Tunnel barrier design in donor nanostructures defined by hydrogen-resist lithography

A four-terminal donor quantum dot (QD) is used to characterize potential barriers between degenerately doped nanoscale contacts. The QD is fabricated by hydrogen-resist lithography on Si(001) in combination with $n$-type doping by phosphine. The four contacts have different separations ($d$ = 9, 12, 16 and 29 nm) to the central 6 nm $\times$ 6 nm QD island, leading to different tunnel and capacitive coupling. Cryogenic transport measurements in the Coulomb-blockade (CB) regime are used to characterize these tunnel barriers. We find that field enhancement near the apex of narrow dopant leads is an important effect that influences both barrier breakdown and the magnitude of the tunnel current in the CB transport regime. From CB-spectroscopy measurements, we extract the mutual capacitances between the QD and the four contacts, which scale inversely with the contact separation $d$. The capacitances are in excellent agreement with numerical values calculated from the pattern geometry in the hydrogen resist. We show that by engineering the source-drain tunnel barriers to be asymmetric, we obtain a much simpler excited-state spectrum of the QD, which can be directly linked to the orbital single-particle spectrum.


INTRODUCTION
Using degenerately doped silicon is the most common way to contact the active region of many types of semiconductor devices. As device scaling requires ever smaller, more abrupt and more highly doped contacts, access resistance and dopant diffusion are critical issues in device performance and device variability. Hydrogen-resist lithography in combination with gas-phase doping provides a way to pattern degenerately doped contacts down to the nanometer scale with atomically abrupt dopant profiles [1][2][3][4][5]. This technique uses the tip of a scanning tunneling microscope (STM) as a lithography tool to locally remove a hydrogen passivation layer on the Si(001):H surface [2]. Exposure of those patterns to phosphine gas enables fabrication of extremely shallow and abrupt junctions with sheet densities on the order of 2 × 10 14 cm −2 [3,4]. This enabled fabrication of phosphorus-doped wires with lithographic widths on the order of 1-2 nm that exhibit Ohmic conduction at temperatures < 100 mK [6]. Furthermore, such wires were used to contact donor quantum dots containing a few or even a single dopant atom [7][8][9], with the ultimate goal of fabricating donor-based qubits.
A drawback of the hydrogen-resist technique is that the electronic coupling to the contacted object cannot be tuned much once the device has been fabricated [7,8,10]. Moreover, the reduced dimensionality of the contacts and the random arrangement of dopants within the patterned regions lead to a peaked density of states, which can affect cryogenic transport measurements such as bias spectroscopy in the Coulomb-blockade (CB) regime [11][12][13]. It is thus important to understand how the geometry, separation and width of such degenerately doped contacts affect electrical transport.

CHARACTERIZATION OF THE POTENTIAL BARRIERS
We use a four-terminal quantum dot (QD) as an instrument to characterize potential barriers between highly doped nanoscale regions. Figure 1  with 20 nm of undoped epitaxial Si. The silicon substrate is only lightly phosphorus-doped at 5 × 10 14 cm −3 and becomes insulating below a temperature of 40 K. The sample is then removed from the UHV environment and Ohmic aluminum contacts are fabricated to the burried dopant device by electron-beam lithography and metal lift-off [4,7].
The donor QD is defined by a small (6 nm × 6 nm) island in the middle of the structure in Fig. 1 (a). The four electrodes A,B,C and D are patterned at distances of 9, 12, 16 and 29 nm from the QD, respectively. These distances are designed to cover a range in which contacts A and B act as tunneling barriers, and contacts C and D act as gate electrodes.
Measurements are performed in a dilution refrigerator at a base temperature of 50 mK. A small magnetic field of 100 mT is applied to inhibit superconductivity of the aluminum-bond pads. We use a source-measurement setup that enables us to set a voltage on each contact and simultaneously detect the currents in all four terminals. The breakdown of STM-defined potential barriers can be understood in a Fowler-Nordheim tunneling picture, where the applied bias leads to field-emission across a tilted potential barrier: Here, c 1 and c 2 are constants that depend on the barrier height and charge-carrier mass.
From this we expect a symmetric breakdown voltage that scales with 1/d. However, the planar geometry and the pointed ends of our dopant contacts lead to deviations from the triangular barrier potential that is assumed for field emission. We numerically calculate the electrostatic potential between two 6 nm wide planar contacts for the four different separations with ANSYS Maxwell, an electromagnetic field solver. The electrostatic potential drop in the low-doped region between the contact and the QD is shown in Fig. 1 (c). In addition to this, we expect a sharp step Φ 0 in the potential at the interface between the highly doped contact and the intrinsic region in between. From the calculations in Ref. [11], we estimate this barrier height Φ 0 ≈ E C − E F to be 80 meV, where E C labels the energy of the conduction band edge and E F denotes the Fermi energy of the degenerately doped contact. For small bias (V QD ≈ V G ) the barrier is therefore roughly square-shaped. At negative gate voltages V G − V QD < −Φ 0 , the tunnel barrier potential becomes more triangular with a width determined by the electric field near the gate (see Fig. 1 (e)+(f)). Because of the planar geometry, the electric field lines are denser near the gate contact and the tunnel barrier is thinner than for the same situation in a conventional Fowler-Nordheim tunneling picture with a triangular barrier.
The observed asymmetry in barrier breakdown voltage (see, e.g., trace D in Fig. 1 (b)) was previously attributed to the depletion of thin dopant contacts under strong fields [7]. While this may play a role for extreme bias conditions, the asymmetry is more likely an effect of the point-like shape of the gate contacts. In Fig. 1 (c), the dashed and dotted black lines are calculations for a situation where either gate D or the QD is made 5× wider than the other contact (see lower right inset). In the patterned structure, the QD has the same lithographic width as contact D. However, as all other gates are grounded, the electrostatic potential distribution looks like the one shown in Fig. 1 (d), which is strongly asymmetric.
Choosing V D = 0.4 V it specifically also describes the situation at the two points I and II marked by circles in Fig. 1 (b). Figure 1 (e) shows the potential barrier for I at negative gate bias. Here, the tunnel barrier is effectively thinner. Figure 1 (f) corresponds to situation II for a positive gate bias and a thicker barrier. From our measurements, we find that the field at which contacts with a width of 6 nm start showing barrier breakdown is about 15 ± 4 mV/nm, with the sign depending on the polarity of the applied voltage.
We note that there are additional effects that will modify the shape of the barrier. The boundary of the doped regions will be blurred on the order of the effective Bohr radius, a * B ≈ 3 nm. This will reduce the effective barrier width significantly for the smaller barriers. In close proximity to the doped regions, we may also expect a reduction of the potential due to image force effects. These will, however, be much smaller than for non-planar tunnelbarrier arrangements.

CURRENT FLOW IN THE COULOMB-BLOCKADE REGIME
Keeping in mind the properties of individual STM-defined potential barriers, we now probe current flow in the CB regime of the QD. For this, we fix V B = −10 mV and V C = 10 mV, while sweeping V A as source drain bias and V D as gate voltage. Figure 2 shows the corresponding maps of the four simultaneously measured currents. Red denotes an electron current flow out of the contact and blue into the contact. We use a distorted color scale that highlights small currents; gray indicates regions in which the current exceeded the range of (d) Leakage current through contact D (e.g. region marked with a triangle).
our measurement setup.
In Figure 2  An interesting situation occurs for leakage into contact C as shown in Fig. 2 (c): In the voltage range near the cross symbol, electrons flow from contact A to both contacts B and C even though the coupling is expected to be significantly lower for contact C because of the larger separation. At the same time, I C is clearly modulated by the electron occupation of the QD determined by alignment with µ A , the electro chemical potential of contact A. As contact B is the rate-limiting contact, the average electron occupation of the QD increases whenever an additional electron is able to enter the QD from A. With increasing electron number the QD states that are being filled are a bit less localized and coupling to the contacts increases. This is seen clearest in the small leakage current into contact C. For the configuration marked with the cross symbol, electrons therefore tunnel sequentially from contact A into the QD and then into B or C with a probability ratio Γ B /Γ C ≈ I B /I C = 10 3 .
In the opposite current direction, for positive V A , we observe no leakage in contact C. Here, the smaller electron occupation of the QD and the reduced effective gate voltage between D and the QD lead to a diminished Γ C .

PROBING DIFFERENT CONTACT CONFIGURATIONS
Instead of probing all four currents, we now focus our attention on current I A . We choose contact A with the smallest gap to achieve the highest sensitivity to current flow through the QD. In Fig. 3 the differential conductance G X = ∂I A /∂V X for all measurement configurations with two variable voltages is shown. Fig. 3 (a)   Here, the stronger capacitive coupling of V A to the QD also rescales the bias voltage axis in comparison to (b) and (f). We have indicated the Coulomb diamond with N electrons in all six panels of Fig. 3. If the different lever arms of the four contacts are taken into account the four diamond plots look nearly identical, with the same excited state lines appearing.
We therefore perform a more quantitative analysis of this using a constant interaction picture. The chemical potential of the QD is given by Here, C Σ = C QA + C QB + C QC + C QD is the sum over all mutual capacitances between gates and the QD. Using the experimental data, we find C Σ = e/E C = 3.2 aF, which is similar to the value in other dopant QD devices [8,9,[14][15][16]. Setting µ Q N = µ A = eV A or µ Q N = µ B = eV B allows us to express the diamond boundary line slopes ∂V X ∂V Y using the capacitances above and, by comparison with the experimentally observed slopes, to extract the mutual capacitances. These are shown as red dots in Fig. 4 (a) as a function of the separation d between contacts and the QD. The error bars quantify the variation of the slopes in the experimental data. For comparison, we use the pattern in the STM image in Fig. 1 (a) as an input to the electromagnetic field solver and calculate the capacitance matrix numerically. For this, we assume metallic electrodes with a thickness of 1 nm. The result is shown as blue squares in Fig. 4 (a). We find excellent agreement between the capacitances extracted from the measured diamonds and those calculated using the geometry of the STM pattern. This indicates that the nanoscale geometry of the donor device remains intact throughout all in − situ and ex − situ processing steps.
With the capacitive coupling of each of the contacts we can convert the applied voltages to chemical potential energies. This is shown in Fig. 5 (a) Fig. 3 (e), with the Coulomb-blockaded regions highlighted in green. We find three diamonds with a size close to E C and one that is about 10 meV larger. This difference is indicative of the level spacing in the QD with some uncertainty due to the fact that the dot capacitance C Σ decreases with decreasing electron occupation of the QD [7,8]. The excited-state lines beyond the boundary of the CB region are due to alignment of QD states with µ B as the latter is the current-limiting contact. The blue regions between the red excited state lines are regions of negative differential conductance (NDC). This is clarified by a vertical trace for N electrons on the QD (see arrow in Fig. 5 (a)). Figure 5 (b) shows the corresponding current trace in I A . When the barrier potential is tilted by changing µ B the transmission is tuned asymmetrically as discussed above. This is indicated by the roman numerals referring to the two situations shown in Figs. 1 (e) and (f) but now for contact B.

as in
In region I, the higher transmission of B renders the source-drain coupling of the QD more symmetric, allowing a larger current through the QD. The systematic appearance of NDC in this region is explained by a peaked density of states (DOS) in the 6 nm wide source-drain contacts [8,[11][12][13]15]. For a contact with a flat DOS, we would expect a stepwise increase in the current except for cases in which a long-lived higher-lying excited state of the QD blocks transport [17]. In our case we see a reduction of the current by about 20% immediately after each step, which is indicative of a peak in the DOS of contact B near µ B . This is also consistent with calculations for 6.1 nm wide leads in Ref. [11] in which the DOS is found to vary on the same order of magnitude.
For the excited states in Fig. 5 (a), we first note that the asymmetric coupling of contacts A and B is expected to greatly simplify the observed excitation spectrum of the QD. The average occupation probability of the QD is tuned predominantly by µ A because of its much stronger coupling to the QD. As the electron number N is thus practically constant along vertical traces, we expect to directly probe the single-particle energy spectrum of the QD, except where it is interrupted by the Coulomb gap. In other words, above the CB diamond with N electrons, we observe excitations of the N-electron QD state as usual. However, below the same diamond we expect to see hole-like excitations of the N-electron QD state rather than electron excitations of the (N-1)-electron state.
With this in mind we label the reappearing orbital levels in the single-particle spectrum of the QD with colored lines. We find clear evidence of paired levels below the CB diamonds with a splitting of about 10 meV and indicate this with two shades of the same color. For the electron excitations above the CB diamonds the pairing is not as clear and other pair assignments than the ones chosen may be possible. Places in which an orbital level is expected to reoccur without observing an excited state line are marked with a dot of the expected color. The observed pairing is interesting in view of a possible valley-orbit splitting in donor-based Si QDs [8,16,18]. The magnitude of the splitting in our case is three times larger than the valley splitting previously observed for a donor cluster [16].
We note that for the N+1 and N+2 electron state we also see vertical excited-state lines indicative of alignment with µ A (highlighted by the dashed lines in Fig. 5 (a)). This could be explained by orbitals that are localized more closely to contact B and thus exhibit a more symmetric tunnel coupling to both leads.

CONCLUSIONS
We used an STM-defined four-terminal dopant QD with four different tunnel barriers to characterize the capacitive and the tunnel coupling of nanoscale dopant contacts. We showed that the field enhancement near the apex of narrow dopant leads is an important effect that influences both barrier breakdown and the magnitude of the tunnel current in the CB transport regime.
Using CB spectroscopy, we were able to determine the capacitive coupling of the contacts to the QD and found excellent agreement with capacitances simulated from the geometry patterned by STM. Furthermore, we showed that by engineering the source/drain tunnel barriers to be asymmetric, we obtain a much simpler excited-state spectrum of the QD, which can be directly linked to the orbital single-particle spectrum.
Our experiments demonstrate that although electrical tunability of STM-defined dopant devices may be more limited than that for top-gate-defined quantum structures, the atomic precision of STM patterning makes up for this. The STM allows us to engineer dopant devices at the sub-nanometer scale, which are shown to preserve their properties throughout the subsequent processing steps.