Exploiting bistable pinning of a ferromagnetic vortex for nitrogen-vacancy spin control

The ability to control ferromagnetic (FM) vortices may lead to advances in spin-based data storage, sensing, or quantum information processing. Precise translations of vortex domain walls in nanowires have been proposed as a platform for non-volatile memory.² Vortex pinning presents an opportunity for the construction of magnetic logic elements by bistable magnetic switching of a vortex between two pinning sites.³ We have recently shown that a vortex state in a ferromagnetic disk can be driven into proximity with a nitrogen-vacancy (NV) spin in diamond to provide the strong, local, rapidly tunable magnetic field gradient needed to address spins on the nanoscale, for applications in sensing or quantum information.¹ In all these cases, manipulation of the vortex state requires an understanding of the motion of the vortex core through a complex potential landscape produced by defects in the FM material.

Thin micron-scale FM disks and wires produce magnetic configurations known as vortex states or vortex domain walls.^4–6 At the center of the vortex resides the vortex core, a region of large magnetostatic energy density, which produces a large dipole-like magnetic fringe field.⁷ Pinning of the vortex core occurs when it encounters defects which effectively lower the local energy of the vortex state.^8–10 The interaction between the pinning sites and the vortex state has been characterized for a variety of pinning sites both artificially produced^11–14 and intrinsic to the fabrication process.^15–18 It has also been shown that by measuring the hysteretic displacement of the vortex core, an effective pinning potential can be constructed that describes the vortex's motion.^19,20

Previous work¹ has shown that the coupling between a vortex state in a thin FM disk and individual NV spins produces a large splitting Δ between the m_s = ±1 sublevels of the NV ground state as the vortex is driven into proximity with the NV center using an applied magnetic field. As the applied magnetic field is swept, Δ does not evolve continuously, but instead changes by a series of discrete steps. In this letter, we will demonstrate that the non-continuous evolution of Δ arises from pinning of the vortex, and that the pinning landscape can be sufficiently characterized to exploit the resulting vortex dynamics for controlling the NV spin resonances on nanosecond timescales. First, we will measure the displacement of the vortex as it is raster scanned across the sample, and construct a map of the underlying effective pinning potential. Next, we will monitor the NV spin splitting Δ using optically detected magnetic resonance (ODMR), as the vortex is swept along a path. The jumps in Δ can be understood from the vortex motion in the pinning potential. With this behavior mapped out, we then put it to use by identifying bistable points in the potential which allow fast switching of NV resonances.

The vortex state is characterized by a curl of in-plane magnetization circulating about a central core.^4–6 The vortex core is a region with a half-width set mainly by the material's exchange length, typically ∼10 nm in which the magnetization is mostly out of the plane of the FM disk.²¹ This out-of-plane magnetization produces a strong localized field with a predicted B_z ∼ 100 mT, 20 nm above the surface of the disk. The field is strongest at the center of the vortex core and falls off as ∼r⁻³.

The vortex core can be translated through a FM disk with the application of an in-plane magnetic field B. As the field increases, the component of magnetization along B increases, and the vortex core is displaced from the center of the disk perpendicular to B. For small displacements, the motion of the vortex core can be described using the rigid vortex model with a potential $U0(x,B)≈12k|x|2−kχ0(xBy+yBx)$⁠, where k is the stiffness coefficient determined by the demagnetization factor of the disk, χ₀ is the displacement susceptibility, and x = (x, y) is the position of the vortex core. Following the path of minimum energy, the equilibrium position x₀ = χ₀(B_y, B_x) of the vortex is displaced linearly with applied field.^7,22 The effects of pinning can be included in this model by introducing localized potential minima $Up(x)=∑nUp,i(x−xi)$⁠, where U_p,i(x – x_i) are the peaked functions describing the pinning potential produced by a defect located at a position x_i.^19,20

FM vortices were prepared in Permalloy (Py) (Ni_0.81Fe_0.19) disks with a diameter d = 2 μm and thickness t = 40 nm (dimensions were checked with atomic force microscopy). The samples were fabricated via electron-beam lithography, electron-beam evaporation, and liftoff atop a gold co-planar waveguide (CPW). Diamond nanoparticles (DNPs, DiaScence, Quantum Particles), with diameter ∼25 nm, each containing zero to several NVs, are mixed with a 1.0% solution of polyvinyl alcohol, M_w 85 000–124 000 (Aldrich 363146) in de-ionized water. The DNPs are spun coat onto the CPW/Py disks yielding a film thickness of ≈35 nm. A photoluminescence (PL) map of the NV-vortex system discussed in this letter is shown in Fig. 1(a), with the NV-containing DNP located in the lower right quadrant of the Py disk. From two-photon correlation measurements (not shown), we conclude that there are several NVs in this DNP.

We use a differential magneto-optical Kerr effect (MOKE) technique to measure the relative vortex displacement Δx₀ in response to an AC magnetic field B₁ produced by the CPW along the y-direction.²³ By sweeping B₁ and an additional field B₀ produced by a translatable permanent magnet at 45° to B₁, we can raster scan the vortex core across the Py disk as is illustrated in Fig. 1(b); shifting the vortex core towards and away from the DNP. Using the method described in Ref. 20, we map the pinning potential seen by the vortex across the Py disk. First, we measure Δx₀ vs. B₁, at different values of B₀. For example, Fig. 1(c) shows Δx₀ at B₀ = 2 mT, with the sweep of B₁ both ascending $Δx0+$ (blue) and descending $Δx0−$ (red). The hysteretic jumps in Δx₀ are transitions between the free path described by U₀ and various local minima produced by pinning sites U_p. The most visible jumps are marked as A1–A7 for ascending B₁ and D1–D6 for descending B₁. The positions of the jumps in Δx₀ reveal the positions of inflection points of the vortex pinning potential. From the known inflection points, still following Ref. 20, we construct an effective 1D pinning potential u_p(x) along the path of the vortex core. Figure 1(d) shows u_p(x) constructed from the data in Fig. 1(c). To confirm that u_p(x) captures the observed vortex behavior, we simulate Δx₀ vs. B₁ (Fig. 1(d) inset) using gradient descent in the potential U₀ + u_p, and find that the plateaus and jumps are reproduced. By constructing u_p(x) at different values of B₀, we can create a pinning map (Fig. 1(e)) that shows how the vortex is trapped in different pinning sites as its equilibrium position is shifted towards or away from the NVs. The region of the disk shown in the map in Fig. 1(e) is indicated by the boxed region in Fig. 1(b). We will now focus on the NV spin splitting as the vortex is swept along the path at B₀ = 2 mT. The same set of measurements has been performed at several other values of B₀, with similar results.

NV spins are initialized and read-out via a standard confocal ODMR technique at room temperature.^24,25 A microwave signal is added to the CPW current at frequency f_MW. An ODMR spectrum is obtained by sweeping f_MW and measuring the PL intensity. A reduction in the PL intensity reveals a transition from the m_s = 0 to m_s = ±1 sublevels of the NV ground state. Application of a magnetic field results in a splitting of the m_s = ±1 sublevels about the zero-field splitting f₀ = 2.87 GHz.

Figure 2(a) shows a set of ODMR spectra taken as B₁ is swept from −3 to 3 mT, with B₀ = 2 mT. Two pairs of resonances are visible, arising from NVs with symmetry axes in two different orientations. The splitting Δ is given by the frequency difference between a pair of resonances. The total field which contributes to Δ is B_net = B₀ + B₁ + B_v, where B_v is the magnetic field produced by the vortex core at the position of the NV. The close proximity of the vortex core results in increased B_v, and thus a larger impact on Δ at positive values of B₁.

As B₁ is swept, we observe a discontinuous evolution of Δ. At multiple locations along B₁, we observe large jumps in Δ from one spectrum to the next. The discontinuous evolution of Δ can be understood by considering the path taken by the vortex Δx₀(B₁). The marks A1–A7 are shown in Fig. 2(a) at the same values of B₁ as in the vortex displacement data in Fig. 1(c). We see that the marks also align with the jumps in Δ as a result of the coupling between the vortex and NV spin via B_v. For example, the jumps in Δ at A4 and A5 correspond to the transitions into and out of the large pinning site near Δx₀ = 0 shown in Fig. 1(d).

We can also observe a hysteretic response in Δ by conducting the same ODMR scans but with descending B₁ (Fig. 2(b)). The general behavior is similar to that observed along the path taken with ascending B₁, but as expected we see a shift in the jumps in Δ which now align with the marks D1–D6 from Fig. 1(c). One example of the hysteresis is near f_MW = 3.2 GHz and B₁ = 2.1 mT, which is outlined by a dashed oval in Figs. 2(a) and 2(b). This region exhibits bistable existence of a resonant transition in the ODMR spectrum as a result of the differing paths taken by the vortex core with ascending and descending B₁.

To highlight regions of hysteresis in the NV spin splitting, the difference between the maps in Figs. 2(a) and 2(b) is shown in Fig. 2(c). In this map, black indicates no difference between the two ODMR maps, red indicates the presence of resonant transitions with ascending B₁ but not descending B₁, and green indicates the reverse. At the bottom, we plot the difference $Φ=Δx0+−Δx0−$ between the position of the vortex core for ascending and descending B₁. We observe a direct correlation between the features in the ODMR difference map and the non-zero regions of Φ. In the regions where the vortex core follows the same path (Φ ≈ 0) we find little difference between the ODMR maps. On the other hand, at values of B₁ where $Δx0+≠Δx0−$ (Φ > 0) we see a shift in the appearance and disappearance of the resonant transitions between the ODMR maps.

In Fig. 3(a), we show a line cut of the OMDR spectrum at B₁ = 2.1 mT, where B₁ has been increased to this value (red) and decreased to this value (blue) demonstrating the bistable existence of the resonant transition at this point. To gain a better understanding of this bistability, we return to the effective one-dimensional pinning potential u_p seen by the vortex core shown in Fig. 1(d). Fig. 3(b) shows the sum of u_p and the rigid vortex potential U₀ at B₁ = 2.1 mT in the region around the potential minimum. At this field, we see the existence of two local minima. The position of the vortex core is set by the approach to B₁ = 2.1 mT. For ascending B₁, the vortex core rests at Δx₀ = 140 nm (pinning site A) and for descending B₁ the vortex core rests at Δx₀ = 220 nm (pinning site B). The difference in position of the vortex core results in a difference of B_v, shifting the spin transition in and out of resonance with the applied MW field.

We can use the bistability presented by the presence of the pinning sites to manipulate the NV spin in a controlled bimodal fashion. Using a single, fast pulse through the CPW, we can switch the vortex core between pinning sites A and B shown in Fig. 3(b) and as a result switch the NV spin in and out of resonance with the MW field at f_MW = 3.2 GHz. Figure 3(c) shows an ODMR measurement taken over 30 min, with static fields B₀ = 2 mT and B₁ = 2.1 mT. Every t_d = 75 s an “on” or “off” pulse is applied, alternately. The “on” pulse increases to B₁ = 2.82 mT for τ = 50 ns. The “off” pulse decreases to B₁ = 1.74 mT τ = 1 μs. Application of an “on” pulse switches the vortex core from pinning site A to pinning site B, and bringing the NV spin on resonance. The “off” pulse switches the vortex back to its previous stable minimum at pinning site A and switches the NV spin off resonance. From Fig. 3(c), we observe the vortex core is stable in both equilibrium positions at pinning sites A and B over a period of t_d = 75 s at room temperature for multiple switches.

The probability P_s of switching the NV spin transition from “off” to “on” depends on the duration, τ, of the “on” pulse, as is shown in Fig. 3(d). Here, P_s is measured as the fraction of successful switches over 200 attempts. Clearly, there is a threshold value of τ_c ≈ 27 ns below which P_s falls to near zero. On the nanosecond timescale of the switching pulses, the dynamics of the vortex core must be considered. The dominant response of the vortex to a step in magnetic field is gyrotropic precession at frequency f_g of the vortex core about the new equilibrium position, which decays over a time τ_r as energy relaxation takes place. For a Py disk with 2 μm diameter and 40 nm thickness, we expect f_g ≈ 0.2 GHz, and typical relaxation time τ_r ∼ 50 ns.^15,26

The timescales of vortex precession and relaxation provide insight into the dependence of P_s on pulse time τ. The value of τ_c is similar to the expected relaxation time τ_r, suggesting that reliable switching between bistable states requires allowing enough time for damping of the gyrotropic precession in the intermediate state. Though P_s is quite low at τ < τ_c, there are small, repeatable spikes at τ = 5 and 10 ns. These values are close to multiples of the gyrotropic precession period 1/f_g and 2/f_g, which may indicate that faster switching is possible by exploiting the dynamic motion of the vortex core as it moves through the pinning potential.

In conclusion, we have demonstrated that by coupling an NV spin to a vortex state in a thin FM disk, we imprint the complex pinning landscape of the FM material onto the spin transitions. This interaction can be understood by mapping the path of the vortex core through the pinning potential using magneto-optical microscopy. Once the pinning potential is known, we can then use this pinning landscape to manipulate the NV spin transitions, switching transitions on and off resonance with a driving field, on nanosecond timescales. These results provide insights into how precise control of FM vortices may be used for addressing and controlling single spins, as well as opening a window into the dynamics of these pinned vortices on fast timescales.

This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Award No. DE-SC008148.