Frequency-based nanoparticle sensing over large field ranges using the ferromagnetic resonances of a magnetic nanodisc

We firstly compute the eigenmodes of the STO's elliptical free layer in the absence of a MNP. In the absence of an external field the obtained resonance frequencies of the first five modes (in order of frequency) are ${f}_{1}=1.60\,$ GHz, ${f}_{2}=3.09\,$ GHz, ${f}_{3}=5.09\,$ GHz, ${f}_{4}=5.55\,$ GHz and ${f}_{5}=7.77\,$ GHz. f₁ is of the same order as the excited mode reported by Zeng et al [46] when extrapolating their data to the zero-current case. Representations of the spatial profiles of mode 1 (most relevant for STOs) as well as the next four higher order modes are shown in figure 3. The shading encodes the amplitude of the magnetisation precession for each mode, with dark regions representing high amplitude resonant oscillations.

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The fundamental N = 1 mode (figure 3(a)) consists of an in-phase precession of all magnetic moments. The precession amplitude is largest at the disc's centre and decays in amplitude towards the boundaries of the free layer. The N = 2 mode (figure 3(b)) is characterised by two regions of large precession amplitude in the left/right halves of the disc, with the magnetic moments in the two parts precessing out-of-phase. Modes 3 and 5 (figures 3(c), (e)) both have multiple nodal axes parallel to the short axis of the ellipse. Mode 4 (figure 3(d)) is similar to mode 2 but with a nodal axis along the long axis of the disc.

In figure 4 we show the dependence of each mode frequency on the magnitude of a spatially uniform magnetic field applied along $+z$ (aligned with ${{\bf{m}}}_{0}({\bf{r}})$). The extracted field sensitivities (frequency shift per unit field) of all modes are consistent with one another to within 0.8% and have a value of 28.162 ± 0.108 MHz mT⁻¹. As will be shown below, however, the situation is more complex in the presence of a small MNP due to the combination of the localised MNP stray field and spatially non-uniform mode profiles. This will generate a clear dependence of the frequency shifts on the position of the MNP relative to the disc (both in the lateral and vertical directions).

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We now introduce a magnetic nanoparticle to the system as per figure 1(c). It is magnetised in the $+z$-direction, parallel to the applied external field.

In the equilibrium configuration of the bare disc, the disc's free layer magnetisation is parallel to the (out-of-plane) z-axis everywhere due to the strong perpendicular anisotropy. This is illustrated in figure 5(a), which shows the magnetisation vector field along the disc's long x-axis. If a MNP is present however, its stray field locally modifies the magnetisation. Examples of this are shown in figures 5(b) and (c) where the MNP is located at $x=-30\,\mathrm{nm}$ and $x=-60\,\mathrm{nm}$, respectively, above the major axis (y = 0) of the disc at d = 5 nm. The presence of the localised MNP field results in a slight localised canting of the magnetisation towards the MNP due to the in-plane components of the stray field. Note, however, that the actual change in the magnetisation orientation is very small. For example, at separation d = 5 nm the spatially averaged, normalised out-of-plane magnetisation is 0.99874 and at d = 20 nm it is 0.99981. These are both very close to 1, which is the value expected for a perfectly out-of-plane magnetised disc. However, despite only inducing a very small change in the magnetisation configuration, the stray field from the MNP can generate strong changes in the resonant frequency (up to ∼350 MHz for N = 1 at d = 5 nm), as will be shown below.

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Beyond modifying the resonance precession frequency of the magnetisation, the non-uniformity of the MNP field can also noticeably modify the spatial profile of the eigenmode. An example for the N = 1 mode is shown in figure 6(a), with a MNP at d = 5 nm. Although the mode excitations remain in-phase over the entire disc there is a reduced oscillation amplitude directly beneath the MNP, consistent with a MNP-induced, local stiffening of the magnetic moment. This also occurs for the N = 3 mode (figure 6(b)), which is the only other mode studied here that has an antinode (= location of maximum oscillation amplitude) at the disc's centre. Similarly, a left displaced particle (figure 6(c)) will lead to larger oscillation amplitudes for the N = 1 mode on the opposing (right) side of the disc. Somewhat analogously for vortices, a localised out-of-plane field at the core of a vortex can stiffen the core, increasing the frequency of its gyrotropic mode in the small displacement limit [36].

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We now look more closely at the influence of the position of the MNP on the frequency shifts, ${\rm{\Delta }}f$. In the case of small MNP-disc separations ($d\lessapprox 15$ nm), there is an intense, localised and predominantly $+z$ oriented field directly beneath the MNP (figure 2). As a result, at these small separations the effect of the MNP on a mode is determined by where the mode's dynamics are concentrated relative to the MNP's location. For MNPs directly above regions of the disc where there are high precession amplitudes, there is a frequency upshift (${\rm{\Delta }}f\gt 0$) due to the increased local $+z$-field below the MNP. This result is qualitatively consistent with that seen for uniform $+z$ fields in figure 4. However, when dynamics are occurring in a region which is laterally offset from the particle and the particle is very close to the disc, the precessing moments can be subject to a field oriented in the $-z$–direction (see again figure 2), leading to a reduction in the frequency (${\rm{\Delta }}f\lt 0$). We discuss these behaviours below.

Figure 7 shows the ${\rm{\Delta }}f$ values observed for all five eigenmodes ($N=1,\ldots ,5$) when shifting the MNP along the disc's long axis at vertical separations of d = 5, 20 and 50 nm, for lateral MNP positions of y = 0 nm and y = 20 nm. If the MNP is close to the surface of the disc (d = 5 nm, red lines in figure 7) then ${\rm{\Delta }}f$ closely follows the spatial mode pattern, as can be seen by comparing the red lines in figure 7 with figures 3(a)–(e). This reflects the fact that in this case the MNP stray field has a very localised influence on the underlying precessing moments. For example, for N = 1 there is a single, broad sensitivity peak near the centre of the disc (x = 0 nm) for both values of y, which mirrors that mode's spatial profile. Likewise, the N = 2 mode exhibits two sensitivity peaks near $x\approx \pm 40\,\mathrm{nm}$, i.e. near the locations of the mode's antinodes (see figure 3(b)). The curve for N = 3 shows three such peaks, with the outer ones slightly higher, reflecting the fact that the outer antinodes of this mode have a larger amplitude than the middle one (see figure 3(c)).

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This pattern continues for N = 4 and N = 5, but while the modes $N=1,2,3$ show fairly similar sensitivities for y = 0 nm and y = 20 nm, those for N = 4, 5 are quite different for both values of y, reflecting the fact that these two modes are less uniform along the short axis of the disc. This is very obvious for N = 4 where the mode antinodes are located at $y\approx \pm 25\,\mathrm{nm}$ (figure 3(d)), leading to a large positive ${\rm{\Delta }}f$ for the laterally y-offset MNP (bottom plot for N = 4 in figure 7).

Comparing the curves obtained for different d values in figure 7, it is evident that with increasing disc-MNP separations, the shifts are reduced and the curves become more homogeneous and indeed comparable for each mode. For an intermediate separation of d = 20 nm (solid blue lines), the influence of the mode profiles on the MNP-position-dependent frequency shifts are still visible. However, at a larger separation of d = 50 nm (dashed black lines), the ${\rm{\Delta }}f$ curves for each mode become very similar (those for N = 1, 4 show a flat, broad peak around the centre of the disc whereas those for $N=2,3,5$ have a more plateau-shaped profile, but the difference is small). This is consistent with two factors: (i) a much more uniform MNP field across the disc at large d (figure 2) (with a net $+z$-orientation, leading to a frequency increase); and (ii) almost identical sensitivities for each mode in a uniform $+z$ field (figure 4), leading to similar ${\rm{\Delta }}f$ values for each mode. We note that in the presence of multiple MNPs at similarly large separations (as might occur in real devices) this will likely result in a ${\rm{\Delta }}f$ that is (roughly) proportional to the number of nanoparticles present, due to the fact that all of them induce a similar frequency change, independent of their exact location above the disc.

Figure 8 shows how ${\rm{\Delta }}f$ varies as the vertical separation d is varied for a laterally centred MNP (i.e., one which is laterally positioned at $(x,y)=(0,0)$). Consistent with the d = 5 nm data in figure 7, the N = 1 and N = 3 modes, which both have dynamics concentrated below the centred MNP, exhibit the largest ${\rm{\Delta }}f$ at small separations. Furthermore, due to the concentration of dynamics below the MNP, the shifts are positive for these two modes for all d values. This is because as d is varied there is no sign change in the out-of-plane component of the MNP stray field, ${H}_{{\rm{MNP}}}^{{Z}}$, which acts on the central precessing moments in the disc (see blue vertical line in figure 2). At large distances, all other modes also exhibit a positive ${\rm{\Delta }}f$ which decreases with increasing d, again consistent with what is seen in figure 7 and discussed above. Note that although the mode N = 3 also has an antinode at the disc centre, its ${\rm{\Delta }}f$ is smaller than that seen for N = 1 because the the N = 3 dynamics are distributed amongst three antinodes (see figure 3(c)), with the outer antinodes being less strongly affected by the centralised MNP.

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One can also see in figure 8 that the frequency shift for modes 2, 4 and 5 decreases as d approaches zero. This is because these modes have dynamics concentrated away from the centre of the disc and they are thus exposed to a weaker or even a negative ${H}_{{\rm{MNP}}}^{{Z}}$ at small MNP-disc separations when the MNP is laterally centered above the disc. For example, the frequency shift for N = 2 is zero at $d\approx 10\,{\rm{nm}}$ (blue triangles in figure 8). At this value of $d,{H}_{{\rm{MNP}}}^{{Z}}$ is indeed ≈0 near the location where the mode dynamics are concentrated ($x\approx \pm 41\,{\rm{nm}};$ see the vertical red lines in figure 2). Note that the exact value of $| x|$ where ${H}_{{\rm{MNP}}}^{{Z}}$ is zero at d = 10 nm is slightly smaller than 41 nm. This is however not unexpected because the frequency shift will result from what is effectively a convolution between the mode profile and the particle stray field whose magnitude is non-uniform across the disc [39]. We also note that at $x\approx \pm 41\,{\rm{nm}}$ ${H}_{{\rm{MNP}}}^{{Z}}$ becomes negative for values of d smaller than $\approx 15\,{\rm{nm}}$ (figure 2). This is consistent with the observed negative frequency shift for the N = 2 mode at very small d in figure 8. Regarding the N = 4, 5 modes, because they have antinodes located closer to the lateral centre of the disc than the N = 2 mode (figure 3) they do not experience a null or negative ${H}_{{\rm{MNP}}}^{{Z}}$ until even smaller MNP-disc separations (as per figure 2). This is consistent with their ${\rm{\Delta }}f=0$ crossing in figure 8 occurring at smaller d. Finally, we note that for a particle shifted in the y-direction (and thus lying above the antinode of the N = 4 mode), it is the N = 4 mode which exhibits the highest ${\rm{\Delta }}f$ (see purple diamonds in inset of figure 8).

For sufficiently large read-out signals, detection based on identifying changes to the resonant frequency of a device will ultimately be limited by the resonance linewidth as well as the MNP-induced ${\rm{\Delta }}f$, with a small linewidth and large ${\rm{\Delta }}f$ being optimal. In the STO study of Zeng et al [46], the minimum observed linewidth of the primary mode (corresponding to the N = 1 mode here) was on the order of 30 MHz, suggesting that detection could be realised for d up to ∼50 or 60 nm (based on the ${\rm{\Delta }}f$ values shown in figure 8). This assumes however that such linewidths can be maintained under the fields required to magnetise the MNP and that passivation layers and/or upper contacts can be made sufficiently thin (this is potentially achievable given that typical passivation layers are on the order of 30–50 nm thick [6, 8] and coating layers for MNP biofunctionalisation can be made very thin, on the order of 2–5 nm [6, 64]). We note that lower linewidths [48] (e.g. 6 MHz full-width-half-maximum [41]) have been observed in out-of-plane magnetised STOs. We also note that oscillators based on magnetic vortices can offer even lower linewidths [65–67], but they also typically have lower field sensitivities, highlighting the need to optimise the sensitivity-to-linewidth ratio if sensing is to be done by directly identifying changes to the frequency. The sensitivity of the specific resonance of the device to localised fields [36] is also critical of course.

There are a number of ways to increase ${\rm{\Delta }}f$ without modifying the properties of the nanodisc that is being used as a detector. For example, one can attempt to engineer particles with higher moments (see figure 9(a), which shows ${\rm{\Delta }}f$ versus particle moment) or increase the size of the particles (see figure 9(b), showing ${\rm{\Delta }}f$ versus particle diameter). In both cases, this increases the MNP-generated magnetic stray fields and thus the resultant shifts. However, oft-used iron-oxide particles will have lower moments and thus generate lower shifts. We note that in contrast to the case of magnetic vortices [36], for this system we saw monotonic increases in ${\rm{\Delta }}f$ when increasing the particle size (and moment).

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The time-averaged moment of superparamagnetic particles can also be increased by increasing the applied field. Thanks to the continued linearity of ${\rm{\Delta }}f$ as a function of the external field strength ${H}_{{\rm{ext}}}$ over a large range of field values (figure 4), the external field can be increased without significantly compromising ${\rm{\Delta }}f$. Indeed, we see good consistency between the calculated ${\rm{\Delta }}f$ values obtained at vastly differing fields of $0\,$, $0.1\,$ and $1\,{\rm{T}}$ (shown for N = 1 in figure 9(c)). Note that here we (unphysically) assume the same particle moment at each field to enable direct comparison of the ${\rm{\Delta }}f$ values. We note that at high field (1 T), the particle still clearly modifies the distribution of the dynamics of the N = 1 mode (shown for a laterally offset particle in the inset of figure 9(c)). The retained sensitivity of this system even in large external fields is in contrast to magnetoresistive sensors whose sensitivities will be almost nil when the magnetisation is (quasi-)uniform in a sufficiently high field.