The development of spintronic technology with increasingly dense, high-speed, and complex devices will be accelerated by accessible microscopy techniques capable of probing magnetic phenomena on picosecond time scales and at deeply sub-micron length scales. A recently developed time-resolved magneto-thermal microscope provides a path towards this goal if it is augmented with a picosecond, nanoscale heat source. We theoretically study adiabatic nanofocusing and near-field heat induction using conical gold plasmonic antennas to generate sub-100 nm thermal gradients for time-resolved magneto-thermal imaging. Finite element calculations of antenna-sample interactions reveal focused electromagnetic loss profiles that are either peaked directly under the antenna or are annular, depending on the sample’s conductivity, the antenna’s apex radius, and the tip-sample separation. We find that the thermal gradient is confined to 40 nm to 60 nm full width at half maximum for realistic ranges of sample conductivity and apex radius. To mitigate this variation, which is undesirable for microscopy, we investigate the use of a platinum capping layer on top of the sample as a thermal transduction layer to produce heat uniformly across different sample materials. After determining the optimal capping layer thickness, we simulate the evolution of the thermal gradient in the underlying sample layer and find that the temporal width is below 10 ps. These results lay a theoretical foundation for nanoscale, time-resolved magneto-thermal imaging.
Spin-based electronics and high-density magnetic storage require precise control of local magnetic moments in devices,1,2 often using either applied magnetic fields3 or spin-transfer torques.4–7 Development of these technologies will be aided by microscopy techniques enabling researchers to characterize dynamical, nanoscale magnetic phenomena8,9 with relevant length scales that are typically 10 nm to 200 nm10 and relevant time scales that are typically 5 ps to 50 ps.11–13 One existing approach is x-ray magnetic circular dichroism-based microscopy, which offers the desired resolution with spot sizes down to 30 nm.14 However, it requires a synchrotron facility15 and thus it cannot be used in a normal laboratory setting. Another approach is magneto-optical Kerr effect (MOKE) microscopy, which allows for table-top, stroboscopic imaging of spin dynamics with straightforward interpretation.16 However, the visible to near-IR light that is typically used fundamentally limits the spatial resolution of MOKE to hundreds of nanometers, set by the diffraction-limited focal resolution of approximately half the wavelength.17
We have recently demonstrated a new form of spatiotemporal magnetic microscopy using the time-resolved anomalous Nernst effect (TRANE).8,18,19 In this technique, a picosecond pulsed heat source generates a spatiotemporal thermal gradient with a component normal to the surface of a ferromagnetic device. The anomalous Nernst effect in the material transduces the component of the magnetization, M, along y within the heated region (with thermal gradient, ) into a voltage, ,20,21 as shown in Fig. 1. The temporal resolution is derived from the picosecond time scale of thermal excitation and relaxation, whereas the spatial resolution is derived from the area of thermal excitation. Recent TRANE studies have used a focused laser pulse for the thermal excitation, and thus the spatial resolution was diffraction limited. However, thermal sources are not inherently diffraction limited, and thus the development and understanding of a nanoscale thermal source based on near-field techniques can enable nanoscale spatiotemporal magnetic microscopy below the far-field optical diffraction limit.
In this work, we theoretically investigate the viability of scanning gold plasmonic antennas as a method to perform nanoscale TRANE microscopy. We consider focusing electromagnetic radiation to a sub-100 nm region by exciting surface plasmon polaritons (SPPs) on conical plasmonic antennas.22–26 By studying the non-local, near-field excitation of charge in a flat surface by a conical antenna, we find that electromagnetic loss and the resultant thermal point spread function (PSF) is more nuanced than for focused laser heating. Through finite element calculations of the tip-sample coupling, we reveal that in the range of realistic sample resistivities, apex radii, tip-sample separation, and film thickness, electromagnetic loss is either peaked directly under the apex with a full width at half maximum (FWHM) on the order of the apex radius, or it takes on an annular profile that peaks in a ring 10 nm from the center. The potentially large variability in the magnitude and profile of thermal gradients generated in different samples is undesirable for a thermal based microscopy technique. Therefore, we also perform heat diffusion calculations in which a 10 nm platinum capping layer on the surface of the sample is used as an intermediate heater. We find that centrally peaked thermal gradients are radially confined to below 100 nm, with temporal FWHM below 10 ps.
Scan-probe microscopy (SPM) using a plasmonically active tip to locally enhance an electric field is well-studied in the context of tip-enhanced Raman spectroscopy (TERS).27–33 In TERS, electric field enhancements at the apex of a scan-probe tip have enabled a × increase of the Raman signal compared to conventional Raman spectroscopy.25,34–36 This dramatic improvement has spurred research into optimization of TERS techniques. A recent development, relevant to this work, is the use of scan-probe tips with gratings patterned on their surfaces to couple far-field radiation into a plasmonic mode that is focused to a nanoscale area as the SPPs propagate to the tip apex.23,36–39 This reduces the artifacts associated with direct illumination, such as heating of the sample outside the area of the tip apex. Along with the enhancement in the electric field, there can also be sample heating of more than 100 K.40,41 This degree of heating is consistent with studies of local heating due to plasmonic structures patterned on surfaces.42,43 While sample heating may have negative consequences for some Raman studies,44 it is beneficial for our application.
As nanoscopic heaters, plasmonic antennas are currently being developed for use in next-generation hard-disk drives that use heat assisted magnetic recording (HAMR) to increase areal density.2,45–47 For HAMR, a near-field optical transducer is used to heat nanoscale areas of the magnetic recording medium as a means to lower the critical magnetic field for writing. The transducers make use of plasmonic losses that, through careful engineering of both the transducer and the recording medium, induce temperature increases of approximately 350 K within a sub-50 nm area.2 At the same time, HAMR-based nanoantennas must be robust to these large temperature changes for the lifetime of a hard-disk drive (10 yr).
Our application—nanoscale TRANE microscopy—motivates the generation of picosecond-scale thermal excitations of 20 K to 50 K, confined to nanoscale dimensions, in a wide range of magnetic sample materials. Heat-based magnetic microscopy, therefore, can build upon the scanning probe techniques and strategies for low-cost plasmonic antenna fabrication, first developed for TERS, with concepts of plasmonic thermal generation, first developed for HAMR.
III. MODEL DESCRIPTION
Our computational model is designed to be representative of an experimental implementation of conical plasmonic antennas in SPM. Antennas with apex radii as small as 10 nm and opening angles near the apex of 6° may be fabricated with an electrochemical etching procedure in hydrochloric acid.27,31,32 We use this opening angle in our calculations and consider tip radii 10 nm. Tip-sample separations (gap values) are based on the SPM feedback mechanism, which leads to distances of closest approach of order 1 nm. Finite element calculations are performed in the COMSOL electromagnetic waves and heat transfer modules. The finite mesh was manually set to achieve a convergence within 1%. More details on the computational model are in Sec. S 1 of the supplementary material.
We consider far-field light coupled into SPPs at the surface of the antenna with diffraction gratings of period
where is the real part of the dielectric constant of gold and is the angle of incidence.48 Numerical simulations suggest that for a sinusoidal grating,49 a grating depth of 180 nm in-couples energy efficiently; we use a grating of this shape in our model to excite SPPs at the antenna boundary with 780 nm light. However, methods based on aperiodic gratings39 and spin-orbit interactions of light50 have been proposed and warrant consideration in future work.
In SPM, grating illumination will not be axially symmetric because the tip is only illuminated from one side. Therefore, we initially simulate the electric field around the apex of a 3D antenna illuminated over , as in Fig. 2(a). We define the field asymmetry as the minimum value of the electric field magnitude, , around the antenna, divided by the maximum field magnitude, at the center of illumination. We find a value of 0.94 near the apex, in sharp contrast to the initial asymmetry of 0.14 at the grating, shown in Fig. 2(b). This justifies the 2D axisymmetric simulation geometry that we use for all subsequent calculations because it is more computationally efficient [Fig. 2(c)].
To calculate the thermal response of the sample to the plasmonic excitations, we consider loss dominated by Ohmic heating, calculated as the product of electric field, E, and current, J, as
where asterisks denote the complex conjugate. Contributions from dielectric loss, arising from the imaginary part of the dielectric constant, are ignored because, for metallic samples, they are insignificant compared to conductive losses as long as the frequency is away from a resonance.51 To simplify our calculations and conserve computational resources, we treat the gold antennas as lossless (, ) but scale the input power by a factor of 0.2 to account for propagation losses. This is justified because the SPP energy dissipation will occur along the antenna length,42,43 but it will not influence the geometry of the electromagnetic field near the tip apex—only its magnitude. We discuss the effects of finite gold conductivity in Sec. S 2 of the supplementary material and quantify propagation losses in Sec. S 3 of the supplementary material. Although we perform our calculations with a cone angle, SPP focusing and dielectric losses are weakly dependent on cone angle. In Secs. S 2 and S 3 of the supplementary material, we justify our approximations and discuss their implications.
A. Heat source point spread function
The spatial distribution of the electromagnetic loss determines the thermal gradient in near-field heating. Depending on the tip apex radius, tip-sample distance, and sample conductivity, loss is either centered under the apex or distributed in an annulus. Figures 3(a)–3(c) present loss profiles for a characteristic tip with a 25 nm radius of curvature above a metallic sample with permittivity (representative of permalloy52) and varying conductivity. In Sec. S 4 of the supplementary material, we show there is a very weak relationship between loss profile and sample permittivity. Thus we conclude that variation in sample permittivity is insignificant for realistic values of sample conductivity. These calculations are performed in the frequency domain with the geometry in Fig. 2(c), with the equivalent of a 0.08 mJ cm−2 laser pulse coupled into an antenna as discussed in Sec. III. We first consider a 30 nm sample grown on a sapphire substrate. This allows us to focus on tip-sample interactions without coupling to the dielectric substrate, which we find can be non-trivial for much thinner samples.
For a realistic tip-sample distance of 2 nm—measured from the center of the tip apex to the top of the sample plane—and a sample conductivity of 1107 S/m (representative of Pt), the electromagnetic loss has a point spread function (PSF) that has its maximum centered under the tip. A centered loss profile will produce a centered thermal gradient profile. However, increasing either the conductivity or the gap size leads to annular loss. The width of the loss PSF is minimized when the profile is centered, leading to the best spatial resolution. The loss is also more confined to the surface for increased conductivity [Fig. 3(b)], while larger gaps noticeably decrease the magnitude of the loss [Fig. 3(c)]. Figure 3(d) presents the transition as a phase diagram, where for each apex radius, the curve separates regions of centered and annular profiles depending on the gap size and the sample conductivity.
The magnitude of loss decreases with increasing gap size, but it has a less straightforward dependence on sample conductivity. Figure 3(e) shows peak loss for varying conductivity at different gaps with a 25 nm radius tip. Peak loss decreases with increasing conductivity until it reaches the centered-to-annular transition, above which the loss profile becomes more confined to the surface but does not diminish. Maximum power dissipation for the centered profiles is comparable to those simulated for laser heating.8
We can understand the profile transition by considering separately the losses excited by vertical and radial electric fields. Figure 4 shows how the shape of the loss profile depends on the relative magnitude of the individual r and z contributions. We look at total loss and decompose it into vertically and radially excited losses, taken along half-line cuts (moving radially out from directly under the apex) at the sample surface. Curves in Figs. 4(a) and 4(b) correspond to representative conductivity (a) or gap (b) values where the profile is centered or annular, according to Fig. 4(d). Vertically excited loss is always centrally peaked because the vertical electric field is concentrated where the apex surface is parallel to the sample plane. Likewise, radially excited loss is always annular because the radial electric field emanates from the sides of the apex. When the conductivity or gap is increased, the peak of the r loss is larger than that of the z loss, and the profiles become annular [Figs. 4(a) and 4(b)]. The r loss is also more extended in the radial direction than z loss; therefore, their relative magnitude will non-trivially determine the width of the PSF.
To understand the origin of these loss profiles better, we plot the peak electric field along the same radial line cut for varying conductivity and gap distance [inset in Figs. 4(a) and 4(b), respectively]. In each case, the point where the vertical component drops below the radial component corresponds to the point in Fig. 3(d) where the loss profile transitions from centered to annular. Physically, a larger conductivity leads to larger charge concentration at the sample surface, which screens vertical oscillations more effectively than radial ones. In the case of a large gap, the charge distribution decreases in magnitude and spreads, leading to the generally steeper decrease of z losses seen in the inset of Fig. 4(b).
Next we investigate the effect of apex radius on profile size. We consider a Pt sample with a Au tip of varying radius placed between 1.3 nm (for 10 nm apex) and 2.6 nm (for 80 nm apex), as in Fig. 2(c). We choose to vary the distance with the radius to model the FWHM in a realistic experimental setting, where gap distance depends on the interatomic forces between the apex and the sample53 and is sensitive to apex radius.54 In Fig. 4(c) the position of the peak loss remains at zero (centered) until an apex radius of 55 nm, where it moves off-center. This is due to a wider charge concentration in larger tips. It is useful to look at the FWHM for centered profiles, shown in Fig. 4(d), which shows two regimes of increasing width. The FWHM grows with apex radius at 0.95(9) nm/nm for radii below 30 nm. For radii above 30 nm, the FWHM grows as 3.73(3) nm/nm.
In samples with thicknesses comparable to the penetration depth of electromagnetic loss, the loss profile is significantly altered. We fix the apex radius at 25 nm and the gap at 2 nm, then consider a range of realistic sample conductivities at several sample thicknesses. The results are plotted in Fig. 5(a). For a 2 nm sample, loss is radially annular for all values of conductivity shown [Fig. 5(c)]. As thickness increases [moving to the right in Fig. 5(a)], loss profiles approach the centered distribution shown in Fig. 3 for the 30 nm sample for thicknesses above 10 nm [Fig. 5(d)]. The dielectric constant of the substrate material also impacts loss profiles for films thinner than 10 nm [Fig. 5(b)]. The large variation in loss profiles for film thicknesses of interest for TRANE microscopy further complicates the uniformity of a plasmonic antenna probe.
B. Heating with a capping layer
To reduce the sample-to-sample variability demonstrated above, we propose the use of an intermediate heater layer. We examine platinum because (1) it has conductivity commensurate with centered losses for achievable tip radii, and (2) it is often already present in spin-Hall devices.
We first determine an optimal capping size that is thick enough to decouple the Pt losses from the material underneath but thin enough that most of the thermal energy is transferred to the sample. By varying sample conductivity for various cap thicknesses, we find a 10 nm layer is the thinnest for which there is negligible effect from sample conductivity [Fig. 6(a)]. By examining loss profiles at the top of the Pt cap for a constant sample conductivity (lines of the same color), we reveal that sample thickness variations below 10 nm affects the loss confinement, most notably at low conductivity. The peak of loss shows slight dependence on conductivity for thin samples but a trivial dependence on sample thickness. In Fig. 6(b) we plot the 2D loss profile for a 2 nm sample; loss is still significant at the bottom of the capping layer, so the underlying sample influences the profile. This is in contrast to the case shown in Fig. 6(c), where the loss in a 10 nm sample is not affected by the sample. Above 10 nm the profiles no longer change with sample thickness. Furthermore, line cuts with varying conductivity converge to the same centered loss profile, suggesting that a 10 nm thick capping layer can provide the same heat source to a variety of samples of interest. This is significant for developing a reliable microscopy technique.
To investigate thermal gradient generation, we simulate loss in the Pt capping layer when an antenna (15 nm apex radius) is excited with a Gaussian pulse of 3 ps FWHM centered at 9 ps in the calculation. We ignore direct heat transfer between the gold tip and Pt layer. While electromagnetic loss may heat the gold tip by 50 K, the thermal conductivity of air—1 10−8 W m−1 K−1 at the nanometer scale55—is low enough such that there is negligible heat transfer across the air gap compared to the heat deposited through electromagnetic loss in the Pt. For temperature changes in excess of 500 K in nanoscale gold structures, there is significantly more energy transferred to the environment;42 however, we do not approach this regime. Near-field heat transfer between Au and Pt for similar geometries has also been shown to be orders of magnitude smaller than our simulated Pt losses.56
We now investigate the heating of a characteristic 10 nm permalloy sample with a 10 nm Pt capping layer. The geometry is displayed in Fig. 2(c), except the sample layer is replaced with the cap/sample stack. For focused laser heating, we expect a depth-dependent thermal gradient because light is absorbed in the sample on a scale set by its skin depth. Here, the near-field electromagnetic loss within the cap generates thermal gradients in the magnetic layer, thus the expected profile is less obvious. The thermal gradient varies strongly with depth in the sample. In particular we see a more than 50% decrease in the thermal gradient magnitude halfway through the depth of the 10 nm film and a more than 75% decrease in the lower quarter, shown in Fig. 7(a). The TRANE voltage is proportional to the in-plane magnetic moment; the depth dependence could present complications for non-uniform moment distributions, where the final signal is a weighted average of the varying in-plane moments. However, typical thin-film magnetic materials have uniform magnetization along their thickness direction, an assumption we will apply in our following analysis.
To determine the spatial resolution of scanning TRANE microscopy with a Pt cap, we calculate the width of the vertical thermal gradient, [Fig. 7(a), inset]. We find that the average FWHM is 45.3 nm, weighted by peak gradient value, calculated when the thermal gradient is at its maximum value. As seen in Fig. 7(a), inset, the width increases more rapidly as we approach the interface with the substrate. This is likely due to the interfacial thermal resistance, which is approximately 1 108 K m2 W−1 for a metal-sapphire boundary.57 However, the thermal gradient magnitude is also diminished as a function of distance from the surface, thus the wider profile at larger depths does not significantly impact the weighted average width.
We also consider the time evolution of . In Fig. 7(b), is plotted for a selection of times before and after the center of the laser pulse. The spatial width (inset) significantly increases at 12 ps, where it is 25% larger than the FWHM at the peak value of . In general, the width increases with time. We do not calculate the width at 15 ps because by then the thermal gradient peak is off-center. This temporal broadening of the PSF is offset by the fact that is less than 10% of its peak value by this time, and thus it will have a small effect on the spatial resolution.
The thermal gradient is centrally peaked for a majority of its duration, so we calculate the temporal width based on the evolution of at different depths along a central vertical line in the sample, in Fig. 7(c). For reference, the shape of the laser pulse is presented as a solid black curve, with the peak time denoted with a vertical dashed black line. The thermal gradient peaks around 10.7 ps, 1.7 ps after the peak of the pulse. The temporal FWHM shows depth variation (inset) of a couple picoseconds. Due to the transfer of heat down through the sample, at a certain time (around 20 ps, but it is depth-dependent), the thermal gradient reverses sign. Although the dip is much smaller than the peak, this will reduce the overall TRANE signal depending on how it is measured.
The profiles in Figs. 7(a)–7(c) were calculated with a 15 nm apex radius; we now consider the effects of varying tip radii on the spatial extent of the thermal PSF. Figure 7(d) shows the FWHM of the along the sample top at its temporal peak. For apex radii of 45 nm and below, the gradient profile is centered with a FWHM of 14.0(12) nm + 2.3(1) ra, where ra is the apex radius. There are no separate discernible regions as with the FWHM of the loss profile [Fig. 4(c)]. Due to the radial spread of heat, the thermal width is in all cases wider than the loss width; however it remains confined to a region approximately the size of the apex. For a 90 nm diameter apex—the largest apex size with a centered profile—the FWHM is 115 nm.
The nanofocusing of light with conical plasmonic antennas offers a means to access nanoscale electromagnetic heating with far-field excitation, making it a realistic path to improving the spatial resolution of TRANE microscopy to nanoscale lengths. We have shown that, when a sub-100 nm antenna apex is within nanometers of a sample surface, strong near-field coupling produces electromagnetic loss in the sample that is confined to a region comparable to the size of the apex. Additionally, the loss profile is dependent on the sample conductivity and the tip-sample distance. As these parameters increase, vertically excited loss decays more quickly than radially excited loss. Radially excited loss is peaked off-center, and eventually it overtakes vertical loss, forming an annular loss profile. For films with thicknesses similar to or below the penetration depth of loss, the bottom interface can have a profound effect on the loss profile. In the case of a 2 nm film on sapphire, the loss profile entirely loses its central peak. This thickness sensitivity persists until the film is thicker than 10 nm.
To avoid sample-to-sample variation of resolution and thermal gradient amplitude, we propose the use of a Pt capping layer. We find that 10 nm of Pt is sufficient to act as an intermediate heater for the underlying material. Because losses in the capping layer are separated from the electromagnetic properties of the sample, the cap may heat uniformly across different samples. These results point the way toward accessible spatiotemporal magnetic microscopy with sub-100 nm spatial resolution and sub-10 ps temporal resolution—capabilities that can enable the next breakthrough in emerging spintronic technologies.
See the supplementary material for a description of the computational model; discussion of the inclusion of antenna conductivity and losses and heating in the antenna; and consideration of sample permittivity.
This research is supported by the U.S. Air Force Office of Scientific Research under Contract No. FA9550-14-1-0243.