Abstract
Substitutional donor atoms in silicon are promising qubits for quantum computation with extremely long relaxation and dephasing times demonstrated. One of the critical challenges of scaling these systems is determining interdonor distances to achieve controllable wavefunction overlap while at the same time performing high fidelity spin readout on each qubit. Here we achieve such a device by means of scanning tunnelling microscopy lithography. We measure anticorrelated spin states between two donorbased spin qubits in silicon separated by 16 ± 1 nm. By utilising an asymmetric system with two phosphorus donors at one qubit site and one on the other (2P−1P), we demonstrate that the exchange interaction can be turned on and off via electrical control of two inplane phosphorus doped detuning gates. We determine the tunnel coupling between the 2P−1P system to be 200 MHz and provide a roadmap for the observation of twoelectron coherent exchange oscillations.
Introduction
Controlling the interaction strength between two quantum particles lies at the heart of quantum information processing. One must have access to classical control fields that, while tuning the environment of quantum particles, are sufficiently decoupled from them as to not disturb their quantum states^{1}. Physical systems ranging from trapped ions^{2}, single photons^{3}, superconducting circuits^{4} and semiconductor quantum dots^{5} have demonstrated this exquisite level of control. In 1998, Loss and Divincenzo^{6} proposed the use of a controllable exchange interaction in semiconductor quantum dots to perform a twoqubit logic gate. In the same year, Kane^{7} proposed how this could be achieved in donorbased devices. Here, the wavefunction overlap between two electrons on neighbouring donor atoms placed ~20 nm apart is controlled using an exchange gate between them. Harnessing this exchange interaction to perform a universal twoqubit quantum logic gate is the next step for donorbased architectures.
Three approaches exist for donor qubits: a controlledphase (CZ) gate^{8}; the controlledrotation (CROT) gate^{9} and a direct twoelectron SWAP operation^{10}. While the first two protocols require the use of high frequency microwave fields for electron spin resonance^{11}, a direct twoelectron SWAP necessitates the ability to turn on and off the exchange interaction between the electrons over orders of magnitude for high fidelity twoqubit operations. Notably, while the extent of a single donor wavefunction is well understood^{12,13,14}, modelling the exchange coupling between two donor electrons is more complex^{12,13,14,15,16,17} due to multivalley interference effects^{18}. To this end, a critical challenge for donorbased architectures is to know the distance required between the donors in order to turn the exchange interaction on and off with external gates^{15,16}.
To date, two main methods for donor placement in silicon exist: ion implantation^{19} and atomic manipulation via scanningtunnellingmicroscopy (STM) hydrogen lithography^{20}. Despite much success in accessing randomly placed donor spins, ion implantation has yet to demonstrate donor placement precision below ~6 nm, while STM lithography has demonstrated donor placement down at the atomic scale^{21}.
In this paper we use STM lithography that allows both the precision placement of donor atoms for direct and independent spinmeasurement of electrons near a readout structure and, most importantly, the control of the exchange interaction between them. We measure the anticorrelated spin states that arise due to the formation of twoelectron singlettriplet states as a function of their wavefunction overlap, which is controlled by inplane detuning gates. By observing the onset of these anticorrelated spin states as a function of detuning pulse voltage and time, we estimate the magnitude of tunnel coupling between the two donor qubits, and provide a roadmap towards coherent exchange gates for future devices.
Results
Independent spin readout of a 2P−1P system
In the original Kane proposal an exchange gate between the donors was suggested to directly tune the exchange coupling between the qubits^{7}. Recent tight binding simulations have shown that it is difficult to tune the exchange energy in a 1P−1P donor configuration using such a gate^{22}. Instead, it has been proposed that the exchange energy could be tuned over five orders of magnitude^{22} by confining electrons in an asymmetric 2P−1P configuration and by utilising ‘tilt’ control using two opposing detuning gates rather than a central Jgate; see Fig. 1a. Motivated by these predictions with estimates for the required interdonor separation, in this paper we demonstrate the ability to control exchange coupling in donorbased qubits at the (1,1)–(2,0) charge region using a 2P−1P donor system.
The device, shown in Fig. 1a, was patterned using STM hydrogen lithography. The qubitsL and R (left and right) are composed of two donors and one donor respectively, determined by examining the size of the lithographic patches (insets in Fig. 1a) as well as their measured charging energies^{23,24} (see Supplementary Figs. 1, 6). Three gates {G_{L}, G_{M}, G_{R}} control the electrostatic environment of the qubits which are tunnel coupled to a larger readout structure made up of approximately 1000 P atoms which serves as a singleelectrontransistor (SET) charge sensor. The SET quantum dot is operated with a sourcedrain bias of 2.5 mV, has a charging energy of ~6 meV and is controlled predominantly via gate G_{S}. All data in this paper was taken in a dilution refrigerator with a base temperature of ~100 mK (electron temperature ~200 mK).
Figure 1b shows the charge stability map of the 2P−1P device. Current peaks running diagonally correspond to charge transitions of the SET island. Two sets of breaks in the SET current peaks are observed with different slopes and correspond to electron transitions from either L or R to the SET island. An avoided crossing (triplepoint) between these two transitions (dashed white line) indicates the region where electrons can tunnel between L and R, in this case at the (1,1)–(2,0) charge transition. Only one more charge transition corresponding to L is observed at lower gate voltages leading to the assignment of the charge regions.
The direct measurement of anticorrelated electrons hinges upon the ability to independently measure their spin states^{25}. To measure the spin of R we employ an energyselective tunnelling technique^{23} where the electrochemical potential of the singleelectron transition from the 1 → 0 charge state is split by the Zeeman energy in a static magnetic field B_{z}; see blue arrows in Fig. 1c. Whether the electron is able to tunnel to the SET reservoir therefore depends on its spin state, i.e. the readout is a spindependent unloading mechanism from the qubit to the SET. This readout technique is employed for the electron at R because the (1,1) region for this qubit borders the 1 → 0 charge states.
For L we use a variant of this method, first reported Watson et al.^{26}. The charge transition for this qubit borders the 1 → 2 charge states, but because the chemical potential from the oneelectron spinup and down states to the twoelectron singlet state are also split by the Zeeman energy, a similar readout method is allowed (red arrows in Fig. 1c^{25}). In this case, we utilise a spindependent loading mechanism from the SET to L. The combination of these two distinct readout techniques avoids the need to pulse over large voltages in order to reach the (1,1)–(2,0) charge transition. Both readout methods are equivalent and give rise to a current ‘blip’ through the SET which is used to discriminate between spinup and down electrons. The average readout fidelity of spinup and down are estimated to be 96.2 ± 1.1% and 97.6 ± 2.1% for qubitL and R respectively (see Supplementary Figs. 2, 3 and Table 1 for full analysis).
Importantly, the readout of each electron must be completely independent of the spinstate of the other. That is to say, the exchange energy at the detuningposition where readout is performed must be vanishingly small, such that no spin flipflops occur during the readout window. This is demonstrated in Fig. 1e, f. For these measurements we prepare one of two states,
where \(\left {i,j} \right\rangle\) indicates the spin state i and j on qubitL and R respectively. Loading spindown for one qubit is performed deterministically as a result of the spin readout protocol. Spinup cannot be loaded deterministically; instead a random mixture of spin up and down is loaded by plunging the qubit far below the SET fermilevel. After initialisation we pulse inside the (1,1) charge region midway between the two readout positions (green square in Fig. 1b) and wait for up to 10 s for the randomly loaded electron spin to decay to spin down. Sequential spinreadout of L and then R is performed, in that order, to minimise the effect of the shorter T_{1} of qubitL. The spinup fractions show relaxation of the qubit initially loaded with random spin, with T_{1} times measured to be 2.9 ± 0.5 s and 9.3 ± 2.4 s for electrons on L and R respectively at B_{z} = 2.5 T. Importantly, the electron initially loaded as spindown shows no significant spinup fraction during this time, demonstrating that at these readout positions there is no significant spin−spin interaction over ~10 s.
Controllable exchange of precision placed donors
The realisation of a twoqubit logic gate hinges on the ability to controllably turn on and off an interaction between quantum particles. We show this here by pulsing towards the (1,1)–(2,0) charge transition where an exchange interaction between the two electrons arises as a consequence of the Pauliexclusion principle^{27}. The Hamiltonian is given by H_{ex} = JS_{L} ⋅ S_{R}, where S_{L} and S_{R} are the left and right electron spin vectors and J is the strength of the exchange interaction^{5}. The magnitude of J is given by the energy difference between the symmetric and antisymmetric twoelectron states \(\left {T^0} \right\rangle = \left( {\left { \uparrow \downarrow } \right\rangle + \left { \downarrow \uparrow } \right\rangle } \right){\mathrm{/}}\sqrt 2\) and \(\left S \right\rangle = \left( {\left { \uparrow \downarrow } \right\rangle  \left { \downarrow \uparrow } \right\rangle } \right){\mathrm{/}}\sqrt 2\) respectively. Similar to gate defined quantum dots, it has been shown that the exchange between donors can also be parameterised in terms of the tunnel coupling and detuning between the (1,1) and (2,0) charge states^{28},
where \(\epsilon\) is the detuning and t_{c} is the tunnel coupling (such that J(0) = t_{c}). The detuning axis \(\epsilon\) is applied along V_{GL} = −0.9 V_{GR} (along the SET Coulomb blockade) and is shown by the white arrow in Fig. 1a. It effects a tilting from the (1,1) towards the (2,0) charge state, shown schematically in Fig. 1d. The detuning energy, \(\epsilon\), is related to the applied gate voltage V_{GL} via the lever arm \(\alpha _\epsilon\) = 0.071 eV/V, such that \(\epsilon\) = \(\alpha _\epsilon\) V_{GL}.
We start by initialising either state from Eq. (1) by loading one qubit randomly and deterministically down on the other, and subsequently apply a 50 ms pulse along the axis \(\epsilon\) to control the strength of the exchange interaction^{5}, shown by the open black circles in Fig. 2a. This time is long enough to allow for a significant exchange interaction, but much shorter than any electron spin relaxation such that readout is not hindered. Upon pulsing back into the (1,1) charge region we perform independent spin readout of L and then R. In addition to the single spin outcomes for each qubit we also determine the joint probabilities P_{ ij } for ij ∈ {↑↑, ↑↓, ↓↑,↑↑}, as shown in Fig. 2b–d.
In the case where ρ_{↑↓} is initialised, after pulsing to \(\epsilon \ll 0\) we observe P_{↑↓} ~ 0.5 and P_{↓↑} ~ 0, indicating no spin flipflops have occurred during the 50 ms pulse duration. However, anticorrelated spins can be seen in Fig. 2b, c as we pulse closer to the (1,1)–(2,0) charge transition, at \(\epsilon\) = 0 where both P_{↑↓} and P_{↓↑} → 0.25. Furthermore, we see that both P_{↑↑} and P_{↓↓} remain constant at approximately 0 and 0.5 respectively as they represent populations of the triplet states \(\left { \uparrow \uparrow } \right\rangle\) and \(\left { \downarrow \downarrow } \right\rangle\) and are not subject to the exchange interaction. Statistical analysis (see Supplementary Fig. 4) of these results indicates a correlation coefficient of ϕ = −0.243 ± 0.028 with a pvalue \(\ll 0.01\) for 0 < \(\epsilon\) < 2.4 meV, demonstrating the presence of statistically significant spin anticorrelations in this region.
Estimate of interdonor exchange coupling
To ascertain the value of the interdot tunnel coupling, t_{c}, we repeat the same pulsing scheme as above while modifying the detuning pulse duration from 0.1 to 2 ms and compare our results to a spinlevel theoretical model; see Fig. 3. The quantum mechanical behaviour of a donorbased twoqubit system is described by the following terms in the Hamiltonian:
where H_{ze} and H_{zn} are the electron and nuclear Zeeman energies, with γ_{e} ≈ 28.024 GHz/T and γ_{n} ≈ 17.235 MHz/T gyromagnetic ratios respectively^{29}. The hyperfine term, H_{hf} is separated into two components as it has been predicted that the hyperfine constants will be different for varying donor cluster configurations^{23,30}. Here for simplicity we assume the bulklike value of A_{L} = A_{R} = A = 117.53 MHz^{29} and define the static field to be \({\bf{B}} = \left( {0,0,\left {B_{\rm z}} \right} \right)\). We numerically calculate the time evolution of the density matrix via a fourthorder Runge−Kutta method with the inclusion of relevant decoherence channels (see Supplementary Fig. 5).
For the theoretical data shown in Fig. 3b we prepare the initial state ρ_{↑↓} from Eq. (1) and simulate nonadiabatic pulses to detuning positions for varying pulse durations, τ_{w}. For this simulation we use a tunnel coupling, t_{c} = 200 MHz, assume dynamic P nuclear spins as well as a singlespin dephasing time of \(T_2^ \ast = {55\,\mathrm {ns}}\) due to the constantly fluctuating Overhauser field of the ^{29}Si nuclear spins. The equivalent measured data set is shown in Fig. 3a with cuts at 0.3 and 1 ms shown in Fig. 3c and compared with the theoretical predictions for t_{c} = 50, 200 and 500 MHz. From these results we can estimate the tunnel coupling within an order of magnitude accuracy to be t_{c} ~ 200 MHz for this device. Following Eq. (2), this result places an equivalent bound on the achievable exchange energy J < 200 MHz inside the (1,1) charge region (see Fig. 4a).
In ref. ^{22} the authors investigated multiple different 2P intradot configurations, and found that disorder at the latticesite level had little effect on the final exchange energy. They showed that the exchange energy for a 2P−1P system with a 15 nm separation was tunable over five orders of magnitude for electric field strengths \( 2 < \left {\bf{E}} \right < 2\,{\mathrm {MV/m}}\) at the donor sites. The voltage applied to the inplane gates in our device amount to a potential difference of ~ 100 mV between V_{GL} and V_{GR} at the (1,1)–(2,0) interdot transition. From an electrostatic model of our device we estimate \(\left {\bf{E}} \right = 0.49 \pm 0.10\, {\mathrm {MV/m}}\) at the donor sites. Our estimate of t_{c} = 0.2 GHz (which is equal to J at \(\epsilon\) = 0) is within an order of magnitude of the theoretical prediction for J in a 2P−1P system given this \(\left {\bf{E}} \right\)^{22}.
Requirements for coherent control of exchange
In this final section we investigate the potential for achieving coherent exchange between two electrons confined to donors in natural silicon. Based on Eq. (2) the plot in Fig. 4a shows the obtainable exchange energies, J, as a function of detuning and tunnel coupling, where the vertical blue dashed line shows t_{c} = 0.2 GHz for our device. Importantly, the boundary where the difference in magnetic field at the two qubit sites ΔB_{z} is equal to the exchange energy J, separates the twoelectron product eigenstates and singlettriplet eigenstates. For donor qubits ΔB_{z} is dominated by the phosphorus nuclearspin hyperfine, A. The exact value of ΔB_{z} varies depending on the number of donors at each qubit site and their nuclear spin orientations: For a 2P−1P device with random nuclear spin configurations ΔB_{z} fluctuates between 3A/2 or A/2 with a 1:3 ratio.
It can be seen from Fig. 4a that for the device studied here there exists only a small range in detuning (approximately 10 μV in gate voltage) over which one could implement coherent exchange oscillations inside the (1,1) charge region (negative \(\epsilon\)). When one takes into account any voltage noise on gates (which influences \(\epsilon\) and ultimately J) this makes the operation of coherent oscillations challenging^{31}. Indeed, in this particular device we measured gate RMS voltage noise of 50 μV from shot to shot, equivalent to detuning noise of δ\(\epsilon\) = 850 MHz, indicating that pulsing repeatedly to the same exchange energy would not be possible for \(\epsilon\) < 0. For the same reasons, charge noise also destroys coherence when adopting the approach to pulse \(\epsilon\) > 0. We carried out experiments with pulses down to 10 ns for both \(\epsilon\) < 0 and \(\epsilon\) > 0 but were unable to observe a coherent exchange phenomenon.
Figure 4b shows the predicted number of exchange oscillations (~15) that would be observed in a device with t_{c} = 2.5 GHz after pulsing to a detuning \(\epsilon\) = −25 GHz (circle marker in Fig. 4a). Conversely, using the same model we estimate that a noise floor of <6 μV (~100 MHz in detuning or ~50 mK, much lower than the electron temperature) would be required to observe the signature of coherent oscillations in the present device. Note that in addition to the realistic detuning noise we have also included the effect of a constantly fluctuating ^{29}Si Overhauser field expected in natural silicon^{32} as well as randomised P donor nuclear spins of the donor atoms themselves.
From these simulations we can extract the frequency of oscillations, ν as well as the dephasing time τ_{d}, allowing us to determine the figure of merit ντ_{d} as a function of tunnel coupling and detuning pulse position; see Fig. 4c. Interestingly, the product ντ_{d} only becomes significant beyond the boundary ΔB_{z} = 3A/2 for values of t_{c} > 2 GHz, providing a lower bound on the required tunnel coupling for coherent control. Figure 4d gives ντ_{d} as a function of tunnel coupling for a detuning pulse to the boundary ΔB_{ z } = 3A/2. These results indicate that at high tunnel coupling, the observation of exchange oscillations will ultimately be hindered by the presence of the fluctuating ^{29}Si Overhauser field. In the case where qubits exist in a spin vacuum, as in ^{28}Si, only charge noise is relevant and ντ_{d} can be seen to increase monotonically as a function of tunnel coupling (dashed line in Fig. 4d).
Discussion
In summary, we have demonstrated a controllable exchange interaction resulting in twoelectron spin anticorrelations on precision placed 2P−1P donors qubits in Si using inplane ‘detuning’ gates. The results are consistent with the exchange interaction behaviour expected at the (1,1)–(2,0) charge transition and represent the first direct measurement of correlated electron spins in donorbased devices. While the small tunnel coupling (0.2 GHz) in the present device prohibited measurement of coherent oscillations, we show our results agree with recent studies^{15} in which much smaller distances than previously predicted are required to achieve a sufficiently large exchange coupling for coherent control. Furthermore, while detuning noise presents a problem for devices with a small tunnel coupling, we theoretically predict that for larger tunnel couplings of t_{c} > 2 GHz it can be overcome. Theoretical work on coupled donor systems suggest a separation of 13–14 nm between a 1P−2P system will be required to achieve this magnitude of exchange coupling^{22}. Importantly, there is no reason to believe that this small change in donor site separation will lead to a significant reduction in electrical controllability based on previous experimental works^{24,25,28,33}. This benchmark for a larger interaction strength between neighbouring donorbased qubits provides the focus for future experiments.
With the atomic precision placement of donors using STM lithography it will be possible to further optimise the interdonor distance to control the coherent coupling between two donor qubits with orderofmagnitude accuracy^{34}. While extensive studies have been conducted for deterministic single P donor incorporation^{35}, similar studies will need to be developed to determine the optimal lithographic patch for deterministic 2P incorporation. Crucially, recent theory predicts that the 2P−1P configuration we present in this paper both increases the tunability of the tunnel coupling and at the same time suppresses the ‘exchange fluctuations’ known for two single donors, and may therefore be less sensitive to the exact atomistic donor positions than two coupled single donors^{22}. Furthermore, our ability to directly place the donor with <1 nm accuracy along with the reproducible demonstration of high fidelity singleshot spinreadout in multiple devices^{25}, bode well for the future scalability of donor qubit quantum computers.
Methods
Device fabrication
The device, shown in Fig. 1, was fabricated using scanningtunnellingmicroscopy hydrogen lithography to selectively remove hydrogen from a passivated Si(100) 2 × 1 reconstructed surface. The lithographic mask is subsequently dosed with PH_{3} and annealed (320 °C) to incorporate P atoms into the silicon substrate^{36} with ~1/4 monolayer density (2 × 10^{14}/cm^{2}) allowing for quasimetallic conduction in all electrodes^{37}.
Measurement setup
For all electrical measurements, the device was mounted on a highfrequency printed circuit board within a copper enclosure, thermally anchored to the cold finger of a dilution refrigerator with a base temperature of 50 mK. Voltage pulses were applied to gates G_{L} and G_{R} by an arbitrary waveform generator (Agilent 81180A), connected via a bias tee to the gate along with a constantvoltage source. The SET current, I_{SET}, was amplified and converted into a voltage signal at room temperature, lowpass filtered to 1 kHz bandwidth, and acquired with a fast digitising oscilloscope.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We thank Sven Rogge for enlightening discussions. This research was conducted by the Australian Research Council Centre of Excellence for Quantum Computation and Communication Technology (project no. CE110001027) and the US National Security Agency and US Army Research Office (contract no. W911NF0810527). M.Y.S. acknowledges an ARC Laureate Fellowship.
Author information
Author notes
M. A. Broome and S. K. Gorman contributed equally to this work.
Affiliations
Centre of Excellence for Quantum Computation and Communication Technology, School of Physics, University of New South Wales, Sydney, NSW, 2052, Australia
 M. A. Broome
 , S. K. Gorman
 , M. G. House
 , S. J. Hile
 , J. G. Keizer
 , D. Keith
 , T. F. Watson
 , W. J. Baker
 & M. Y. Simmons
Centre of Excellence for Quantum Computation and Communication Technology, School of Physics, The University of Melbourne, Parkville, VIC, 3010, Australia
 C. D. Hill
 & L. C. L. Hollenberg
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Contributions
M.A.B., S.K.G., J.G.K. and W.J.B. fabricated the device. M.A.B., S.K.G., M.G.H. and S.J.H. obtained all measurements. Calculation of donor charging energies was performed by S.J.H. and S.K.G. The data was analysed by M.A.B., S.K.G. and M.G.H. and was discussed critically with all authors. Theoretical modelling of coherent exchange coupling was carried out by M.A.B. and D.K. with inputs from S.K.G., C.D.H. and L.C.L.H. Spinreadout analysis was performed by M.A.B., D.K. and T.F.W. The manuscript was written by M.A.B., S.K.G and M.Y.S. with inputs from all other authors. M.Y.S. supervised the project.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to M. Y. Simmons.
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