A low-disorder metal-oxide-silicon double quantum dot

Electron spins in silicon devices are promising qubits for a quantum processor, defining a natural two-level system and demonstrating long spin coherence times.^1–3 Recently, 2-qubit quantum controlled-not (CNOT) operations have been demonstrated in both Si/SiGe^4,5 and metal-oxide-silicon (MOS) quantum dot systems,⁶ establishing a crucial building block for a universal quantum computer.⁷ While the MOS system allows for electron-donor interactions (enabling an avenue for quantum memories in donor states)^8,9 and demonstrates larger valley splittings (critical for high fidelity spin-selective operations),^10–14 it suffers from high disorder compared to its Si/SiGe counterpart.¹⁵ Indeed, because of disorder, the scaling-up of MOS quantum dots to multi-qubit systems has lagged behind Si/SiGe systems, where an array of nine uniform quantum dots have been achieved¹⁶ as well as the coupling of a single spin to a superconducting resonator.^17,18 Furthermore, in MOS, disorder may be unintentionally introduced to the Si/SiO₂ interface during high-energy processes like electron-beam lithography,^19,20 an essential fabrication process for quantum dot devices produced by research labs. In this work, we address the issue of disorder in MOS quantum dots by engineering and characterizing a low-disorder MOS quantum dot device where the disorder parameters critical for quantum dot devices, i.e., density of shallow traps and critical density,^19,21 approach the low-disorder levels demonstrated in Si/SiGe systems.^13,22

The metric typically cited to characterize disorder in quantum dot devices is the low-temperature electron mobility (μ), where Si/SiGe devices routinely report mobilities from 10⁵ to 10⁶ cm²/Vs, up to two orders of magnitude higher than the best MOS devices.^{13,19,20,22–26} However, the peak mobility is an insufficient metric for quantum dot quality because the peak mobility occurs at high electron densities (∼10¹² cm⁻²) where enough electrons are present in the system to screen defects and disorder.²⁷ Quantum dot devices operate in the single or few electron regime (∼10¹⁰ cm⁻²) where the peak mobility value is less applicable. Instead, the more appropriate metrics of disorder in quantum dot devices are (1) the critical density (n₀), i.e., the lowest electron density required to support a conducting pathway, and (2) the density of shallow electron traps, i.e., electrically active electron traps within a few meV of the conduction band edge and present at cryogenic temperatures.^19,21

We have fabricated and characterized a low disorder MOS quantum double-dot device, leveraging a previously published process yielding very low critical and shallow trap densities (8.3–9.5 × 10¹⁰ cm⁻²), and simultaneously very high mobilities (1.4–2.3 × 10⁴ cm²/Vs), despite exposure to high-energy processes like electron-beam lithography.¹⁹ We note that the critical densities of this starting gate stack are within a factor of 2–3 of critical densities reported in Si/SiGe devices (4.6 × 10¹⁰ cm⁻²)¹³ and are on par with the lowest critical densities reported in MOS²³ (which used thick-oxide devices). We adopt a reconfigurable device architecture pioneered in Si/SiGe devices,¹³ with three overlapping layers of gates defining two parallel conduction channels (Fig. 1). Notably, our device's first layer of gates is fabricated from degenerately doped poly-silicon instead of the typical aluminum (Fig. 1). A 25 nm layer of atomic layer deposition (ALD) aluminum oxide is then deposited over the poly-silicon gates and two more layers of aluminum gates, oxidized in an O₂ plasma for electrical insulation, are deposited to complete the device. The device is then annealed in forming gas (95% N₂, 5% H₂) for 30 min at 400 °C to passivate the interface states.¹⁹ 

For this work, we bias the device to form two quantum dots in the upper conduction channel and define a charge sensor dot in the center of the lower conduction channel. All measurements were taken at a base temperature of 305 mK in a Janis ³He cryostat with a corresponding electron temperature of ∼305 mK estimated by the line width of the Coulomb blockade peaks. Conductance data for bias spectroscopy and charge sensing measurements were taken using an Ithaco 1211 transimpedance amplifier and a Stanford Research SR830 lock-in amplifier at a frequency of 229 Hz and an AC excitation amplitude of 50 μV.

We first characterize each quantum dot individually with low-frequency bias spectroscopy measurements to extract the charging energies and lever arms of each dot. We observe regular Coulomb blockade diamonds over a wide electron occupation range which monotonically decrease in size with increasing electron occupation, reflecting the increase in the quantum dot size with increasing voltage applied to the plunger gate [Fig. 2(a)]. In this regime, the charging energies (E_C) for both dots are measured to be ≈5 meV, and we extract lever arms of 0.059 meV/mV for the right dot and 0.067 meV/mV for the left dot, respectively [Figs. 2(b) and 2(c)].

In order to interrogate the single-electron regime, we define a charge sensor dot in the center of the lower conduction channel beneath TM₂. When the charge sensor dot is biased to the edge of a Coulomb blockade peak, small changes to the local electrostatic environment result in a measurable change in current through the charge sensor dot. Figure 3 shows the charge stability diagrams measured by the charge sensor dot of the upper two quantum dots as a function of the corresponding quantum dot plunger gate voltage and an adjacent tunnel barrier voltage. Three notable features are visible in Figs. 3(a)–3(c): a set of parallel electron transitions corresponding to the quantum dot (QD), three isolated individual lines corresponding to the charging of nearby defect states (highlighted in red), and a persistent background signal from the charging of poly-silicon grains.

The set of QD transitions are strongly coupled to the plunger gate voltages and their charging energies are consistent with the charging energy measured by bias spectroscopy. In addition, they show regularity over a large number of electron transitions, indicative of low interface disorder. At sufficiently negative tunnel gate voltages, the tunnel rate can no longer keep up with the measurement scan rate and “latching” behavior is seen in the lower region of the left-most transitions. Monitoring the transition point of the latching behavior for each electron transition ensures that the tunnel rate of each electron transition remains fast enough to be detected during the scan.¹³ Thus, we demonstrate the depletion of each quantum dot to zero electrons.

Utilizing the charge sensor, we may now extract the charging energies for the first electron transition, converting the plunger gate voltage difference between the first and second electron transitions to energy using the lever arms measured from the Coulomb blockade diamonds. We extract charging energies of 7.9 meV for the right dot and 7.6 meV for the left dot. Treating each dot as a metallic disc at the Si/SiO₂ interface, we can estimate the radius (R) of each dot in the single-electron regime from the dot's measured charging energy (E_C) and calculated self capacitance: $EC=e2Cdisc$⁠, where $Cdisc=8ϵ¯ϵ0R$⁠.¹³ Here, $ϵ¯$ is given by $12(ϵSi+ϵox)$⁠, the arithmetic mean of the relative permittivity of silicon (ϵ_Si = 11.7) and silicon dioxide (ϵ_ox = 3.9).²⁷ This analysis yields radii of 37 nm for the right dot and 38 nm for the left dot, in good agreement with the effective lithographic radius of the dots (50 × 80 nm² → R = 35.7 nm).

In addition, bias-spectroscopy measurements of the left dot in the single-electron regime (not shown) reveal a conductance resonance 1.2 meV above the ground state which we attribute to the first excited orbital state. Treating the dot now as a 2D square box,¹³ we estimate the energy of the first excited state as $Eorb=3ℏ2π22m*L2$⁠, where $m*=0.19m0$ and L is taken to be $(πR2)1/2$⁠. This analysis yields E_orb = 1.3 meV, in good agreement with the measured resonance. From these data, we conclude that these dots are lithographically defined and not dominated by random disorder at the Si/SiO₂ interface.

In the background of the charge sensing signal is a persistent signal caused by the charging of individual grains in the poly-silicon depletion gate.²⁸ These charging events are only evident in the charge sensing signal and not the bias spectroscopy signal, indicating that the object being charge does not originate from the Si/SiO₂ interface. The slope of these transitions turns positive when a charge sensing measurement is performed with the poly-silicon gate and any other gate. A positive slope of these transitions is only possible when the object being charged is present on one of the gates being scanned. We emphasize that the poly-silicon transitions do not interact with or otherwise affect the charge transitions of the QDs, but simply add a noisy background signal. These “phantom dot” signatures have been observed in other similar devices and can be eliminated by keeping the poly-silicon layer thicker than the poly-silicon grain size.

The charge sensor in conjunction with the quantum dot can also be used as a local probe of defect states in the vicinity of the quantum dot. Defect states are evidenced by a single transition line slanted away from the set of parallel quantum dot transitions, highlighted in Figs. 3(a)–3(c), creating avoided crossing with the QD transitions. Scanning the QD gate voltages over a wide parameter range (limited to 4 V to avoid leakage between gates) to search for defects, we identify three distinct defect states, two in the vicinity of the right dot [highlighted in Figs. 3(a) and 3(b)] and one in the vicinity of the left dot [highlighted in Fig. 3(c)]. These defect states are observed to hold only a single electron and the location of the defect state transition relative to the QD transitions is consistent from cool down to cool down. These observations suggest that the origin of these defects is a fixed positive charge in the oxide near the interface, as opposed to a mobile ionic charge which can freely migrate through the oxide at room temperature.²⁹ Using the lithographic size of the dots, a rough estimate for the defect density can be obtained yielding $≈3×1010$ cm⁻². The estimate of the defect density from this method is consistent with the order of magnitude of the defect density measured by ensemble electron spin resonance measurements of the shallow trap density and conductivity measurements of the critical density of this material.¹⁹ 

Finally, we demonstrate the controllable formation of a quantum double dot [Fig. 3(d)]. We observe the classic “honey-comb” structure and triple-points. In this regime, there are several electrons in each quantum dot. The double-dot structure shown here demonstrates relatively weak inter-dot coupling, but we note that by tuning the tunnel barrier between the dots (TM₁), we are able to tune the inter-dot coupling from a single large dot to a double dot. We note that this double quantum dot enables a promising platform for future two-qubit and singlet-triplet qubit operations.

Another important device characteristic is charge noise. Charge noise can be measured by a quantum dot by biasing the dot to the edge of a Coulomb blockade peak, where the dot is maximally sensitive to fluctuations in the local electric field environment. By comparing the noise spectrum data at the edge of the Coulomb blockade peak with the noise spectrum taken at the top of the Coulomb blockade peak (where the device is minimally sensitive to local noise), the noise generated by the local environment can be extracted from noise generated by all other noise sources in the device and measurement circuitry.^30,31

Figure 4(a) shows the derivative of the quantum dot drain current with respect to the plunger voltage (⁠$dISD/dVPR1$⁠) across a Coulomb blockade peak. Overlaid on the same plot is the magnitude of the noise spectrum at 0.5, 1.0, and 1.5 Hz. As expected, the magnitude of the noise correlates with the absolute value of the derivative of the device current, indicating that the noise spectra are dominated by the fluctuations in the dot's chemical potential, ϵ, from local environmental noise sources.³⁰ The noise spectra at the maximum and minimum points of $dISD/dVPR1$ [indicated in Fig. 4(a) by $VSImax$ and $VSImin$] are plotted in Fig. 4(b) from 0.2 Hz to 59 Hz. The noise spectra follow a low frequency 1/f dependence, frequently observed in electronic devices and indicative of an ensemble of two-level fluctuators in the vicinity of the device.³² 

We convert the measured current noise to the equivalent potential noise felt by the quantum dot, Δϵ, using the relation $ΔIϵα=|dIsd/dVPR1|Δϵ$⁠.^30,31 Here, $ΔIϵ=SImax−SImin$⁠. For Δϵ evaluated at 1 Hz at 300 mK, we calculate a noise value of 3.4 μeV/Hz^1∕2, consistent with other reported noise values reported at 1 Hz at 300 mK in MOS and Si/SiGe quantum devices.³⁰ This noise figure is likely to decrease at dilution refrigerator temperatures as has been observed in other work.^30,33

Finally, we perform magneto-spectroscopy measurements of the N = 0 → 1 and N = 1 → 2 electron transitions to measure the valley splitting, Δ_{V S}. In silicon QD devices, electrons populate the two valley states perpendicular to the interface. For high fidelity spin selective quantum operations, the valley splitting must be large in relation to k_BT and the qubit energy.^11,34 By detecting the transition from a spin singlet configuration to a spin triplet configuration, we may experimentally measure the valley splitting.

In a perpendicular magnetic field, we tune the upper left dot to the first and second electron transitions and then monitor the position of each transition's Coulomb blockade peak as a function of the magnetic field by fitting the conductance peak to a cosh⁻² function.¹⁴ We convert the peak position from the plunger voltage to energy using the measured lever arm. In this experiment, we measure the conductance of the quantum dot instead of utilizing our charge sensor in order to bypass the noisy poly-silicon charging events which are apparent in the charge sensing signal. Figure 5(a) plots the evolution of the conductance peaks of the first two electron transitions as a function of the magnetic field. The evolution of these peaks contains contributions from the orbital and spin energies (which are both magnetic field dependent) as well as the charge offset drift (which is time dependent, but independent of the magnetic field and the same for both electron transitions). By taking the difference of the conductance peak positions, we may subtract out the orbital and charge offset drift components, leaving the difference in spin energies of the two electrons which is used to determine the valley splitting.

Figure 5(c) shows the difference in addition energies of the first and second electrons. Near B = 0, we fit out magneto-spectroscopy data to a slope of gμ_B = 116 μeV/T, the expected Zeeman spin splitting for a spin singlet configuration. Here, the g-factor is taken to be 2 (electron in Si) and μ_B is the Bohr magneton. In this configuration, both electrons occupy the lower valley state [Fig. 5(b)]. As the magnetic field increases, the difference in spin energy saturates at a value of 110 ± 26 μeV, corresponding to the electron transition to a spin triplet configuration. In this configuration, the second electron has loaded into the upper valley state. Thus, we demonstrate a valley splitting large enough in this device to support spin-selective operations at typical dilution refrigerator temperatures (∼100 mK = 8.7 μeV).³⁴ 

In conclusion, we have engineered a low-disorder MOS quantum dot device and demonstrated a promising platform for electron spin qubits. We demonstrate the controllable formation of lithographically defined individual and double quantum dots with charging energies consistent with the lithographic size of the dots. The local defect density around the quantum dots is low enough to support single electron occupation and correlates with ensemble measurements of the defect density as measured by ESR and percolation thresholds. Charge noise spectroscopy measurements show a 1/f power spectral density yielding a value of 3.4 μeV/Hz^1∕2, comparable to other Si MOS and Si/SiGe devices measured at 300 mK, and demonstrating a quiet noise environment. Finally, we measure a valley splitting of 110 ± 26 μeV, large enough to support high-fidelity spin operations. This work represents a platform for optimizing quantum dot disorder in MOS and a promising architecture for spin qubits.

See supplementary material for complete device fabrication details.

We gratefully acknowledge D. M. Zajac and J. R. Petta at Princeton University and R. M. Jock, M. Lilly and M. S. Carroll at Sandia National Laboratories for many useful discussions on device fabrication, instrumentation, and interpretation of data.

This work was supported by the NSF through the MRSEC Program (Grant No. DMR-1420541). This work was performed, in part, at the Center for Integrated Nanotechnologies, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science by Los Alamos National Laboratory (Contract No. DE-AC52-06NA25396) and Sandia National Laboratories (Contract No. DE-NA-0003525).