Introduction

The development of quantum computers based on spin in quantum dots1,2 has made tremendous advances, from the demonstration of initialization3,4,5, coherent control5,6,7,8 and readout3,9,10,11 of the single spin quantum state, via two-qubit operations12,13, to architecture design14,15 and manufacturing with advanced semiconductor technology16. For large-scale arrays of multi-terminal quantum dot (QD) qubits, the control electronics should be integrated on-chip to avoid an interconnect problem14,15. This creates another challenge17,18,19, because balancing the heat generated by the control electronics is difficult at temperatures below 100 mK in a dilution refrigerator, which has limited cooling power. In a liquid He bath at 4.2 K the cooling power is three orders of magnitude higher. Hence, it is indispensable that qubits can operate at higher temperatures17,18,19 while at the same time maintaining error rates well below the threshold (10−2) for fault-tolerant quantum computing20,21.

The traditional method3,4,5 of initialization/readout of a QD spin qubit, based on Zeeman spin splitting of the dot’s energy levels in a magnetic field B0 (Fig. 1a), yields limited fidelity and operation temperature T < 100 mK because the Zeeman splitting is <0.1 meV for practical values of B0. Therefore, only at very low T selective tunneling of an electron from the lead into the lowest-energy spin state (initialization) occurs. Thermal broadening in the leads at elevated T makes the higher-energy spin state accessible, thus producing initialization errors. A similar issue occurs for readout. Attention has therefore necessarily shifted to the only other available method for initialization/readout, which is based on Pauli spin blockade6,7,22,23 in double QD qubits. This method, which has been around for more than two decades, has a larger energy scale that is set by the singlet-triplet splitting. For electron-based quantum dots, the singlet-triplet splitting is often comparable to the valley splitting, with reported values of 0.3–0.8 meV for Si quantum dots24 and up to about 0.3 meV for the SiGe/Si system25. The readout window is enhanced25 for certain multi-electron configurations in which the orbital excited state energy comes into play (an orbital excited state energy of 1 meV was measured for a 4-electron Si quantum dot26). With Pauli spin blockade, qubit operation at ~1 K was achieved for Si quantum dots with electrons17,18 and holes19 with fidelities that are approaching the fault-tolerance threshold, although some degradation for higher T up to 4 K was also observed19. Fidelities exceeding the fault-tolerance threshold have recently been reported27,28,29, but only at temperatures below 100 mK. While these are significant advances, one already operates close to the limits of the Pauli spin blockade technique. What is desirable is a method for initialization/readout that can provide a leap forward.

Fig. 1: Quantum dot spin qubit with reservoir spin accumulation.
figure 1

a Comparison between the Zeeman spin splitting of the quantum dot states (the traditional method to achieve spin selectivity in a quantum dot) and the method described here, which employs a reservoir with a spin accumulation Δμ, denoting the spin-dependent occupation of the electronic states. b Basic device layout showing the quantum dot defined by electrostatic gates (grey strips). Tunnel barriers (white areas) separate it from the source and drain reservoirs, which consist of a two-dimensional electron gas (2DEG). In the drain reservoir a spin accumulation Δμ is created by an electrical spin current across a ferromagnetic tunnel contact (drive current IFM). The induced spin accumulation spreads out (pink area) in the reservoir away from the magnetic contact by spin diffusion (length scale: spin-diffusion length LSD). c Energy diagrams showing the spin orientation of a single electron on the quantum dot, as it is determined by the spin accumulation. Positive (negative) Δμ yields a single electron trapped on the quantum dot in a spin-up ∣↑ > (spin-down ∣↓ > ) quantum state. The sign of Δμ is controlled by the direction of the magnetization (white arrows) of the ferromagnet (green, label FM) and/or by the polarity of the drive current across the tunnel barrier (light grey, label I), which yields spin injection into or spin extraction from the drain reservoir. The spin-polarized flow from the FM tunnel contact to the drain reservoir is indicated by pink (spin up) and violet (spin down) arrows.

Full size image

Results And Discussion

Principle of the approach

The scheme for the high-temperature initialization and readout of QD spins we describe here employs a drain reservoir with a spin accumulation \(\Delta \mu ={\mu }_{D}^{\uparrow }-{\mu }_{D}^{\downarrow }\). The electrochemical potentials \({\mu }_{D}^{\uparrow }\) and \({\mu }_{D}^{\downarrow }\) for the spin-up and spin-down electrons describe the spin-dependent occupation of the electronic states of the reservoir (Fig. 1a). Because Δμ can be much larger than the thermal broadening, in a wide energy interval between \({\mu }_{D}^{\uparrow }\) and \({\mu }_{D}^{\downarrow }\), the reservoir states are completely empty or full, depending on the spin. This provides a thermally robust spin selectivity. The Δμ can be created by electrical30,31,32,33,34,35,36,37,38, dynamical39,40,41 or thermal42,43 spin injection from a ferromagnetic (FM) contact, or via the spin Hall effect44,45,46. We consider spin injection by a spin-polarized current IFM across a FM tunnel contact to the reservoir (Fig. 1b). This allows routine creation of spin accumulation, with values as large as 13 meV at 10 K (3.5 meV at 300 K) for n-type Si with Fe/MgO contacts35,36,38, and 4.1 meV at 300 K in graphene34. The Δμ is proportional to the tunnel spin polarization and to IFM and is thus electrically tunable. The sign of Δμ can be reversed by either reversing the magnetization of the contact, or the current polarity36 (Fig. 1c). Since Δμ decays with distance from the injection contact on the scale of the spin-diffusion length LSD, which is typically a few microns31,32,37, the magnetic contact is to be placed near the quantum dot (Fig. 1b). Since a large spin accumulation of holes in p-type semiconductors cannot be achieved due to the much shorter spin-relaxation time for holes47,48,49,50, the method we describe here is applicable to electron-based qubits.

The spin accumulation determines the spin occupation of the QD as follows. The energy Ed of the dot’s ground state (twofold degenerate at B0 = 0) is positioned between \({\mu }_{D}^{\uparrow }\) and \({\mu }_{D}^{\downarrow }\) with an electrostatic gate (Fig. 1c). If Δμ > 0, a spin-down electron can tunnel from the source to the dot, and escape into the drain because \({E}_{d}\, >\, {\mu }_{D}^{\downarrow }\). However, as soon as a spin-up electron tunnels onto the dot, it is trapped, because there are no empty spin-up states available in the drain owing to the spin accumulation (\({E}_{d}\, <\, {\mu }_{D}^{\uparrow }\)), while the empty state the electron left behind in the reservoir is quickly filled up due to fast energy relaxation. If the charging energy of the dot is sufficiently large, Coulomb blockade prevents tunneling of a second electron onto the dot. The current is then blocked and the dot is occupied by a single electron in a ∣↑ > quantum state (Fig. 1c). For negative spin accumulation (\({\mu }_{D}^{\uparrow } < {\mu }_{D}^{\downarrow }\)), the situation is similar, except that now the single electron has a ∣↓ > quantum state. The spin accumulation thus determines the spin state of a single electron on the dot. Exchange coupling between the dot and the reservoir is not required.

Model description

In order to confirm the described behavior and quantify the characteristics, we compute the current ISD through the quantum dot and the spin-dependent occupation probabilities N↑ and N↓ of the dot using rate equations51,52 (Supplementary Note 1). Single-electron tunneling events induce transitions between the QD states (empty dot / single spin-up electron / single spin-down electron / doubly-occupied singlet state). The transition energy ΔS from the ground to the singlet state includes the Coulomb charging energy. The various tunneling rates are functions of the spin-independent tunnel coupling ΓS and ΓD of the dot to the source and drain reservoir, respectively, and of the Fermi-Dirac distribution factors. The latter depend on Ed relative to the source (μS) and the spin-dependent drain electrochemical potentials (\({\mu }_{D}^{\uparrow }\), \({\mu }_{D}^{\downarrow }\)). The source reservoir is taken to be at ground potential. The source-drain voltage VSD shifts the drain electrochemical potentials by − eVSD and shifts the quantum dot level according to Ed = ϵd − xeVSD, with ϵd the dot level energy at VSD = 0, and e the electron charge. The fraction x is determined by the relative values of ΓS and ΓD (we have taken x = 1/2). Spin flips in the quantum dot are included with a rate Γ1 = ℏ/(2eT1), as governed by the longitudinal spin-relaxation time T1, which is set to 1 ms (ℏ is the reduced Planck’s constant). In some cases, a Zeeman splitting of the quantum dot level is included. The formalism considers quantum dot spin states that are collinear with the magnetization of the ferromagnet, which sets the quantization axis of the spin accumulation. This is sufficient for the calculation of the spin selectivity and the error rates. Transverse spin components are needed to describe the non-collinear spin states that appear during qubit spin manipulation by spin resonance. Also, orbital excited states are not included (as explained in Supplementary Note 1).

Spin selection with reservoir spin accumulation

The calculated ISD exhibits a typical Coulomb staircase, which is modified when Δμ is non-zero (Fig. 2a). More importantly, in the presence of a spin accumulation, a spin-dependent dot occupation is produced in a wide VSD interval, where N↑ and N↓ are either 1 or 0. A more complete picture is obtained from the stability maps produced by varying VSD and ϵd (Fig. 2b). Without spin accumulation, the current is zero for large positive ϵd, where the dot energy level is not accessible, and in the central so-called Coulomb diamond (Fig. 2b, dotted white lines) due to Coulomb blockade. However, because Δμ = 0 and B0 = 0, we have N↑ = N↓ everywhere. In the presence of a spin accumulation, the boundaries of the central Coulomb diamond are changed, and two regions labeled α and β appear in which the dot occupation is highly spin-dependent (Fig. 2b). For Δμ = + 2 meV, we obtain two regions with N↑ = 1 and N↓ = 0 while also ISD = 0, implying that a single electron in a ∣↑ > quantum state is trapped continuously on the dot. When Δμ is reversed to −2 meV, a single electron with the opposite spin state (∣↓ > ) is trapped on the dot (N↑ = 0 and N↓ = 1). The calculations establish that the reservoir spin accumulation determines the spin quantum state of a single electron on the dot, which provides a scheme for spin qubit initialization and readout. Noteworthy is that Δμ is controlled by the magnetization of the FM contact. This means that a classical bit of information, represented by the magnetization direction, is mapped onto the quantum state of a single electron spin.

Fig. 2: Current and spin occupation of a quantum dot with reservoir spin accumulation.
figure 2

a Calculated current and dot occupations N↑ and N↓ as a function of the source-drain voltage VSD for Δμ = 0, +2 meV and −2 meV. The energy ϵd of the singly-occupied quantum dot ground state at VSD = 0 was taken to be −1 meV. b Stability maps of the current, N↑ and N↓ as a function of ϵd and VSD. With spin accumulation, there are two regions (labeled α and β) in which N↑ = 1 and N↓ = 0 (for Δμ > 0) or N↑ = 0 and N↓ = 1 (for Δμ < 0), while ISD is zero. In these regions, a single electron is trapped on the quantum dot in a spin-up ∣↑ > or a spin-down ∣↓ > quantum state, depending on the sign of Δμ. Calculations performed at T = 0.5 K and without Zeeman splitting (B0 = 0) using a ground state to singlet state transition energy ΔS of 4 meV. Transitions to the triplet states are not included here. On the horizontal axes we have used ϵd (the dot level energy at VSD = 0). Using the actual dot level energy ϵd − xeVSD would skew the stability maps.

Full size image

While triplet states play no role for region α, they are required to properly describe region β, as explained below. We therefore included triplet states in the calculations (Supplementary Note 1). The stability maps for different values of Δμ (Fig. 3a) show that the boundaries of region α depend on Δμ. The upper right boundary of the central Coulomb diamond, which is defined by the dot level energy Ed relative to the drain electrochemical potential, splits into two boundaries (solid pink lines in Fig. 3b) because \({\mu }_{D}^{\uparrow }\,\ne\, {\mu }_{D}^{\downarrow }\). Spin selectivity then appears when the ground state lies between \({\mu }_{D}^{\uparrow }\) and \({\mu }_{D}^{\downarrow }\), as already discussed above. For region β, the origin of the spin selectivity is different (Fig. 3c). For region β, the ground state lies below the reservoir electrochemical potentials, while a transition to the doubly-occupied singlet state (label S) requires an energy between \({\mu }_{D}^{\uparrow }\) and \({\mu }_{D}^{\downarrow }\). This also produces a single electron in a ∣↑ > quantum state (for Δμ > 0). A spin-down electron is not trapped in the ground state because a spin-up electron can tunnel in from the drain. From the resulting singlet state, an electron of either spin can tunnel out. If a spin-up electron tunnels out, we return to the starting configuration, and the process repeats itself until the spin-down electron tunnels out from the singlet state. This leaves behind an electron in a ∣↑ > ground state. This is trapped if (i) the singlet state is energetically not accessible for a spin-down electron, and (ii) a spin-up electron cannot tunnel into the dot to create a triplet state (label T), which requires an additional energy given by the singlet-triplet transition energy ΔST. If Δμ > ΔST, as we consider here, the lower boundary of region β occurs when \({\mu }_{D}^{\uparrow }\) crosses the triplet energy and a non-zero current appears. The spin selectivity for region β then results from the combination of the spin accumulation and Pauli spin blockade. Therefore, the relevant energy scale is ΔST, whereas it is Δμ for region α.

Fig. 3: Operation window for spin selection in a quantum dot with reservoir spin accumulation.
figure 3

a Stability maps of N↑ as a function of ϵd and VSD, showing the evolution of the high spin-selectivity regions α and β as the spin accumulation is increased from +0.25 meV to +4 meV (calculated for T = 0.5 K, B0 = 0, ΔS = 4 meV and ΔST = 0.5 meV). b Sketch of the regions α and β in the presence of a spin accumulation. The two boundaries marked by the solid black lines do not change, whereas the two boundaries marked by the dotted black lines are modified (pink lines) for a non-zero Δμ. The actual boundaries of α and β extend beyond the pink lines, as indicated by the dashed pink areas (Supplementary Note 2). c Energy diagrams showing the spin orientation of a single electron on the quantum dot for regions α and β. Indicated are the energies ΔS and ΔST for transitions from, respectively, the ground to the singlet state (label S) and from the singlet to the triplet state (label T). In region α, the quantum dot spin is determined by the spin accumulation. For region β the mechanism is different, and the quantum dot spin is determined by a combination of the spin accumulation and Pauli spin blockade.

Full size image

High-temperature spin selection

The Δμ provides a large energy scale that makes the spin selectivity robust against temperature. Region α with N↑ ≈ 1 and N↓ ≈ 0 persists even at 6 K for a Δμ of + 3 meV (Fig. 4a). These results were obtained with a 70 μeV Zeeman splitting that favours the spin-down state, opposite to the spin accumulation. But because Δμ > > 70 μeV, the spin accumulation defines the quantum dot spin. For region β, the N↑ decays more rapidly with increasing T because the relevant energy scale is ΔST, which was set to 0.5 meV. The decay of N↑ is accompanied by a finite current (of spin-up electrons via the triplet states), making region β unsuitable for qubit operation at 4 K. The boundary of region β for N↓ is determined by the relative position of \({\mu }_{D}^{\downarrow }\) and the singlet state, and thus by Δμ, so that N↓ for region β remains ≈ 0 even at 6 K. For a more quantitative analysis, we extract the error rates from region α. The error rate is defined here as the probability for finding the spin state opposite to what was desired, i.e., the error rate is N↓/N↑ for Δμ > 0 and N↑/N↓ for Δμ < 0. Without spin accumulation, the error rate rises quickly and surpasses the fault-tolerance threshold for T > 100 mK (Fig. 4b). With a spin accumulation, the error rate remains much smaller at higher T. At 4.2 K, a Δμ of about 2 or 3 meV yields an error rate of 10−2 or 10−3, which is below the fault-tolerance threshold (red line). For larger spin accumulation, higher T and/or smaller error rates are possible. The latter reduces the qubit array size required for computation20,21.

Fig. 4: Robustness of spin selectivity at higher temperature.
figure 4

a Quantum dot spin occupations N↑ and N↓ for various temperatures (0.5, 1, 2, 4 and 6 K), showing that with a reservoir spin accumulation of +3 meV, the high spin selectivity is maintained above 4 K despite the thermal broadening. Calculations for ΔS = 6 meV and B0 = 0.6 T, producing a Zeeman splitting that opposes the spin selected by the spin accumulation. The Zeeman splitting controls the spin occupation in a narrow region near the lower right edge of the Coulomb diamond (white arrows) at T = 0.5 K, but this disappears quickly at higher T (see also the stability maps vs. T for Δμ = 0 in the Supplementary Note 3). The white dots in the top panels are guides to the eye and are in the exact same location for each panel. Note that the lower boundary of region β is different for N↑ and N↓ (see Supplementary Note 3). b Temperature variation of the error rate, represented as the ratio of the occupation probabilities for the non-desired and the desired spin, as extracted from a location near the white dot in region α (at least 0.5 mV away from the boundary). When Δμ = 0, only the small Zeeman splitting for a typical 0.6 T magnetic field is active and the error rates quickly rise above the threshold (red line) for fault-tolerant quantum computing, thus restricting the operation temperature to <100 mK. In contrast, with a reservoir spin accumulation of 2–3 meV, the error rates remain below the threshold even at the liquid He bath temperature of 4.2 K. Even higher temperatures and lower error rates are possible for larger spin accumulation.

Full size image

Implementation in spin qubits

We describe a basic sequence for the initialization, control and readout of a quantum dot qubit with a reservoir spin accumulation (Fig. 5). In the simplest scheme, the magnitude and sign of the spin accumulation are kept constant throughout the entire sequence. After emptying the dot, VSD and ϵd are adjusted so as to reach region α in the stability diagram and thereby initialize a single electron with a well-defined spin on the dot. The selected spin state is determined by the sign of Δμ. A large operation window (region α) exists for VSD and ϵd, making the system insensitive to errors produced by potential variations. In the next step, the quantum dot level is moved down so as to obtain a deep Coulomb blockade for spin manipulation by established spin resonance methods5,7,8. Care has to be taken that during this stage the qubit spin is not disturbed by the spin accumulation in the drain reservoir (see the Supplementary Discussion). In the final step, VSD and ϵd are adjusted to their readout values. The spin accumulation enables readout because only one type of electron can tunnel out into the drain (spin down for the case of Δμ > 0 depicted in Fig. 5). The tunneling event can be detected via established charge sensing techniques3,9,10,11.

Fig. 5: Basic operation sequence of a quantum dot spin qubit with reservoir spin accumulation.
figure 5

In the simplest scheme, the magnitude and sign of the spin accumulation are kept constant throughout the entire sequence. The initial spin state of the quantum dot is set by the reservoir spin accumulation and by choosing appropriate settings for ϵd and VSD. For the next step, spin control by established electron spin resonance (ESR) techniques for spin qubits, ϵd is adjusted so as to obtained a deep Coulomb blockade. For readout, the spin accumulation is used to produce spin selectivity. If after the spin control by spin resonance a ∣↑ > quantum state is obtained, the electron cannot tunnel into the drain reservoir owing to the non-zero Δμ. If a ∣↓ > quantum state is obtained after spin resonance, the electron can tunnel into the drain, and the associated charge movement can be detected by established charge sensing techniques.

Full size image

The large spin accumulation offers high spin selectivity for initialization and readout. For the overall readout fidelity, other factors also come into play, including errors that might occur in the charge sensing of the readout transition. Such errors depend on the charge sensing method and on other design details, which are beyond the scope of this work. Since Δμ is adjustable electrically (via the current through the magnetic contact to the drain), the spin selectivity can be tuned to levels for which the overall fidelity is limited only by the spin control process. Variations of the operation sequence presented in Fig. 5 are possible. For instance, the spin that is selected for initialization and readout can be changed by reversing the magnetization of the ferromagnetic contact to the drain. However, local magnetization control for each reservoir contact requires additional elements and increases the complexity and footprint of the qubit design. One might also consider active control of Δμ, for instance switching it off during the spin resonance manipulation. This needs to be done in a way that does not disturb the spin qubit (see the Supplementary Discussion for more information). It is emphasized that reversal of the spin accumulation does not constitute coherent spin control because the originally trapped electron tunnels out and is replaced by a different electron with opposite spin. It has been proposed that coherent manipulation of a qubit spin by scattering off ”flying” spins in the reservoir is possible if an exchange coupling exists between the qubit and the reservoir spins53.

Surely, the use of a drain reservoir with a ferromagnet to create the spin accumulation introduces additional complexity, increases the size of the spin qubit, affects the scalability, and may also modify the noise during qubit operation. At the same time, the described method enables high spin selectivity, enhanced robustness at higher temperature, with a wide operation window for VSD and ϵd. Also, the approach is conceptually simple as it uses one quantum dot with a single electron, instead of the multi-electron double-dot devices used for Pauli spin blockade6,7,17,18,19,22,23,25. In future work, it is of importance to investigate the required modifications to the design of the qubit array and assess whether the envisioned benefits of the described method outweigh the potential drawbacks. Some initial considerations are given in Supplementary Note 4 and the Supplementary Discussion, which also describe aspects related to the practical implementation. This includes the feasibility to induce a sufficiently large spin accumulation in the 2DEG reservoir, the applicability to different qubit material systems (not only quantum dots in Si/SiO2 2DEG’s but also SiGe/Si with a buried 2DEG, or dopant-based spin qubits), the expected dimensions of a reservoir with a ferromagnetic contact, and the drive current IFM (~3 nA) and power (~20 pW) required to achieve the desired Δμ. We note that nanomagnets are already widely used5,8,17,25,54 in spin qubits to produce an inhomogeneous magnetic field for electrically driven spin resonance (EDSR). It is of interest to examine whether the ferromagnetic tunnel contact to the drain reservoir can have a dual function (create spin accumulation and assist in producing the magnetic field gradients required for EDSR).

In summary, we have described a scheme for high-temperature initialization and readout of a spin qubit. It is based on a reservoir with a spin accumulation, which controls the spin state of a single electron on a quantum dot. The spin accumulation is induced electrically by spin injection from a nearby ferromagnetic tunnel contact. The method enables operation at elevated T, even above 4.2 K, with error rates below 10−4. The error rates are electrically adjustable and the operation window for VSD and ϵd is wide, making the approach insensitive to potential variations. Via the spin accumulation, a classical bit of information (magnetization direction) is mapped onto the quantum spin state of a single electron. This connects spin qubits and quantum computing to spintronics.