) relates the tunneling rates to the tunneling resistance values where the tunneling matrix elements, A, represent the tunnel couplings for our system. However, it should be noted that the term “tunnel coupling” is also widely used in the context of quantum dots, where the tunnel coupling is a measure of level broadening of the energy eigenstates on the quantum dots and can lead to a loss of electron localization on the dot in the strong tunnel coupling regime. The term is also used in analog quantum simulation where the tunneling coefficient t in the Hubbard Hamiltonian denotes the hopping energy or tunnel coupling strength between adjacent sites.
In the following, we show that the total tunneling resistance \(R_{\mathrm{S}} + R_{\mathrm{D}}\) of an SET can be extracted by measuring, at zero drain–source direct current (DC)bias, the peak amplitudes of the differential conductance Coulomb oscillations, as shown in Fig. 4b (see Supplementary Note 1 for typical I–V characteristics of the gates and source/drain leads of STMpatterned SETs). At \(V_{\mathrm{{DS}}} = 0\,{\mathrm{V}}\), the differential conductance Coulomb blockade oscillations reach peaks at \(V_{\mathrm{{GS}}} = V_{\mathrm{{GS}}}^{\mathrm{{peak}}} = \left( {N + \frac{1}{2}} \right)\frac{e}{{C_{\mathrm{G}}}}\), where \(N\) is an integer and \(\left( {N + \frac{1}{2}} \right)e\) represents the effective gating charge when the island Fermi level \(\mu _{\mathrm{{IS}}}(N)\) aligns with \(\mu _{\mathrm{S}}\) and \(\mu _{\mathrm{D}}\). At low temperatures and in the metallic regime, \(\Delta E \ll k_{\mathrm{B}}T \ll E_{\mathrm{C}}\), where \(E_{\mathrm{C}} = e^2/C_{\mathrm{\Sigma }}\) is the charging energy, and \(C_\Sigma = C_{\mathrm{S}} + C_{\mathrm{D}} + C_{\mathrm{G}}\) is the total capacitance (see Supplementary Table 1), and assuming energy independent tunnel rates and density of states in a linear response regime, Beenakker and coworkers^{23,24} have shown that the peak amplitude of the zerobias differential conductance oscillations in an SET reduces to the following temperatureindependent expression for arbitrary R_{S} and R_{D} values,
where G_{S} and G_{D} are conductances through the source and the drain tunnel barriers, \(\rho\) is the density of state in the metallic island, and the density of states in the leads is embedded in the tunneling rates.
In Fig. 4b we observe Coulomb blockade oscillations in all SETs except SETB. The small gap separation (~\(7.4\) dimer rows \(\approx 5.7\,{\mathrm{nm}}\)) in SETB is comparable to twice the Bohr radius, \(r\sim 2.5\,{\mathrm{nm}}\), of an isolated P atom in bulk Si^{25}, indicating significant wave function overlap within the gap regions between the island and the source/drain reservoir. Given that SETB does not exhibit single electron tunneling behavior (Coulomb oscillations), we estimate the resistance at the junction gaps in this device using 4point I–V measurement. As shown in Fig. 4a, SETB has a linear I–V behavior with the 4point resistance of \(136.7\,{\mathrm{k}}{\Omega}\). Subtracting the resistance contribution from the source/drain leads (~74 squares) using the estimated sheet resistance (~\(1.7\,{\mathrm{k}}{{\Omega }}\)) from WireA, we obtain a junction resistance value of \(\sim 5.5 \pm 4.5\,{\mathrm{k}}{{\Omega }}\) per junction in SETB, which does indeed fall below the resistance quantum (\(\sim {\!}26\,{\mathrm{k}}{{\Omega }}\)), and explains the absence of Coulomb blockade behavior. In SETB (Fig. 3b), the impact on electrical transport properties that results from having two contacts straddle a single atomic step edge is negligible given the small vertical offset (1 monatomic layer = 0.138 nm) compared with the expected electrical density distribution in the same direction (~2 nm for an ideal, saturationdoped Si:P monolayer)^{26,27}. We emphasize that, due to the absence of the Coulomb blockade effect, the estimated resistance at the junctions in SETB is an ohmic resistance, which should not be confused with the tunneling resistance.
For the rest of the SETs, we extract the total tunneling resistance, \(R_{\mathrm{S}} + R_{\mathrm{D}}\), from the Coulomb oscillation peak heights following Eq. (1). Figure 4c summarizes the measured junction resistance values (after sheet resistance correction from the source and drain leads) as a function of the averaged gap separations. The tunneling resistance follows a clear exponential relationship with the gap separations. It is notable that a change of only nine dimer rows gives rise to over four orders of magnitude change in the junction resistance. Increasing the gap separation over a small range (from ~7 dimer rows in the gap to ~12) dramatically changes the SET operation from a linear conductance regime (no sign of Coulomb oscillations at ~7 dimer rows separation in SETB) to a strong tunnel coupling regime (at ~9.5 dimer rows separation in SETC) to a weak tunnel coupling regime (at ~12 dimer rows separation in SETF). The relatively strong tunnel coupling in SETC (see Fig. 4d) blurs the charge quantization on the island and introduces finite conductance within the Coulomb diamonds through higher order tunneling processes (cotunneling)^{28}. In the weak tunnel coupling regime in SETF (see Fig. 4e), the Coulomb blockade diamonds become very well established. Tuning the tunnel coupling between strong and weak coupling regimes in atomic devices is an essential capability: e.g. for simulating nonlocal coupling effects in frustrated systems^{29}.
It has been found essential for capacitance modeling (see Supplementary Table 1) to add a lateral electrical seam^{30} and a vertical electrical thickness^{26} to the STMpatterned hydrogenlithography geometry (Fig. 3) to account for the Bohr radius and yield the actual “electrical geometry” of the device. We fit the total tunneling resistance (\(R_{\mathrm{S}} + R_{\mathrm{D}}\)) from SETB to SETH as a function of the tunnel gap separation by simulating a single tunnel junction’s tunneling resistance (multiplied by two to account for the presence of two junctions) using a generalized formula for the tunnel effect based on the Wentzel–Kramers–Brillouin (WKB) approximation^{31} (for detailed WKB formulation, see Supplementary Note 2: Modeling the Tunnel Barriers Using the WKB Method). Due to the linear dependence of the WKB tunneling resistance on the tunnel junction crosssectional area, we ignore the small variations in the STMpatterned junction width, \(w\), (see column 3 in Table 2) and adopt an averaged value of \(w = 12\,{\mathrm{nm}}\) in the WKB simulation. We account for the “electrical geometry” of the devices by assuming an electrical thickness of \(z = 2\,{\mathrm{nm}}\)^{26}, while treating the lateral electrical seam width, \(s\), and the mean barrier height, \(E_{\mathrm{{barr}}}\), as fitting parameters. We obtain \(100 \pm 50\,{\mathrm{meV}}\) as the bestfit barrier height (uncertainty represents two σ), which is in good agreement with the theoretically predicted range of Fermi levels below the Si conduction band edge in highly \(\delta\)doped Si:P systems, ~80 to ~130 meV, from tightbinding^{26} and density functional theory^{25} calculations. A similar barrier height value (~80 meV) has also been experimentally determined in a Fowler–Nordheim tunneling regime by Fuhrer’s group using a similar STMpatterned Si:P device^{8}. We obtain \(3.1 \pm 0.4\,{\mathrm{nm}}\) as the bestfit seam width (uncertainty represents two σ), which is in good agreement with the Bohr radius of isolated single phosphorus donors in bulk silicon (\(r\sim 2.5\,{\mathrm{nm}}\))^{25}. Using the bestfit seam width from the WKB simulation, we also find good agreement between the experimental and simulated capacitance values from the SETs (see Supplementary Note 3: Comparison between the Measured and Simulated Capacitances in STMpatterned SET Devices).
Figure 4c is a key result of this study, clearly demonstrating an exponential scaling of tunneling resistance consistent with atomic scale changes in the tunneling gap. The devices shown in Fig. 3 were fabricated in series from two different ultrahigh vacuum (UHV) STM (UHVSTM) systems with similar but nonidentical hardware platforms using the same nominal methods and processes.
Atomicscale asymmetry in precisionpatterned SET tunnel gaps
Having demonstrated atom scale control of the tunneling resistance, we now take an additional step to characterize the junction resistance difference in a pair of nominally identical tunnel junctions in SETG, where both the tunnel gaps have irregular edges and the tunnel gap separations are less welldefined when compared with the tunnel gaps in the other SETs, representing a lower bound of controllability among the SET devices in this study. We present the measured Coulomb diamonds and finite bias Coulomb oscillations in Fig. 5a and b. In Fig. 5b, the Coulomb oscillation peaks are asymmetric across the gate voltage. For positive drain–source bias, at the leading edge of the Coulomb oscillation peak of \(N \leftrightarrow N + 1\) transition, the island spends most of the time unoccupied (\(N\)). So, the total tunneling rate is limited by tunneling from the source to the island, and thus the total tunneling resistance is dominated by \(R_{\mathrm{S}}\). The other three cases are analogous. Figure 5c takes \(V_{\mathrm{{DS}}}\, > \, 0\) for instance and shows a numerical simulation (at \(T = 0\,{\mathrm{K}}\)) of \(I_{\mathrm{{DS}}}\,{\mathrm{{vs}}}.\,V_{\mathrm{{GS}}}\) at different drain–source bias. The dashed and dotted lines in Fig. 5c illustrate the asymptotic slopes at the leading and trailing edges of the Coulomb oscillation peaks at \(V_{\mathrm{{DS}}} = 0.8E_{\mathrm{C}}/e\), which also represent the tunneling current through the ratelimiting source and drain tunnel junctions, respectively, while ignoring the other junction in series. At \(T = 0\,{\mathrm{K}}\), the source and drain junction resistances can be derived from the right derivative at the leading edge, where \(V_{\mathrm{{GS}}} = V_{\mathrm{{GS}}}^{\mathrm{L}} = \left( {N + \frac{1}{2}} \right)\frac{e}{{C_{\mathrm{G}}}}  \frac{{C_{\mathrm{D}}}}{{C_{\mathrm{G}}}}V_{\mathrm{{DS}}}\), and from the left derivative at the trailing edge, where \(V_{\mathrm{{GS}}} = V_{\mathrm{{GS}}}^{\mathrm{T}} = \left( {N + \frac{1}{2}} \right)\frac{e}{{C_{\mathrm{G}}}} + \frac{{(C_{\mathrm{S}} + C_{\mathrm{G}})}}{{C_{\mathrm{G}}}}V_{\mathrm{{DS}}}\), of a Coulomb oscillation peak in \(I_{\mathrm{{DS}}}\). This is shown in Eq. (2) (for mathematical derivations, see Supplementary Note 4: Quantifying Individual Junction Resistances in a Metallic SET), again, taking positive drain–source biases for example,
See Supplementary Table 1 for the gate and total capacitances, \(C_{\mathrm{G}}\) and \(C_\Sigma\). To estimate the drain and source tunneling resistances from the Coulomb oscillation peaks that are measured at finite temperatures (Fig. 5b), we approximate the asymptotic slopes at the leading and trailing edges by fitting the leading and trailing slopes of the measured Coulomb oscillation peaks and average over a range of \(V_{\mathrm{{DS}}}\) bias (see Fig. 5d). We find a factor of approximately four difference in the source and drain tunneling resistances. Possible contributions to this resistance difference include atomicscale imperfections in the hydrogen lithography of tunnel gaps, the randomness in the dopant incorporation sites within the patterned regions, and unintentional, albeit greatly suppressed, dopant movement at the atomicscale during encapsulation overgrowth. Field enhancement near any pointed apex due to atomicscale edge nonuniformity/roughness in the dopant distribution profile can also be an important effect that influences the tunnel current in the Coulomb blockade transport regime, as has been previously suggested by Pascher et al.^{8}. Other factors that can affect the tunnel barrier and therefore cause tunnel resistance variability include changes to the local potential landscape due to buried charge defects near the device region in either the substrate or the overgrowth layer. From the exponential dependence in Fig. 4c, a factor of four corresponds to an uncertainty in the gap separation of only about half of a dimer row pitch distance, which represents the ultimate spatial resolution (a single atomic site on the Si(100)2 × 1 reconstruction surface) and the intrinsic precision limit for the atomically precise hydrogen lithography.
Discussion
The results presented here are of interest where critical device dimensions and pattern fidelity or tunnel coupling play a direct role in device performance. Complex devices such as arrays of quantum dots for analog quantum simulation have stringent requirements with respect to sitetosite tunnel coupling. While the details of the tunneling characteristics are different than an SET in the metallic regime, the fabrication methods described here are applicable to fabrication of single or few atom quantum dots and should aid in achieving a higher degree of reproducibility in those devices^{32}.
In summary, we have demonstrated the ability to reproducibly pattern devices with atomic precision, and that improved lockinglayer methods coupled with meticulous control over the entire donorbased device fabrication process resulted in STM patterned devices with predictable tunneling properties. By using the natural surface reconstruction lattice as an atomic ruler, we systematically varied the tunneling gap separations from 7 dimer rows to 16 dimer rows and demonstrated exponential scaling of tunneling resistance consistent with atomicscale changes in the tunneling gap. We emphasize that, critical fabrication steps, such as a defect and contaminantfree silicon substrate and hydrogenresist formation, atomically abrupt and ultraclean hydrogen lithography, with dopant incorporation, epitaxial overgrowth, and electrical contact formation that suppress dopant movement at the atomic scale, are all necessary to realize devices with atomic precision. This study represents an important step towards fabricating key components needed for highfidelity silicon quantum circuitry that demands unprecedented precision and reproducibility.
Methods
STMpatterned donorbased device fabrication
The Si:P SETs are fabricated on a hydrogenterminated Si(100)2 × 1 substrate (\(3 \times 10^{15}\, {\mathrm{cm}}^{  3}\) boron doped) in an UHV environment with a base pressure below \(4 \times 10^{  9}\,{\mathrm{{Pa}}}\) (\(3 \times 10^{  11}\,{\mathrm{Torr}}\)). Detailed sample preparation, UHV sample cleaning, hydrogenresist formation, and STM tip fabrication and cleaning procedures have been published elsewhere^{11,18,33}. A low 1 × 10^{−11} Torr UHV environment and contaminationfree hydrogenterminated Si surfaces and STM tips are critical to achieving highstability imaging and hydrogenlithography operation. The device geometry is defined by selectively removing hydrogenresist atoms using an STM tip in the lowbias (3–5 V) and highcurrent (15–50 nA) regime where the small tipsample separation allows for a spatially focused tunneling electron beam under the atomicscale tip apex, creating hydrogen lithographic patterns with atomically abrupt edges. For complete hydrogen desorption within the patterned regions, the typical tip scan velocity and scanline spacing are 100 nm sec^{−1} and 0.5 nm line^{−}^{1} respectively. We then saturationdose the patterned device regions with PH_{3} followed by a rapid thermal anneal at 350 °C for 1 min to incorporate the P dopant atoms into the Si surface lattice sites while preserving the hydrogen resist to confine dopants within the patterned regions. The device is then epitaxially encapsulated with intrinsic Si by using an optimized lockinglayer process to suppress dopant movement at the atomic scale during epitaxial overgrowth^{12,14}. The sample is then removed from the UHV system and Ohmiccontacted with ebeamdefined palladium silicide contacts^{15}.
Lowtemperature transport measurements
Lowtemperature transport measurements are performed using either a closedcycle cryostat at a base temperature of 4 K or a dilution refrigerator at a base temperature of ~10 mK. For SETB to SETG, the zeroDC bias differential conductance (\({G}_{0}\)) are measured using 0.1 mV AC excitation at 11 Hz. For SETH and SETI, \(G_0\) is numerically estimated from the measured DC Coulomb diamonds. We calibrate the zero drain–source bias level by mapping out complete Coulomb diamonds, where the intersections of the Coulomb diamonds represent the true zerobias condition across the source–drain leads. We extract the zerobias conductance curves (as shown in Fig. 4) from the measured Coulomb diamond diagrams. Since the effect of gate voltage compensation on the SET island’s chemical potential is insignificant under our measurement conditions at 4 K, we did not compensate \(V_{\mathrm{{GS}}}\) when measuring or calculating \({\mathrm{{d}}}{I}_{\mathrm{{DS}}}/{\mathrm{{d}}}{V}_{{\mathrm{{DS}}}}\) at the zero drain–source bias for extracting the tunnel resistance values. The gate leakage currents are on the order of ~10 pA or less within the gating range used in this study.
Characterization of STM lithographic pattern dimensions
We estimate the critical dimensions of the STMpatterned tunnel junctions in an SET from the STM topography images in Fig. 3, where the gap distance, \(d\), is the average across the full junction width, \(w,\) using both junctions. The junction width is the average over the island and the first 15 nm of the source and drain leads near the island. The hydrogen lithography and STM imaging are carried out using different tips and/or under different tip conditions. To eliminate the STM imagebroadening due to the convolution between the wave functions of the tip apex and Sidangling bonds and extract the boundary of the hydrogendepassivated surface lattice sites, we estimate the imagebroadening, \(\Delta b\), from the difference between the imaged single dangling bond size, \(b\), (fullwidth at halfmaximum) and the size of a single dangling bond lattice site, \(b_0\), where we have assumed \(b_0\) equals half a dimer row pitch (see Fig. 2c). The imagebroadening, \(\Delta b = b  b_0\), is then used to correct the critical dimensions that are read out from the halfmaximum height positions in the STM topography images.
Theoretical modeling of SETs
The theoretical analysis of the transport through SETs is based on an equivalent circuit model (see Fig. 2d) under a constant interaction approximation. The analytical expressions regarding the equilibrium drain–source conductance are derived using the standard Orthodox theory under a twostate approximation^{21,34}.
Data availability
All relevant data are available upon request from the authors.
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Acknowledgements
This work was sponsored by the Innovations in Measurement Science (IMS) program at NIST: Atombased Devices: single atom transistors to solidstate quantum computing. This work is also funded in part by the Department of Energy Advanced Manufacturing Office Award Number DE‐EE0008311. We thank Neil Zimmerman, Josh Pomeroy, Garnett Bryant, Daniel Walkup, and John Kramar for valuable comments and discussions. This work was performed in part at the Center for Nanoscale Science and Technology NanoFab at NIST.
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X.W. and J.W. conceived of the experiment under the guidance of R.M.S. In situ device fabrication was performed by X.W. and J.W. Ex situ sample preparation and electrical contact fabrication was carried out by P.N., S.W.S, M.D.S. Jr, J.W., and X.W. Lowtemperature transport measurement was carried out by R.V.K., A.M., and, X.W. Data analysis and calculations were carried out by X.W., J.W., and R.M.S. The manuscript was prepared by X.W., J.W., and R.M.S. with input from all the authors.
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Wang, X., Wyrick, J., Kashid, R. et al. Atomicscale control of tunneling in donorbased devices. Commun Phys 3, 82 (2020). https://doi.org/10.1038/s4200502003431
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DOI: https://doi.org/10.1038/s4200502003431
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