Robust micromagnet design for fast electrical manipulations of single spins in quantum dots

Electronics based on spin degrees of freedom, e.g., spintronics and spin-based quantum computing,^1,2) has attracted much attention recently as an approach to overcome the scaling limit of conventional semiconductor microelectronics. Such spin-based processing requires the ability to rapidly manipulate electron spins before spin information is lost. A canonical method to control spins in solid-state nanoscale devices is electron spin resonance (ESR).^1–3) Recent progress in nanoelectronics science has led to the observations of ESR at the single-electron level by several different mediating mechanisms in semiconductor quantum dot (QD) devices.^4–7) However, control strengths are often bounded by material properties and are smaller than noises in the condensed-matter environment, which destroy the quantum nature of spin during manipulation. For coherent spin rotations with no unintentional phase shift, one needs fast ESR (>50 MHz, e.g., for GaAs⁸⁾), which acts on timescales shorter than the dephasing time (∼40 ns, e.g., in GaAs). As two-spin entanglement can be gated on much faster timescales,^9,10) a generic approach to enable fast ESR would provide means to the high-fidelity universal operations of spin qubits in a quantum processor³⁾ and the control of the coherent superposition state of solid-state electron spins possibly interacting relatively strongly with the environment.

In this letter, we focus on the electrically driven ESR in QD devices with micromagnets (MMs). The integration of MMs gives the following advantages for the local control of single spins.^7,8,10–14) First, ESR can be electrically driven in a slanting magnetic field induced by MMs (MM-ESR).^7,11) For practical applications, control by electric fields is particularly appealing, since they are more readily generated locally than magnetic fields. MM-ESR has been shown to generate the fastest electrical spin rotations so far in semiconductor QDs⁸⁾ and can also boost ESR speed mediated by the spin–orbit interaction.^5,15,16) Second, the technique is highly scalable and can be extended to more than 25 spin qubits.¹⁵⁾ Third, it is material-independent^11,13) and applicable to QDs fabricated in various materials, e.g., isotopically purified C- or Si-based semiconductors with longer spin coherence times. However, for the application of the MM approach to fast spin control, a careful design of the MM is mandatory. In what follows, we show comprehensively how the local magnetic field produced by MMs can be tailored to facilitate a fast addressable ESR necessary for QD-spin-based spintronics and quantum information processing.

To explore the possibility of high-fidelity spin rotations with the MM, it is of primary importance to clarify the required conditions of its local field. In MM-ESR, two magnetic field properties are mainly utilized: the linear gradient of the magnetic field component normal to the spin quantization axis (b_sl) and the difference in Zeeman field between QDs (ΔB_Z). Here, the Zeeman field is defined as the magnetic field component parallel to the spin quantization axis. As formulated in Ref. 11, b_sl couples the electron's spin and orbital degrees of freedom, and allows the electrically driven ESR. ΔB_Z, on the other hand, gives different Larmor frequencies for spins in different QDs, allowing access to a single spin without affecting others. To quantify the required field strengths, we first analyze the conventional MM designs that were employed previously in GaAs double QD (DQD) devices^7,10,17) (Fig. 1). For each device, two ESR peaks were observed, which demonstrates the control addressability. From the ESR peak separation, ΔB_Z can be measured directly. On the other hand, the exact values of b_sl are difficult to extract from the experimental data. Although the ESR peak height reflects the Rabi frequency f_Rabi, f_Rabi is not solely determined by b_sl, but proportional to the product of both b_sl and the amplitude of the ac driving electric field E_ac relative to the leading order of b_sl and E_ac.¹⁸⁾ Therefore, we numerically simulate the MM properties,¹⁹⁾ b_sl and ΔB_Z, using a boundary integral method (Table I). In our simulation, we take the device-dependent geometrical parameters into account, which may modify the MM properties. We also allow for the 75 nm misalignment of the MM pattern with respect to the QDs, which is possibly present in real QD devices. Other sources of parameter error, e.g., the interdot distance estimation, can also have an effect but not significant. The consistency of the simulated MM properties and experimental observations (Table I) supports the validity of this simulation. All MMs satisfy the conditions b_sl $\gtrsim$ 0.4 mT/nm and ΔB_Z $\gtrsim$ 10 mT, which we consider sufficient for discerning individual ESR peaks in GaAs QDs.

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However, we argue that fast ESR, with spin-flip frequencies sufficiently high for the demonstration of nontrivial two-spin operations in GaAs QDs as a benchmark platform,^2,3) will require the improvement of the MM field properties in the following ways. First, b_sl needs to be $\gtrsim$0.8 mT/nm to further increase the ESR rotation speed. Currently, ESR is the most time-consuming operation in the universal gate set. Given this field gradient and assuming typical experimental parameters in GaAs QDs,¹²⁾ f_Rabi of $\gtrsim$50 MHz is accessible. This allows several spin rotations within the ensemble phase coherence (or dephasing) time $T_{2}^{*}$ of a few tens of ns in GaAs QDs,^8,22) with an improved gate fidelity.

Secondly, for fast ESR, ΔB_Z has to be increased in proportion to f_Rabi. The reason is that to address single spins independently, the ESR peak separation |g|μ_BΔB_Z (g is the Landé factor and μ_B is the Bohr magneton) must be greater than the ESR peak width $\Delta f \sim f_{\text{Rabi}} + T_{2}^{*}{}^{ - 1}$.^8,23) While for relatively slow ESR, Δf is dominated by the fluctuation in Larmor frequency, mostly due to the Overhauser field; the contribution of f_Rabi to Δf becomes no longer negligible when the spin flip rate exceeds the dephasing rate, i.e., $f_{\text{Rabi}} > T_{2}^{*}{}^{ - 1}$. Assuming that f_Rabi is more or less the same in all QDs, fast addressable ESR is only feasible when |g|μ_BΔB_Z > 2f_Rabi. That is, for f_Rabi = 50 MHz in GaAs QDs with |g| = 0.40, ΔB_Z has to be >18 mT.

Third, MM field properties should be tolerant of the relative misalignment between MM and QDs. Such misalignment is usually present in real QD devices as it can arise from overlay fabrication errors and inaccurate estimations of QD positions (QDs may not be formed exactly at positions expected from gate geometry, owing to imperfect simulation and/or dopants in semiconductor wafers). It is difficult to reduce this error distance d_err, especially for a multiqubit system. In reality, d_err is typically 50–100 nm even with state-of-the-art semiconductor processing technology. Not surprisingly, this amount of misalignment can spoil the MM properties (Table I), since the stray field of the MM tends to be strongly position-dependent.

In the following, we optimize the MM design relying on the simulation to meet all the following requirements clarified above: (1) b_sl $\gtrsim$ 0.8 mT/nm, (2) ΔB_Z > 18 mT, and (3) misalignment robustness, i.e., conditions (1) and (2) are met in the presence of d_err = 75 nm MM misalignment.

In general, the MM designs for QD devices can be categorized into two types: a "single" MM and a "paired" MM. A single MM can be specified by the length l(x) in the magnetization axis y, where the x-axis is in the two-dimensional electron gas (2DEG) plane and orthogonal to the y-axis [see Fig. 1(a)]. A paired MM [see Figs. 1(b) and 1(c)], on the other hand, can be characterized by the gap width g(x). If l(x) = g(x), these different types of MMs are almost equivalent in that they produce in principle roughly the same field properties away from the MM edges, with the stray field directing in the opposite direction. Therefore, although in what follows, we will restrict ourselves to the paired MM, the same type of reasoning will hold for the single MM as well.

First, we examine how to realize a large b_sl in the presence of finite misalignment. We distinguish between the two main configurations of the QDs and MM, namely, "parallel" and "perpendicular", which are related to the angle of the QD alignment axis with respect to the MM magnetization axis (||B₀ and ||y). b_sl becomes largest when we minimize the displacements of QDs from the MM gap center (y = 0) and this is realized when QDs are on the MM center line. Therefore, the perpendicular configuration [see Fig. 1(b)] is more favorable than the parallel one to make b_sl more robust against the misalignment. We note, however, that in the parallel configuration [see Figs. 1(a) and 1(c)], where the QD axis is aligned with the magnetization axis, a large ΔB_Z can be readily obtained with asymmetrical MMs.

Next, we discuss how a good balance between a large b_sl and a large ΔB_Z is achieved. A conventional method of producing ΔB_Z in the perpendicular configuration [as in Fig. 1(b)] is to taper the MMs. However, the simulation shows that this can harm the robustness of b_sl. The reason is related to the size of the gap between the paired MMs. When the gap becomes smaller than some optimal value for a given d_err, b_sl gets more susceptive to the misalignment, whereas a larger gap will weaken b_sl at the center [see Fig. 2(a)]. For the robustness of b_sl, it is therefore important not to change g(y) too far from this optimized value, which maximizes b_sl with finite misalignment. This value is ∼260 nm for the geometrical parameters such as t_MM and d_2DEG specified in Table I. Therefore, g(y) should not be tapered much, but the problem with a paired MM with g(y) roughly fixed to some value is that it can produce only a small ΔB_Z.

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We find through the simulation that a "bridge" structure is more favorable than the tapered structure to obtain the necessary ΔB_Z [see Fig. 2(b) inset]. The bridge part creates the Zeeman field (and also the slanting field) opposite to that induced by the rest (the paired MM with a constant gap). This change in sign brings about an abrupt distribution of the Zeeman field and hence a large ΔB_Z among the QDs. Since ΔB_Z decays slower [see Fig. 2(b)] than b_sl, a sufficiently large ΔB_Z can be achieved with only a slight decrease in b_sl using this approach. The advantage of this structure is that unlike the tapered one, the misalignment robustness of b_sl is ensured by the paired MM with an optimal and constant opening.

We note that by putting the bridge closer to the QDs, this structure can also be utilized to supply ΔB_Z exceeding 50 mT. This makes a single-step controlled-phase gate operating at ∼20 MHz within experimental reach in GaAs QDs,^3,24) and may help to construct efficient two-qubit gates other than CNOT and $\sqrt{\text{SWAP}}$.

An example of the optimized, bridged MM design is shown in Fig. 3(a). The constant gap of the split-pair part is chosen to maximize the minimum b_sl within 75 nm from the MM axis, so that b_sl becomes misalignment-proof by design. The position of the bridge is chosen to keep ΔB_Z > 18 mT in the presence of a 75 nm misalignment. Figures 3(c) and 3(d) show the simulation results, where b_sl > 0.9 mT/nm for both QDs and ΔB_Z > 19 mT in a 150 × 150 nm² area. We would like to point out that the bridge design is less susceptible to misalignment than previously presented designs (Table I). To demonstrate the effectiveness of the presented design scheme, we incorporated this design of MM with a DQD device (device A). The ESR spectra measured in device A [Fig. 3(e)] shows ΔB_Z to be around 70 mT. This indicates that the average Zeeman field gradient across the DQD is as large as 0.35 mT/nm (the simulated value is 0.14–0.32 mT/nm, see Table I), even when we assume a slightly large interdot distance of 200 nm to account for a relatively weak tunnel coupling observed in this device. This also ensures an ESR addressability for f_Rabi > 100 MHz.

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For device B, we also used the bridge design. Here, a peak separation ΔB_Z of about 60 mT was experimentally observed. Again, excellent agreement between experiment and simulation is shown [Fig. 4(b) and Table I].

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In summary, we discussed the crucial requirements of the MM stray field to obtain fast spin rotations in QDs using MMs, i.e., a misalignment robust b_sl and ΔB_Z. We proposed a new design to satisfy these requirements by employing a constant gap and a bridge in a paired Co MM. The proposed design gives b_sl > 0.9 mT/nm and ΔB_Z > 19 mT for realistic device parameters up to a misalignment of 75 nm of the MM. This facilitates the realization of fast addressable ESR ($\gtrsim$50 MHz in GaAs QDs)⁸⁾ as well as other quantum gate operations involving ESR such as the CNOT gate operation.³⁾ The MM field properties can be even further enhanced by decreasing the distance between the MM and the QDs as well as using a MM material with stronger magnetization than Co. Since the spin operation scheme is material-independent, the proposed design would be useful for optimizing the performance of nanostructure-based spintronic devices and quantum information processors, and for the precise control of a solid-state electron spin and its superpositions in various materials.

Acknowledgments