Modular nanomagnet design for spin qubits confined in a linear chain Open Access

Single-qubit gates in electron-spin qubits are typically implemented with electron dipole spin resonance (EDSR) enabled by a magnetic field gradient.^1–4 Successful operation of up to six qubits with a common micromagnet has been demonstrated.⁵ Despite its current success, scaling this technique to drive and address thousands of single qubits remains challenging,^6–10 because the magnetic stray field needs to fulfill different requirements at the same time. It should provide: (i) large driving gradients to achieve fast Rabi frequency $fRabi$⁠, (ii) a difference in Larmor frequencies $fL$ between neighboring qubits significantly larger than $fRabi$ for qubit addressability with low crosstalk, and (iii) small gradients of its longitudinal field component to prevent spin dephasing.

A possible strategy to address this challenge is to place individual nanomagnets close to the quantum dots that define the qubits, e.g., by co-integration of the magnets with the hosting structure. For this purpose, platforms offering an inherent confinement into a one-dimensional geometry, such as Si nanowires¹¹ or finFETs,^12,13 are of advantage, since less conventional gate electrodes are needed to control the qubits, freeing up space to place the nanomagnets. In such geometries, the displacement of the electron wavefunction for EDSR driving is preferably induced along the qubit chain. This is different compared to currently established micromagnet designs, where a displacement transverse to the qubit chain is used to drive EDSR.^5,10,13

In this Letter, we present a modular design of nanomagnets aimed at driving a large number of spin qubits arranged in a linear chain and strongly confined in directions lateral to the chain. We have simulated and fabricated “U”-shaped Fe nanomagnets down to a feature size of 50 nm. Their magnetization pattern has been imaged by spin-polarized scanning electron microscopy (spin-SEM),¹⁴ showing excellent agreement with micromagnetic simulations. We discuss sweet spots for reduced influence of charge noise and reduced crosstalk in EDSR driving and investigate how these properties depend on the alignment accuracy between nanomagnets and the spin qubits. The concept presented provides a scalable approach to EDSR driving of electron spins and can be implemented in several spin qubit platforms ranging from FinFETs to nanowires and planar geometries.

Figures 1(a) and 1(c) show top and side views of our design. Individual U-shaped nanomagnets are placed laterally to the one side of the qubit chain, one nanomagnet per two qubits. The two ends of the U point toward the qubit chain. Because of magnetic shape anisotropy, the magnetization in the two arms of each nanomagnet points in an alternating way toward and against the chain even with an external magnetic field $Bext$ applied along the qubit chain. This creates a gradient in the stray field component transverse to the external field. This gradient is used to drive Rabi oscillations of the spin qubits by spatially oscillating the position of the host electron in the direction along the chain. The sign of the longitudinal component of the stray field alternates from qubit to qubit, which together with $Bext$ provides differences in the Larmor frequencies of neighboring qubits and enables addressability with low crosstalk.

Micromagnetic simulations have been performed with the software package MuMax³.¹⁵ We choose Fe as the magnetic material and assume a saturation magnetization of 1.7 MA/m, an exchange stiffness of 21 pJ/m, a cell size of 1 nm, and zero temperature. The $x−y$ plane defines the sample surface with the qubit chain oriented along the x direction. The nanomagnets have arm width $W=50$ nm, length $L=150$ nm, thickness $T=50$ nm, separation $S=50$ nm between the arms, and $G=50$ nm between the magnets, which results in a qubit-qubit pitch of 100 nm. Smaller pitches can be implemented by decreasing the size of S and G, which would further increase the attainable driving gradient at the qubit locations.

The stray field $B=(Bx,By,Bz)$ of the nanomagnet at $Bext=0.05$ T is represented in a quiver plot in Figs. 1(a) and 1(c). The maxima and minima of its longitudinal component (x) coincide with the zero-crossings of the transverse one (y) [Fig. 1(b)], setting optimal locations for the qubits at the center of the nanomagnets $x=0$ nm (“arm”) and in between them at $x=100 nm$ (“gap”). At $z=25$ nm and $y=30$ nm, the driving gradient $dBy/dx$ reaches 3.9 mT/nm at the arm position and −4.8 mT/nm at the gap position, see also Table I. Together with zero-crossing of the dephasing gradient $dBx/dx$⁠, these positions define a sweet spot.

The electric field that induces the oscillatory spatial displacement of the electron wavefunction of an individual quantum dot will also unavoidably affect neighboring qubits. Individual addressability, therefore, requires qubit frequencies $fL$ to be separated far enough to only resonantly excite one qubit, which translates into having distinct magnetic fields among neighboring qubits. In our design, B_x has a similar magnitude but opposite sign at neighboring qubit positions. The difference of f_L between these qubits can be set by applying a finite B_ext. We find that at $Bext=0.05$ T, $fL=1.84$ GHz (arm) and 4.64 GHz (gap), assuming an electron g-factor of 2.0 [Fig. 1(d)]. When set to 0.10 T, $Bext$ matches $|Bx|$ such that $fL≈0$ at the arm location. For $Bext>0.3$ T, the driving gradient decreases significantly due to a reorientation of the magnetization pattern in the arms. At $Bext=2.0$ T, the magnetization almost saturates everywhere into the x direction, and neighboring qubits approach the same Larmor frequency. The magnitude of the driving gradient then approaches the values at low fields.

Hence, in our design $Bext$ can be adjusted to provide large driving gradients and at the same time single qubit addressability. The magnitudes of useful $Bext$ are significantly smaller than in common EDSR experiments and simulations with micromagnets,^{1–5,16–19} so that several advantages arise. First, a lower $fL$ leads to reduced crosstalk between neighboring drive lines due to the frequency dependence of their capacitive coupling.²⁰ Second, less demanding control electronics is required. Third, the spin relaxation time is increased,²¹ also avoiding hot spots when the Zeeman energy coincides with the valley splitting.²² Finally, coherent shuttling of qubits is facilitated at low fields, since the acquired spin phase during shuttling is directly dependent on the magnitude of $Bext$⁠.²³ 

With this nanomagnet design, $fL$ of next-nearest qubits is identical, which may still lead to crosstalk while driving. Different compensation and synchronization²⁴ schemes that were proposed for nearest-neighbor qubits can be applied to solve this issue. In addition, individual qubit frequencies can be engineered by changing the nanomagnet geometry along the chain. For example, an increase in S of every second magnet and G of every second gap to 70 nm (adapting W correspondingly to keep the same quantum dot pitch) leads to a difference of 607 MHz (at $Bext=0.05$ T) between next-nearest qubits. In this case, only every fifth qubit would have the same Larmor frequency.

The design summarized in Fig. 1 has four main advantages with respect to previous micromagnet designs: tunable addressability between neighboring qubits at a low external field (difference in $fL$ is above 2.5 GHz at $Bext=0.05$ T), a stronger driving gradient (⁠$dBy/dx=3.9$ mT/nm), a zero-crossing in the dephasing gradient $dBx/dx$⁠, and a modular design that can be extended to arbitrary long chains of spin qubits.

We have fabricated such U-shaped nanomagnets on a silicon substrate covered by native oxide using e-beam lithography on a PMMA-based bilayer followed by e-beam evaporation and liftoff of 50 nm of Fe and 3 nm of Pd as a capping layer. Before the magnetic measurements, the capping layer was removed in ultrahigh vacuum by Ne ion sputtering. An in-plane magnetic field pulse of 17 mT was applied along $+x$ to orient the bases of the magnets along the same direction. The magnets were then investigated at room temperature and in remanence (i.e., at $Bext=0$⁠) by spin-SEM, where topography and magnetization distribution of the first ∼2 nm of the sample surface are simultaneously determined with a lateral resolution of approximately 10 nm.¹⁴ 

In Fig. 2(a), we show an SEM image of the fabricated nanomagnets. The patterning process caused the edges to be rounded with respect to the geometry used in the simulations. This rounding is also visible in the spin-SEM image in Fig. 2(b), where an arrow-plot combining both in-plane magnetization components is overlaid on top of the topographic data. The magnetization follows the shape of the magnet similar to that obtained in the simulations at $Bext=0$ T shown in Fig. 2(c). Deviations between simulation and experimental magnetization patterns are found in the rounded region at the apices of the arms. In Fig. 2(b), we see that the magnetization rotates its direction at the apex of both arms along the x direction. This is compatible with the vortex visible in the simulation $x−z$ cut at $y=0$ nm in Fig. 2(c).

We investigate the impact of the vortex formation by comparing the simulation described in Fig. 1 (“relaxed”) with a hypothetical nanomagnet with arms uniformly magnetized in opposite directions (“uniform”). We see that the rearrangement of the magnetization pattern reduces the driving gradient as well as $fL$ at both qubit positions (Table I).

A smaller stray field is also found in the simulation where nanomagnets have a rounded apex (“rounded”). We approximate the shape of the nanomagnet apex [Fig. 2(a)] with a semi-cylinder with a diameter of 50 nm. All other simulation parameters are identical to the relaxed case. The comparison between these two geometries shows a reduction of the driving gradients by $≈20%$ at both qubit locations for the rounded case. This reduction is ascribed to the smaller amount of the magnetic material present in the vicinity of the qubit.

Scalable designs of one-dimensional arrays of qubits can be implemented by repeating the nanomagnets along the chain, provided that magnetization patterns among the different nanomagnets are similar. In Fig. 3, we show both in-plane magnetization components of a 3 × 3 nanomagnet array after applying a magnetic pulse of 17  mT along $+x$ and $−x$⁠. We extract the nanomagnets contour from the topographic data and overlay it on the magnetic measurement images. The comparison of the magnetization components between the two measurements demonstrates a symmetrical reversal of the magnetization pattern for opposite magnetic field directions with the bases of the magnets oriented along the direction of the magnetic pulse and the arms alternating between $+y$ and $−y$⁠. Upon closer inspection, we see in the horizontal magnetization images that the magnetization at the apices is partially oriented along the x direction, as already noted in Fig. 2(b). The sign of this component varies among the different apices, suggesting a varying chirality of the vortices, both within and between structures. We believe that the chirality is induced by random local defects, and according to the micromagnetic simulations, the energy difference between nanomagnets with identical or opposite vortex chirality is negligible. The vortex chirality has no significant impact on the driving gradient or addressability at the foreseen qubit positions.

Large driving gradients are accompanied by large dephasing gradients and may limit the fidelity of qubit operations. We, therefore, calculate the dephasing time and quality factor that we expect from the U-shaped nanomagnets. Charge noise has been identified as the main limitation to the gate fidelity of single-qubit gates for micromagnet-based EDSR^16,17 and will be considered as the dephasing mechanism in the following. We assume that the charge noise has a 1/f power spectrum.^17,25,26 The noise in an electric field will translate into noise in the spatial position of the electron wavefunction and through the dephasing gradient into noise in $fL$⁠. The relevant dephasing gradients are $dBx/dx$⁠, $dBx/dy$⁠, and $dBx/dz$⁠, since the y and z stray field components are negligible at the qubit position [Fig. 1(b)]. The gradients are calculated in a 50 × 50 nm² area by fitting the simulated stray field with a quadratic polynomial at each mesh point and taking the first derivative of the fitted polynomial. Higher orders lead to significantly smaller contributions and are neglected in the following.

The dephasing rate for each direction $i∈x,y,z$ is calculated by assuming root mean square displacements $Δi$ along the i direction²⁵ with

where $γe$ is the electron gyromagnetic ratio (a g-factor of 2), h is Planck's constant, and $dBx/di$ is the approximated first derivative of the stray field along the i direction. We assume $Δx=5$ pm (Ref. 2) and $Δy, Δz=0.5$  pm. The larger $Δx$ is justified by the weaker confinement of the quantum dot along that direction. The dephasing time is then $T2*=1/Γ=1/(Γx+Γy+Γz)$⁠.

Figure 4(a) shows the calculated dephasing rates $Γx$⁠, $Γy$⁠, and $Γz$ as a function of the qubit position around the arm position. At $x=0$ nm, $Γx$ is zero, and the same is true for the position gap at $x=100$ nm (data not shown). $Γy$ provides the main source of dephasing, whereas $Γz$ is negligible within the simulated range. We note that because the magnetic field is rotation-free outside the nanomagnets $dBx/dy=dBy/dx$⁠, such that the maxima in the driving field $dBy/dx$ at the arm and gap positions coincide with maxima in $Γy$⁠. Here, the strong confinement of the electron wavefunction in the fin reduces displacements along the y direction, thus mitigating dephasing.

We calculate the quality factor $Q=T2Rabi/τRabi$⁠, an upper bound for the qubit fidelity.³ The Rabi dephasing time is $T2Rabi=π2A2/fRabi$⁠,²⁵ and the Rabi frequency $fRabi=1/τRabi$ is

The A factor defines a $1/f$ power noise spectrum $A2/f$ of the qubit frequency. We obtain A from the relation²⁵ $σ=Alog(fh/fl)$⁠, where $fh$ (⁠$fl$⁠) is the highest (lowest) frequency of noise the qubit is exposed to, and $σ$ is the root mean square of the qubit frequency noise. The latter is obtained from $Γ$ calculated above by the relation $Γ=π2σ$⁠. We assume $fh=100$ MHz and $fl=1/3600$ Hz, similar to the experiment in Ref. 16, and a displacement amplitude $δx$ of 500 pm from the EDSR drive, similar to Ref. 20. Figure 4(b) shows the expected Rabi frequency, highlighting an increase in value with decreasing distance between the qubit and the nanomagnet, similar to the stray field trend visible in Fig. 1(a).

The quality factor $Q=fRabi2/A2π2$ is smoothed by a Gaussian filter with a sigma-width of 5 nm, compatible with a full-width at half maximum of the dot size in the $x−y$ plane of 12 nm [Fig. 4(c)]. We see that the qubits are optimally placed at x positions aligned with the center of the magnet and the gap between them, corresponding to the dephasing minimum positions described before. The weak dependency of Q on the y-coordinate is due to the equality between the driving gradient and the main dephasing gradient (⁠$dBx/dy$⁠), as a consequence of Ampère's law. On the other hand, we see a strong dependence on the x alignment of the qubit, caused by the corresponding increase in $Γx$⁠. Misalignment of 3 nm in the x direction reduces the quality factor by 30%. Nevertheless, quality factors higher than $105$ are achieved within 15 nm of misplacement, which exceeds the alignment precision of standard e-beam lithography.²⁷ Note that the quoted values for Q depend on the assumed displacement noise and scale inversely with $A2$ and, therefore, also inversely with the square of the dephasing gradient.

In conclusion, we have shown a nanomagnet design aimed at driving a linear chain of qubits that are strongly confined in directions lateral to the chain. EDSR can be performed with an external magnetic field below the saturation field of the nanomagnets, which is advantageous for operation of spin qubits at lower frequencies. At the foreseen locations, neighboring qubits differ in the Larmor frequency by more than 2.5 GHz. We found reproducible magnetization of fabricated nanomagnets and excellent agreement with micromagnetic simulations. The qubits are expected to have quality factors greater than $105$ with alignment tolerances relative to the nanomagnet positions that are within reach for e-beam lithography. The modularity of the design opens up the possibility to address a large number of qubits aligned along a linear chain.

We thank the Cleanroom Operations Team of the Binnig and Rohrer Nanotechnology Center (BRNC), Matthias Mergenthaler, Nico Hendrickx, and Felix Schupp for advise on sample processing, Andreas Fuhrer, Patrick Harvey-Collard, and Andrea Ruffino for discussions, Eoin Kelly for proof-reading the manuscript, and Andreas Bischof and Stephan Paredes for support in the lab. This work was supported as a part of NCCR SPIN funded by the Swiss National Science Foundation (Grant No. 51NF40-180604).