Device description

We use a microelectronics technology based on 300 mm silicon-on-insulator wafers. Our qubit device, schematically shown in Fig. 1a, is derived from silicon nanowire field-effect transistors19. It relies on confined hole spins20,21,22,23,24, and it consists of a 10 nm-thick and 20 nm-wide undoped silicon channel with p-doped source and drain contact regions, and two ≈30 nm-wide parallel top gates, side covered by insulating silicon nitride spacers (further details on the spacers are given in Supplementary Note 1). A scanning electron microscopy top view and a transmission electron microscopy cross-sectional view are shown in Fig. 1b,c, respectively. At low temperature, hole QDs are created by charge accumulation below the gates25. The double-gate layout enables the formation of two QDs in series, QD1 and QD2, with occupancies controlled by voltages V g1 and V g2 applied to gates 1 and 2, respectively (Supplementary Fig. 2 and Supplementary Note 2). We tune charge accumulation to relatively small numbers, N, of confined holes (we estimate N≈10 and ≈30 for QD1 and QD2, respectively, as discussed in Supplementary Note 2). In this regime, the QDs exhibit a discrete energy spectrum with level spacing δE in the 0.1–1 meV range, and Coulomb charging energy U≈10 meV.

Figure 1: CMOS qubit device. (a) Simplified three-dimensional schematic of a silicon-on-insulator nanowire field-effect transistor with two gates, gate 1 and gate 2. Using a bias tee, gate 1 is connected to a low-pass-filtered line, used to apply a static gate voltage V g1 , and to a 20 GHz-bandwidth line, used to apply the high-frequency modulation necessary for qubit initialization, manipulation and read-out. (b) Colourized device top view obtained by scanning electron microscopy just after the fabrication of gates and spacers. Scale bar, 75 nm. (c) Colourized transmission electron microscopy image of the device along a longitudinal cross-sectional plane. Scale bar, 50 nm. Full size image

In a simple scenario where spin-degenerate QD levels get progressively filled by pairs of holes, each QD carries a spin S=1/2 for N=odd and a spin S=0 for N=even. By setting N=odd in both dots, two spin-1/2 qubits can be potentially encoded, one for each QD. This is equivalent to the (1, 1) charge configuration, where the first and second digits denote the charge occupancies of QD1 and QD2, respectively. In practice, here we shall demonstrate full two-axis control of the first spin only, and use the second spin for initialization and read-out purposes. Tuning the double QD to a parity-equivalent (1, 1)→(0, 2) charge transition, initialization and read-out of the qubit relies on the so-called Pauli spin blockade mechanism5,26. In this particular charge transition, tunnelling between dots can be blocked by spin conservation. Basically, for a fixed, say ‘up’, spin orientation in QD2, tunnelling will be allowed if the spin in QD1 is ‘down’ and it will be forbidden by Pauli exclusion principle if the spin in QD1 is ‘up’, that is, a spin triplet (1, 1) state is not coupled to the singlet (0, 2) state. This charge/spin configuration can be identified through characteristic experimental signatures27,28,29 associated with the Pauli blockade effect discussed above (Supplementary Fig. 4 and Supplementary Note 3). (We note that deviations from pairwise filling of the hole QD orbitals can occur, especially beyond the few-hole regime30, resulting in more complex spin configurations.)

Electric-dipole spin resonance

We now turn to the procedure for spin manipulation. In a recent work on similar devices with only one gate, we found that hole g-factors are anisotropic and gate-dependent25, denoting strong spin–orbit coupling29. This implies the possibility to perform electric-dipole spin resonance (EDSR), namely to drive coherent hole-spin rotations by means of microwave frequency (MW) modulation of a gate voltage (Supplementary Note 4). Here we apply the MW modulation to gate 1 to rotate the spin in QD1. Spin rotations result in the lifting of spin blockade. In a measurement of source-drain current I sd as a function of magnetic field B (perpendicular to the chip) and MW frequency f, EDSR is revealed by narrow ridges of increased current28. The data set in Fig. 2a shows two of such current ridges: one clearly visible, most likely associated with QD1 (strongly coupled to the rf-modulated gate); and the other one rather faint, most likely arising from the spin rotation in QD2 (which is only weakly coupled to gate 1). Both ridges follow a linear f(B) dependence consistent with the spin resonance condition hf=gμ B B, where h is Planck’s constant, μ B the Bohr magneton and g the hole Landé g-factor (absolute value) along the magnetic field direction. From the slopes of the two ridges we extract two g-factor values g 1 =1.63 and g 2 =1.92 comparable to those reported before25. In line with our plausible interpretation of the observed EDSR ridges, we ascribe these g-factor values to QD1 and QD2, respectively. We have observed similar EDSR features at other working points (that is, different parity-equivalent (1, 1)→(0, 2) transitions) and in two distinct devices (Supplementary Figs 5 and 6 and Supplementary Note 4).

Figure 2: Electrically driven coherent spin manipulation. (a) Colour plot of the source-drain current I sd as a function of magnetic field B and MW frequency f. Electrically driven hole spin resonance is revealed by two enhanced current ridges. The barely visible upper ridge is indicated by a white arrow. Inset: horizontal cut at f=5.4 GHz. (b) Schematic representation of the spin manipulation cycle and corresponding gate-voltage (V g1 ) modulation pattern. (c) Same type of measurement as in a done on a different device. The cycle presented on b is also applied with a MW burst of 20 ns. Coherent manipulations presented in d–f have been carried at the working point indicated by a white arrow, while the black arrow highlights the working point for Figs 3 and 4. (d) Colour plot of Rabi oscillations for a range of microwave powers P MW at f=8.938 GHz and B=0.144 T. (e) Rabi oscillations for different powers taken from c and fitted (solid lines) to A cos(2πf Rabi +φ)/ (ref. 34), current has been averaged for 1 s for each data point. Rabi frequencies are 24, 39 and 55 MHz for P MW =−5, −0.5 and 2.5 dBm, respectively. (f) Rabi frequency versus microwave amplitude, , with a linear fit (solid line). Full size image

Coherent spin control

To perform controlled spin rotations, and hence demonstrate qubit functionality, we replace continuous-wave gate modulation with MW bursts of tunable duration, . During spin manipulation, we prevent charge leakage due to tunnelling from QD1 to QD2 by simultaneously detuning the double QD to a Coulomb-blockade regime4 (Fig. 2b). Following each burst, V g1 is abruptly increased to bring the double dot back to the parity-equivalent (1, 1)→(0, 2) resonant transition. At this stage, a hole can tunnel from QD1 to QD2 with a probability proportional to the unblocked spin component in QD1 (that is, the probability amplitude for spin-up if QD2 hosts a spin-down state). The resulting (0, 2)-like charge state ‘decays’ by emitting a hole into the drain, and a hole from the source is successively fed back to QD1, thereby restoring the initial (1, 1)-like charge configuration. The net effect is the transfer of one hole from source to drain, which will eventually contribute to a measurable average current. (In principle, because not all (1, 1)-like states are Pauli blocked, the described charge cycle may occur more than once during the read-out-initialization portion of the same period, until the parity-equivalent (1, 1)→(0, 2) becomes spin blocked again and the system is re-initialized for the next manipulation cycle.)

We chose a modulation period of 435 ns, of which 175 ns are devoted to spin manipulation and 260 ns to read-out and initialization. Figure 2c shows an EDSR resonance recorded on a second device taken with the previously described gate 1 modulation and a MW burst of 20 ns (a wider f−B range of the EDSR spectrum is shown in Supplementary Fig. 6a). Figure 2d shows I sd as a function of MW power P MW , and at the resonance frequency for B=144 mT (see white arrow in Fig. 2c). The observed current modulation is a hallmark of coherent Rabi oscillations of the spin in QD1, also explicitly shown by selected cuts at three different MW powers (Fig. 2e). As expected, the Rabi frequency f Rabi increases linearly with the MW voltage amplitude, which is proportional to P MW 1/2 (Fig. 2f). At the highest power, we reach a remarkably large f Rabi ≈85 MHz, comparable to the highest reported values for electrically controlled semiconductor spin qubits31. Figure 3a shows a colour plot of I sd (f, ) revealing the characteristic chevron pattern associated to Rabi oscillations13. The fast Fourier transform of I sd , calculated for each f value, is shown in the upper panel. It exhibits a peak at the Rabi frequency with the expected hyperbolic dependence on frequency detuning Δf=f−f 0 , where f 0 =9.68 GHz is the resonance frequency at the corresponding B=155 mT (working point indicated by a black arrow in Fig. 2c).

Figure 3: Frequency dependence of Rabi oscillations and Ramsey fringes. (a) Bottom panel: I sd (f, ) at B=0.155 T and P MW =3 dBm. Each data point was averaged for 600 ms and, for each f, the average current was subtracted. Top panel: Fourier transform of the data in the bottom panel showing the expected hyperbolic dependence of f Rabi (f). (b) Bottom panel: I sd (f, ), where is the waiting time between two 7 ns-long bursts. Each data point was obtained with a 2 s integration time and the average current was subtracted. This data set, taken at B=0.155 T and P MW =8 dBm, shows a characteristic Ramsey-interference pattern. Top panel: Fourier transform of the data in the bottom panel showing the expected linear evolution of the Ramsey fringes frequency. (c) Ramsey sequence manipulation scheme (top), and two I sd ( ) data sets corresponding to vertical cuts in b for f=9.595 and 9.720 GHz. Solid lines are fits to A cos(Δf +φ)exp(−( )2). The data in blue have an upward offset of 200 fA. Full size image

Dephasing and decoherence times

To evaluate the inhomogeneous dephasing time during free evolution we perform a Ramsey fringes-like experiment, which consists in applying two short, phase coherent, MW pulses separated by a delay time . The proportionality between the qubit rotation angle θ and is used to calibrate both pulses to a θ= rotation (see sketch in Fig. 3c). For each f value, I sd exhibits oscillations at frequency Δf decaying on a timescale ≈60 ns (Fig. 3b). Extracted current oscillations at fixed frequency are presented in Fig. 3c. At resonance (Δf=0), the two pulses induce rotations around the same axis (say the x axis of the rotating frame). The effect of a finite Δf is to change the rotation axis of the second pulse relative to the first one. Alternatively, two-axis control can be achieved also at resonance (Δf=0) by varying the relative phase Δφ of the MW modulation between the two pulses. For a Ramsey sequence , the first pulse induces a rotation around x and the second one around x, y, −x and −y for Δφ=0, , π and , respectively. The signal then oscillates with Δφ as shown in the insets to Fig. 4a, and the oscillation amplitude vanishes with on a timescale (Fig. 4a).

Figure 4: Two-axis qubit control and spin coherence times. (a) Amplitude ΔI sd of Ramsey oscillations versus delay time . For each , the phase of the second π/2 pulse is shifted by Δφ (see top diagram), which corresponds to a change in the rotation axis. Insets: full 2π oscillations at short (4.35 ns) and long (69.6 ns) and corresponding sinusoidal fits (solid lines) enabling the extraction of ΔI sd and associated s.d. error bars. The decay of ΔI sd ( ) is fitted to exp[−( ]2) giving =59±1 ns. (b) Results of a Hahn echo experiment, whose manipulation scheme is given in the top diagram. The duration of the refocusing π pulse is 14 ns. Insets: full 2π oscillations at relatively short (57.4 ns) and long (153 ns) and corresponding sinusoidal fits (solid lines). The Hahn echo oscillation amplitude ΔI sd decays on timescale longer than the largest , which was limited to 160 ns to ensure a sufficiently fast repetition cycle, and hence a measurable read-out current. The solid line is a fit to exp(−( /T echo )3) yielding T echo =245±12 ns. Full size image