Abstract
The rapid rise of spintronics and quantum information science has led to a strong interest in developing the ability to coherently manipulate electron spins^{1}. Electron spin resonance^{2} is a powerful technique for manipulating spins that is commonly achieved by applying an oscillating magnetic field. However, the technique has proven very challenging when addressing individual spins^{3,4,5}. In contrast, by mixing the spin and charge degrees of freedom in a controlled way through engineered nonuniform magnetic fields, electron spin can be manipulated electrically without the need of highfrequency magnetic fields^{6,7}. Here we report experiments in which electrically driven addressable spin rotations on two individual electrons were realized by integrating a micrometresize ferromagnet into a doublequantumdot device. We find that it is the stray magnetic field of the micromagnet that enables the electrical control and spin selectivity. The results suggest that our approach can be tailored to multidot architecture and therefore could open an avenue towards manipulating electron spins electrically in a scalable way.
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Main
Magnetic resonance was recently used to coherently manipulate the spin of a single electron^{5} in a semiconductor structure, called a quantum dot^{8,9}, whose tally of electrons can be tuned one by one, down to a single charge^{10,11}. However, producing strong and localized oscillating magnetic fields, which is a necessary step for addressing individual spins, is technically demanding. It involves onchip coils^{5,12}, relatively bulky to couple with a single spin, dissipating a significant amount of heat close to the electrons, whose temperature must not exceed a few decikelvins. In comparison, strong and local electric fields can be generated by simply exciting a tiny gate electrode nearby the target spin with lowlevel voltages. For scalability purposes, it is therefore highly desirable to manipulate electron spins with electric fields instead of magnetic fields.
To benefit from the advantages of electrical excitation, a mediating mechanism must be in place to couple the electric field to the electron spin, which usually responds only to magnetic fields. Spinorbit coupling^{13,14}, hyperfine interaction^{15} and gfactor modulation^{16} work as the mediating mechanism, which attract interest for their physical origins but necessitate refinement in terms of both manipulation speed and scalability. Instead, we controllably mix the spin and charge degrees of freedom in a magneticfield gradient^{6}, very much like the Stern–Gerlach effect^{17}. This allows for greater flexibility, because the method is applicable to any semiconductor material. In addition, the magnetic field profile can be engineered to enable the selective manipulation of several spins using a single electrode.
Thereby, we demonstrate addressable voltagedriven singlespin electron spin resonance (ESR) in a magneticfield gradient. Two electrons are confined and spatially separated from each other in a gatedefined double quantum dot^{18} (Fig. 1a). The a.c. electric field, E_{a.c.}, is generated by exciting a nearby gate that couples to both spins (with presumably smaller strength for the right electron). The magneticfield gradient is obtained by using a ferromagnetic strip that we integrate on top of the doubledot structure. The strip is magnetized uniformly along its hard axis by applying an inplane magnetic field, B_{0}, stronger than the micromagnet’s saturation field (∼2 T). In this condition, the resulting stray magnetic field has an outofplane component that varies linearly with position, pointing in the upward (or downward) direction to the left (or right) of the quantumdot locations (Fig. 1b). In addition, the inhomogeneity of the inplane component yields two different quantumdot Zeeman fields B_{0L} and B_{0R} (Fig. 1c). We use this feature to probe each spin separately.
To achieve ESR, we periodically displace the two electrons around their respective equilibrium positions in the slanting field. In each dot, the spin feels an upward magnetic field when the charge is displaced to the left. Conversely, the electron experiences a magnetic field pointing in the downward direction when displaced to the right. This effective oscillatory magnetic field induces transitions between the electron spin states (pointing in the direction parallel or antiparallel to the external field B_{0}) only when the driving frequency, f, matches the Larmor frequency, f_{0}, of the target spin. The latter is proportional to the corresponding quantumdot Zeeman field (h f_{0L,R}=g μ_{B}B_{0L,R}, where h is the Planck constant, g the Landé factor and μ_{B} the Bohr magneton). By adjusting the frequency, phase and duration of the a.c. electric field burst used to periodically displace the electrons, arbitrary single spin rotations can then be realized in each dot through the ESR effect, a prerequisite for realizing the CNOT gate using the exchange interaction between neighbouring spins^{19}.
To detect the electrically induced spin flips, we operate the double dot in the Pauli spinblockade regime, where no current flows unless spin flips occur in either dot^{5,20} (Fig. 1d). The blockade arises because of Pauli exclusion: once the electron in the right dot forms a spinpolarized triplet with the electron in the left dot (either the T_{+}=↑↑ or T_{−}=↓↓ state), the right electron cannot move to the left dot. The spinblockade regime is identified in the doubledot stability diagram by mapping the dot current, I_{dot}, under large source–drain bias as a function of the left and right quantum dot gate voltages (Fig. 1e).
We now show that electric excitation can induce singleelectron spin flips. We apply a continuous microwave voltage and follow the stability diagram around the resonance condition (Fig. 2a). The resulting electric field modifies the diagram through a process known as photonassisted tunnelling^{21} (PAT). In general, PAT can assist electrons in breaking the spin blockade by, for instance, hopping to the left dot triplet states, usually energetically inaccessible. For our power level (typically −40 dBm taking into account line attenuation), a clear spinblockade region remains with leakage current below the noise floor of the experiment (20 fA). The spin blockade is lifted off by PAT only at much higher power level (−20 dBm, 100 fA leakage current).
The situation is different at the resonance condition for ESR, where a finite leakage current now flows in the spinblockaded region. Starting from the ↑↑ state, an ESR field resonant with the left (right) electron changes the initial state to the ↓↑ (↑↓) configuration. Expressed in the singlet–triplet measurement basis (S, T_{0}, T_{±}), ↓↑ or ↑↓ is an equal superposition of the T_{0} and S states. For the singlet component (S) of this state, the right electron can tunnel immediately to the left dot because the left dot singlet state is energetically accessible. The T_{0} component first evolves into S (owing to the large difference in quantumdot Zeeman fields, Δ_{Z}, compared with the exchange energy J, that is, g μ_{B}Δ_{Z}≫J), and then the right electron can move to the left dot as well^{22}. Thus, by resonantly flipping the spin of the electron residing on either dot to form an antiparallel spin state, an electron charge moves through the dots, thereby lifting the spin blockade. The resonant response is observed clearly as B_{0} and f are varied for constant E_{a.c.} (Fig. 2b). Two equally spaced peaks (with spacing Δ_{Z}=13±2 mT) in I_{dot} are seen at a frequency proportional to B_{0}. Judging from the amplitudes, we attribute the first (second) peak to spin flips of the electron residing on the left (right) dot. This selective addressing is enabled by the inhomogeneous inplane strayfield profile mentioned above. By slightly modifying the micromagnet geometry, the frequency selectivity can, in principle, be used to address individual spins in a scalable way (see Supplementary Information, Note A.1).
The linear dependency of each resonance on the external magnetic field is a key signature of ESR because the Larmor frequency is proportional to B_{0}. From the averaged position of one of the ESR peaks obtained over a wider range of magnetic field (Fig. 2c), we determine g=0.41±0.01, in good agreement for our type of device. Following the peak position below the micromagnet’s saturation field, we have confirmed that B_{0L,R} are smaller than the external field, a feature expected for the stray magnetic field (see Supplementary Information, Fig. S1a).
Evidence for spin–charge coupling induced by the slanting field is revealed in the ESR peak height. This gives information on the effective a.c. magneticfield strength, B_{a.c.}, which is proportional not only to E_{a.c.} but also to the magneticfield gradient, b_{SL}. To estimate B_{a.c.}, we use the nonmonotonic response of the peak height to microwave power. As the power level is raised, the peak amplitude initially increases and then saturates past a certain level corresponding to an electric field E_{a.c.}^{*} (Fig. 3, inset). This response results from the interplay between the ESR and fluctuating Overhauser fields. The Overhauser field arises from hyperfine interaction between the electron and nucleus spins of the host material^{23}. This interaction shifts the Larmor frequency randomly by an amount Δf_{0}=g μ_{B}B_{N}/h, where B_{N} is the amplitude of the nuclear field fluctuations. For B_{a.c.}>B_{N}, power broadening washes out the fluctuations. Every time an electron blocks the transport by spin blockade, the ESR field flips its spin and the current flow is therefore saturated. For B_{a.c.}<B_{N} the resonance condition is met only occasionally. Fewer electron spins are flipped per unit time and the current is consequently lower. Saturation occurs at B_{a.c.}∼B_{N}/2 (refs 15, 22).
The Overhauser field fluctuations are also responsible for the jitter in the peak position visible in Fig. 2b, which enables us to extract B_{N}=2.4 mT. Using this result, we estimate B_{a.c.} to be 1 mT at the onset of saturation. Remarkably, such a magnitude is obtained for microwave power 500 times smaller than for magnetically driven ESR with an onchip coil^{5,12}. By operating deeper in the Coulombblockade region of the stability diagram, fields as strong as 10 mT are possible because stronger PAT is required to lift the spin blockade, yielding a spinflip time as fast as 20 ns. The efficiency can further be improved by increasing the micromagnet thickness and using stronger ferromagnetic materials^{7}.
In Fig. 3, we plot the estimated spinflip rate (Rabi frequency, ν_{Rabi}=g μ_{B}B_{a.c.}/2h) normalized over electric field in a magneticfield range above the micromagnet saturation field. The normalized Rabi frequency does not vary significantly with B_{0}, as expected because b_{SL} should be constant in this regime. A linear fit through the data suggests that a second field also contributes, on a smaller level, to the effective ESR field. We attribute the second contribution to the usual spin–orbit coupling present in most semiconductors. The latter gives rise to an intrinsic slanting magnetic field of slope 2B_{0}/l_{so}, where l_{so} is the characteristic spin–orbit length^{13,24}. The precise profile of this effective magnetic field depends on the relative orientation of B_{0} and E_{a.c.} with respect to the crystal directions. For our geometry, the spin–orbit contribution works in reducing the ESR field when l_{so}>0, a trend observed in the data. To be more quantitative, we derived the following expression for the total effective ESR field strength (see Supplementary Information, Note A.2):
where Δ and l_{orb} are the quantum dot’s confinement energy and orbital spread. The fit to equation (1) yields b_{SL}∼0.8 T μm^{−1}, in good agreement with the expected straymagneticfield profile, and l_{so}∼58 μm, whose magnitude is consistent with recently observed spin–orbitmediated ESR in a similar system^{13}. The fluctuating nuclei field was shown to also enable electrically driven spin flips^{15}, and should, like the spin–orbit case, contribute to our ESR signal. However, the weak ESR response observed at low external magnetic fields (where b_{SL}∼0) implies that the hyperfine effect does not contribute significantly to the effective ESR field (see Supplementary Information, Fig. S1b).
The spin rotations demonstrated here, in combination with experimentally realized spin readout^{25,26} and tunable exchange coupling^{27,28}, fulfil many of the requirements for quantum computing with electron spins in quantum dots^{19} using only electric fields. In contrast to previously reported voltagedriven ESR mediated by spin–orbit coupling or interaction with nucleus spins, our scheme, which is applicable to any material, does not rely on intrinsic properties that are also responsible for degrading the spin coherence in solidstate systems. The coherence time for our hybridized spin is expected to be as long as 1 ms using cleaner materials^{6} such as carbon nanotubes, Si nanowires and SiGe heterostructures. Moreover, micromagnets may simplify the daunting task of integrating many quantum dots into a multiqubit quantum register. The independent addressing of the spin in each of the double dots observed here and in ref. 15 inspires scalability with the help of micromagnets. By engineering the strayfield profile, a common ESR gate could be used to operate on any spin in the register, simply by matching the driving frequency to a positiondependent Zeeman field.
Methods
The measurements were made in an Oxford Instrument Kelvinox 100 dilution refrigerator operating at a base temperature of 40 mK with an estimated electronic temperature of 200 mK. The microwave signal is applied to the ESR gate using a commercial microwave source (Agilent 8360B). The ESR gate is part of an onchip coplanar waveguide, which is wire bonded to the sample holder’s alumina coplanar waveguide. The latter is connected to the highfrequency line of the dilution refrigerator using a microwave bead. These precautions are taken to minimize loss by improving impedance mismatches.
The GaAs–AlGas heterostructure from which the sample was made was purchased from Sumitomo Electric. The twodimensional electron gas has a mobility of 1×10^{2} m^{2} V^{−1} s^{−1} and an electron density of 3×10^{15} m^{−2} measured at 1.4 K. The device used in this experiment suffered initially from a high level of telegraphic noise, associated with the switching of a few background charges. This made the quantumdot behaviour extremely unstable. To improve the charge stability, the device was first cooled down under a positive bias of +0.5 V applied to the quantum dot gates. A negative bias of −2 V was then applied to the micromagnet once the device had reached the base temperature.
The occupation of the double dot by only two electrons was confirmed by first opening the interdot barrier (using the P gate) to form a single dot and then opening the barriers separating the dot from the source and drain reservoirs (with the T gate). Under these conditions, no extra Coulombblockade peaks appeared in the region N_{L}+N_{R}=0 of the doubledot stability diagram.
The stray magnetic field produced by the ferromagnetic strip is calculated numerically using the Mathematica package Radia, available at http://www.esrf.fr, assuming a uniform magnetization and using the saturation magnetization of cobalt (μ_{0}M_{Co}=1.8 T). The strip is 5 μm long.
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Acknowledgements
We thank F. H. L. Koppens and C. Buizert for discussions, I. Mahboob for comments and Y. Sekine for advice. S.T. acknowledges financial support from GrantsinAid for Scientific Research S (No 19104007) and B (No 18340081).
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M.P.L. designed the experiment, fabricated the device and wrote the paper. T.O. ensured proper operation of microwaves. M.P.L. and T.O. carried out the bulk of the experimental work and analysis. Y.T. conceived the theory. Y.S.S. participated in experiments. T.K. participated in the theoretical work. K.Y. assisted with device processing. T.T. assisted with micromagnet technology. S.T. planned the project. All authors discussed the results and commented on the manuscript.
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Supplementary Notes and Supplementary Figures 1–3 (PDF 625 kb)
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PioroLadrière, M., Obata, T., Tokura, Y. et al. Electrically driven singleelectron spin resonance in a slanting Zeeman field. Nature Phys 4, 776–779 (2008). https://doi.org/10.1038/nphys1053
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DOI: https://doi.org/10.1038/nphys1053
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