Tunable coupling and isolation of single electrons in silicon metal-oxide-semiconductor quantum dots

TL;DR

This paper demonstrates the ability to tune the tunnel coupling between single electrons in silicon MOS quantum dots over a wide frequency range, enabling improved control for quantum computing.

Contribution

It introduces a method to achieve and control tunable tunnel coupling in silicon quantum dots, a long-standing challenge in the field.

Findings

Tunable coupling up to 13 GHz achieved.

Tunable tunnel rates below 1 Hz demonstrated.

Charge sensing effectively discriminates charge states.

Abstract

Extremely long coherence times, excellent single-qubit gate fidelities and two-qubit logic have been demonstrated with silicon metal-oxide-semiconductor spin qubits, making it one of the leading platforms for quantum information processing. Despite this, a long-standing challenge in this system has been the demonstration of tunable tunnel coupling between single electrons. Here we overcome this hurdle with gate-defined quantum dots and show couplings that can be tuned on and off for quantum operations. We use charge sensing to discriminate between the (2,0) and (1,1) charge states of a double quantum dot and show excellent charge sensitivity. We demonstrate tunable coupling up to 13 GHz, obtained by fitting charge polarization lines, and tunable tunnel rates down to below 1 Hz, deduced from the random telegraph signal. The demonstration of tunable coupling between single electrons in a silicon metal-oxide-semiconductor device provides significant scope for high-fidelity two-qubit logic toward quantum information processing with standard manufacturing.

Figures

c shows the current through the SET, electrostatically defined using gates ST, LB and RB, that is used as charge sensor and as an electron reservoir. The highly regular coulomb peak spacing indicates a well defined quantum dot, which has a constant charging energy of approximately 0.9 meV. We extract a gate capacitance of 13 aF, in agreement with a simple parallel plate capacitor model. We form a double quantum dot between the confinement barriers CL and CR, using the gates P1 and P2 to tune the quantum dot potentials. Bt and BR are used to control the tunnel coupling between the quantum dots

a and 2b show the double quantum dot charge stability diagrams of the charge sensor response as a function of V*P2* and V*P1* for weak (V*Bt* = 2.9 V) and strong (V*Bt* = 3.6 V) coupling. Horizontal and vertical blue lines indicate the loading of an additional electron from the SET to quantum dots D1 (located under the gate P1) and D2 (located under P2) respectively, while diagonal yellow lines indicate electron transitions between the two quantum dots. We do not observe more charge transitions at voltages lower than the measured range (see Supplemental Figure 1) and we conclude that the doubl

c with $\epsilon>$ 0). This deviation can also not be explained by a modified model that includes valley states, considering an adiabatic detuning sweep and assuming zero temperature [zhao2018coherent ](#bib.bib39). While these measurements were done adiabatically, the elevated temperature of 0.44 K can cause a non-negligible population of valley or other excited states. These excited states can cause a charge transition at a different detuning energy, thereby giving rise to a deviation. A large tunnel coupling can increase the relaxation rate of these excited states and thus decrease their po

Full text

Silicon, tunnel coupling, quantum dots

A first step toward the required control has been the demonstration of tunable coupling in a double quantum dot system operated in the many-electron regime, where gaining control is more accessible owing to the larger electron wave function tracy2010double ; lai2011pauli . More recently, exchange-controlled two-qubit operations have been shown with three-electron quantum dots yang2019silicon . However, tunnel couplings between single electrons that can be switched off and turned on for qubit operation still remain to be shown in SiMOS.

In this work we show a high degree of control over the tunnel coupling of single electrons residing in two gate-defined quantum dots in a SiMOS device. The system is stable and no unintentional quantum dots are observed. We are able to measure charge transitions using a sensitive single-electron-transistor (SET) as charge sensor and characterize the system in the single-electron regime. From a comparison of charge stability diagrams of weakly and strongly coupled double quantum dots, we conclude that we control the tunnel coupling by changing quantum dot location. We show that we can effectively decouple the double quantum dot from its reservoir and control the inter-dot tunnel coupling of the isolated system with a dedicated barrier gate. We quantify the tunability of the coupling by analyzing charge polarisation lines and random telegraph signals, and find tunnel couplings up to 13 GHz and tunnel rates down to below 1 Hz.

I Results and discussion

Figure 1a shows a scanning electron micrograph (SEM) of a SiMOS device nominally identical to the one measured and Fig. 1b shows a schematic cross-section of the quantum dot region along the dashed line in Fig. 1a. A high quality wafer is realized sabbagh2019quantum with a 100 nm 28Si epilayer with an 800 ppm residual 29Si concentration veldhorst2014addressable , covered by 10 nm thermally grown SiO2. Ohmic contacts are made by defining highly doped n*++* regions by phosphorus-ion implantation. We use an overlapping gate integration scheme angus2007gate ; borselli2015undoped ; zajac2016scalable and use palladium (Pd) gates, which have the beneficial property of small grain size brauns2018palladium . The gates are electrically isolated by an Al2O3 layer grown by atomic layer deposition. The sample is annealed at 400 *∘*C in a hydrogen atmosphere to repair e-beam induced damage to the silicon oxide and to reduce the charge trap density kim2017annealing ; nordberg2009enhancement .

Figure 1c shows the current through the SET, electrostatically defined using gates ST, LB and RB, that is used as charge sensor and as an electron reservoir. The highly regular coulomb peak spacing indicates a well defined quantum dot, which has a constant charging energy of approximately 0.9 meV. We extract a gate capacitance of 13 aF, in agreement with a simple parallel plate capacitor model. We form a double quantum dot between the confinement barriers CL and CR, using the gates P1 and P2 to tune the quantum dot potentials. Bt and BR are used to control the tunnel coupling between the quantum dots and from the quantum dots to the SET, respectively.

We characterize the charge readout sensitivity by recording the random telegraph signal (RTS) originating from the tunneling of the electrons between the (2,0) and (1,1) charge states with $\Gamma_{c}\approx$ 48 Hz, with $\Gamma_{c}$ being the inter-dot tunnel rate. The fidelity of the (2,0)-(1,1) charge readout is often limited by the sensitivity of the charge sensor to inter-dot transitions. We have designed and positioned the SET with respect to the double quantum dot in such a way that this sensitivity is maximized. Figure 1d shows a histogram of the measured readout signal, using an integration time $\tau$ = 82 $\mu$s. We fit the counts with a double Gaussian curve with $\mu_{(2,0),(1,1)}$ and $\sigma_{(2,0),(1,1)}$ the mean and standard deviation of the Gaussian distributions corresponding to the two charge states. We find $\Delta\mu_{(2,0)-(1,1)}$ $>$ 16 $\sigma_{(2,0)}$ corresponding to an excellent discrimination between the (2,0) and (1,1) charge states.

To precisely measure charge transitions, we implement charge sensing using a lock-in amplifier and apply a square wave excitation at $f_{ac}$ = 77 Hz on the gate Bt. Figure 2a and 2b show the double quantum dot charge stability diagrams of the charge sensor response as a function of VP2 and VP1 for weak (VBt = 2.9 V) and strong (VBt = 3.6 V) coupling. Horizontal and vertical blue lines indicate the loading of an additional electron from the SET to quantum dots D1 (located under the gate P1) and D2 (located under P2) respectively, while diagonal yellow lines indicate electron transitions between the two quantum dots. We do not observe more charge transitions at voltages lower than the measured range (see Supplemental Figure 1) and we conclude that the double quantum dot is in the single electron regime. In order to highlight the difference between weak and strong coupling, Fig. 2c and 2d show higher resolution maps of the (2,0)-(1,1) anticrossing.

When we set a weak inter-dot coupling, charge addition lines of D2 are barely visible in the charge stability diagram, because of the low tunnel rate between D2 and the reservoir. This indicates that the tunnel rate is significantly smaller than the excitation frequency applied to the gate. Similarly, at the (2,0)-(1,1) inter-dot transition, no transitions between the quantum dots can be observed because of the low inter-dot coupling. The loading of the first electron in D2 can only be observed from the shift of the D1 charge addition line, caused by the mutual capacitance $E_{m}$ of the two quantum dots. Only in the multi-electron regime where the quantum dot wave functions are larger and have more overlap, the coupling is sufficiently high to observe charge transition lines.

When the inter-dot coupling is strong, charge addition lines belonging to D2 are visible near the anticrossings and at high VP1, where $\Gamma_{R2}$ is increased. Additionally, $t_{c}$ and $E_{m}$ are increased and we observe a honeycomb shaped charge stability diagram, with clearly visible inter-dot transition lines, even when only a single electron is loaded on each quantum dot.

We estimate the relative location and size of the quantum dots from the gate voltage differences $\Delta$VP1(2) needed to load the second electron with respect to the first electron. We additionally use the cross-capacitances $\alpha_{r1(2)}$ of the plunger gates, determined by measuring the shift in VP1(2) of the charge transition line of the first electron in D1(2) as a function of a step in VP2(1), where $\alpha_{r1(2)}$ is the ratio between the shift and the step.

When the coupling is weak, we find $\Delta$VP1 $\approx$ 70 mV, $\alpha_{r1}<$ 0.05 for D1 and $\Delta$VP2 $\approx$ 50 mV, $\alpha_{r2}\approx$ 0.33 for D2. We conclude that we have a system of two weakly coupled quantum dots located under P1 and P2.

We now analyse how the locations of D1 and D2 change from the changes in $\Delta$VP and $\alpha_{r}$. For D1, both $\Delta$VP1 and $\alpha_{r1}$ are almost independent of the coupling. For D2, $\Delta$VP2 increases by a factor 11, from $\Delta$V${}_{P2}\approx$ 50 mV for weak coupling to $\Delta$V${}_{P2}\approx$ 550 mV for strong coupling, while $\alpha_{r2}$ increases by a factor 5, from 0.3 to 1.5. The increase in $\alpha_{r2}$ can be explained by a change in the location of D2 toward the gate P1, to a position partly below the gate Bt. This change of quantum dot location will decrease the lever arm and this is likely the cause of the increase in $\Delta$VP2. We conclude that tuning from weak to strong coupling causes the location of D2 to shift from a position mostly under P2 to a position partly below Bt, while D1 is stationary under P1. The ease with which D2 can be displaced additionally suggests that no unintentional quantum dots are formed between barrier gates.

By reducing VBR, the tunnel rate $\Gamma_{R}$ between the the SET reservoir and the quantum dots can be reduced and the loading and unloading of electrons can be prevented, resulting in an isolated quantum dot system bertrand2015quantum ; yang2019silicon . Because the reservoir is connected to room temperature electronics, decoupling the quantum dot from it may provide the advantage of reduced noise rossi2012electron . Figure 3a shows the (2,0)-(1,1) and (1,1)-(0,2) anticrossings as a function of VP2 and VP1 when the coupling is strong. Only inter-dot transition lines are present over a wide range of voltages, much larger than the $\Delta$VP extracted in the previous section. This implies that no additional electrons are loaded, as a result of a negligible coupling to the reservoir. The ability to control the inter-dot transitions of a double quantum dot without loading additional electrons provides good prospects for the operation of quantum dot arrays that are only remotely coupled to reservoirs, as proposed in quantum information architectures veldhorst2017silicon ; li2018crossbar ; taylor2005fault .

We control the tunnel coupling $t_{c}$ with the gate BT. To compensate for the influence of VBt on detuning $\epsilon$ and on-site potential $U$, we implement virtual gates using a cross-capacitance matrix baart2016single ; hensgens2017quantum ; mills2019shuttling and convert VP2, VP1 and VBt to $\epsilon$, $U$ and tc. Figure 3b shows the (2,0)-(1,1) and (1,1)-(0,2) anticrossings as a function of the new set of virtual gates Vϵ and Vtc. For both transitions the inter-dot line vanishes at low Vtc, meaning that the coupling has been largely switched off. We observe that for the (1,1)-(0,2) anticrossing, the transition line disappears at V${}_{tc}<$ 3.1 V, while for the (2,0)-(1,1) anticrossing this happens for V${}_{tc}<$ 2.95 V. The variation may come from a small asymmetry in the system.

We tune the double quantum dot to a significantly coupled regime and quantitatively analyze the system by taking charge polarization lines. Figure 3c shows charge polarization lines at high, intermediate and relatively low tunnel couplings within this regime. We measure the charge sensor response $Q$ as a function of detuning $\epsilon$ and fit the data according to a two level model that includes cross-talk of $\epsilon$ to the charge sensor and the influence of the quantum dot charge state on the charge sensor sensitivity dicarlo2004differential ; hensgens2017quantum . From the thermal broadening of the polarization line at low tunnel coupling, we extract the lever arm of Vϵ for the detuning axis $\alpha_{\epsilon}\approx$ 0.04 eV/mV, by assuming the electron temperature to be equal to the fridge temperature of 0.44 K.

For relatively low tunnel couplings, we observe in the charge polarization lines deviations from the model for a two-level system dicarlo2004differential (see the red curve in Fig. 3c with $\epsilon>$ 0). This deviation can also not be explained by a modified model that includes valley states, considering an adiabatic detuning sweep and assuming zero temperature zhao2018coherent . While these measurements were done adiabatically, the elevated temperature of 0.44 K can cause a non-negligible population of valley or other excited states. These excited states can cause a charge transition at a different detuning energy, thereby giving rise to a deviation. A large tunnel coupling can increase the relaxation rate of these excited states and thus decrease their population. As a consequence, the charge polarization lines are in agreement with the model for a two-level system dicarlo2004differential at larger tunnel couplings.

At tunnel couplings below 3 GHz, the thermal broadening of the polarization line prevents accurate fitting. Instead of the tunnel coupling energy $t_{c}$ we determine the inter-dot tunnel rate $\Gamma_{c}$, which is proportional to the square of the tunnel coupling braakman2013long ; gamble2012two ; korotkov1999continuous . We measure the RTS (Fig. 3e) at the (2,0)-(1,1) transition and fit the counts $C$ of a histogram of the tunnel times $T$ to $C=Ae^{-\Gamma_{c}T}$, where $A$ is a normalisation constant. In the measurements we have tuned Vϵ such that $\Gamma_{c(2,0)-(1,1)}\approx\Gamma_{c(1,1)-(2,0)}$.

Figure 3d shows a $t_{c}$ as a function of Vtc, demonstrating tunable tunnel coupling in the strong coupling regime and Fig. 3f shows the obtained $\Gamma_{c}$ as a function of Vtc from 1 kHz down to below 1 Hz. We note that we can further reduce the tunnel rate to even smaller rates simply by further reducing Vtc.

A change in barrier height or width results in an exponential change in tc and in $\Gamma_{c}$. When the tunnel coupling is low, D2 is located mainly under P2, and a change in Vtc has a significant impact on the barrier. Correspondingly, we observe an exponential dependence of $\Gamma_{c}$ versus Vtc. When the tunnel coupling is high, D2 is located mostly under Bt and the impact of Vtc on the barrier is vanishing. As a result we observe a seemingly linear dependence of tc versus Vtc from 3 up to 11 GHz that saturates around 13 GHz for V${}_{tc}>$ 3675 mV.

II Conclusions

We have demonstrated control over the tunnel coupling of single electrons residing in a double quantum dot in SiMOS. The inter-dot coupling of the (2,0)-(1,1) charge transition can be controlled by a barrier gate which changes the quantum dot location. We have demonstrated control over the tunnel coupling in the strong coupling regime from 3 to 13 GHz, as well as control over the tunnel rate in the weak coupling regime from 1 kHz to below 1 Hz. Achieving this degree of control in an isolated system constitutes a crucial step toward independent control over detuning and tunnel coupling for operation at the charge symmetry point martins2016noise ; reed2016reduced , and reaching the control required for large-scale quantum computation with quantum dots loss1998quantum ; veldhorst2017silicon ; vandersypen2017interfacing ; li2018crossbar ; taylor2005fault ; friesen2007efficient ; trauzettel2007spin . While SiMOS systems are often said to be severely limited by disorder, the excellent control shown here provides great prospects to operate larger arrays fabricated using conventional semiconductor technology.

III Acknowledgements

We thank Max Russ for helpful discussions. H.G.J.E, L.P and M.V. are funded by a Netherlands Organization of Scientific Research (NWO) VIDI grant. Research was sponsored by the Army Research Office (ARO) and was accomplished under Grant No. W911NF- 17-1-0274. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Office (ARO), or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.