Optical generation of intense ultrashort magnetic pulses at the nanoscale

Generating, controlling, and sensing strong magnetic fields at ever shorter time and length scales is important for both fundamental solid-state physics and technological applications such as magnetic data recording. Here, we propose a scheme for producing strong ultrashort magnetic pulses localized at the nanoscale. We show that a bimetallic nanoring illuminated by femtosecond laser pulses responds with transient thermoelectric currents of picosecond duration, which in turn induce Tesla-scale magnetic fields in the ring cavity. Our method provides a practical way of generating intense nanoscale magnetic fields with great potential for materials characterization, terahertz radiation generation, and data storage applications.


I. INTRODUCTION
The study of ultrashort magnetic phenomena has been largely driven by their application to magnetic storage technologies, which are rapidly evolving towards the nanoscale and sub-picosecond regimes, giving rise to significant advances in the understanding of magnetization dynamics [1]. Currently, no easily accessible method is available to generate intense sub-picosecond magnetic pulses localized at the nanoscale. Short electron pulses can be tightly focused and are accompanied by magnetic fields of a few Teslas, but they require access to electron accelerators [2]. Transient magnetization has been also studied via the inverse Faraday effect in cases for which absorption of circularly polarized light leads to a prevailing spin species [3][4][5][6], although this technique is only applicable to highly absorbing materials exhibiting strong spin-orbit coupling, with spatial resolution limited by the light focal spot. Here, we propose a radically new approach towards ultrafast magnetism with nanoscale resolution relying on optically driven generation of magnetic fields.
In this work, we show that bimetallic nanorings can act as nanoscale sources of intense ultrashort magnetic pulses. We rely on the enhanced light absorption associated with the plasmons of metallic rings to generate transient thermoelectric currents that in turn produce sub-picosecond pulses with magnetic fields as high as a few tenths of a Tesla in the vicinity of the rings. The ability to generate strong magnetic fields localized on the nanoscale is of interest for elucidating spin and magnetization dynamics at sub-picosecond time and nanometer length scales [1], and it holds great potential for materials characterization, terahertz radiation generation, and magnetic recording.

II. RESULTS AND DISCUSSION
A scheme of our magnetic source is shown in Fig. 1. A 100 fs laser pulse heats a bimetalic ring consisting of three quadrants of gold and one quadrant of nickel. This choice of materials is a compromise between their thermoelectric response and their electrical resistance in order to optimize the resulting magnetic field. The ring is 100 nm thick, has inner (outer) diameter of 70 nm (170 nm), and is embedded in glass. The incident pulse is polarized along one of the Au/Ni junctions (B in Fig. 1a) and is centered at a wavelength of 920 nm, which corresponds to the lowest-order dipole plasmon resonance of the ring. In a homogeneous gold ring, symmetry would lead to resonant absorption at two diametrically opposed regions. However, the asymmetry of the bimetallic ring polarization leads to stronger absorption near one of the Au/Ni junctions (A in Fig. 1a). This nonuniform absorption produces a steep gradient of the electron temperature along the ring, and consequently, a net displacement of charge carriers occurs from hotter to colder regions, giving rise to a Seebeck electromotive force. In the Au/Ni thermocouple charge carriers in gold and nickel are electrons and holes, respectively, so that the resulting thermoelectric current contributes along the FIG. 1: Schematic description of our nanoscale ultrashort strong magnetic-field source. (a) A femtosecond nearinfrared laser pulse heats a bimetallic ring consisting of one quadrant of nickel and three quadrants of gold. At the plasmonic resonance of the ring, a dipole current oscillating at the incident light frequency is excited (blue/red arrows), leading to absorption and subsequent conduction-electron heating. A strong, non-oscillating thermoelectric current is then produced in this short-circuited thermocouple, driven by the conduction-electron temperature difference (∼ thousands of degrees) between the cold and hot Au/Ni junctions. (b) The thermoelectric ring current generates a strong magnetic field normal to the plane of the ring. The intensity of the magnetic-field component perpendicular to the ring is represented by the density plot, whereas magnetic-field lines are shown in green. same direction on either side of each metal junction [7]. Femtosecond pulses give rise to temperature differences of several thousand degrees in the conduction electron gas. This results in a strong thermoelectric current before thermalization into the atomic lattice takes place within a few picoseconds. Finally, this transient circular current generates a magnetic field, mainly perpendicular to the plane of the ring (green lines in Fig. 1b). We solve the underlying nonlinear heat-transfer/thermoelectric/electromagnetic problem within the two-temperature model [1,7] using a commercial finite-elements tool (COMSOL), and we supplement these results with a semi-analytical model that correctly predicts the magnitude of the generated field and its duration (see Appendix B for more details).
The time evolution of the conduction-electron (T e ) and lattice (T l ) temperatures at the two Au/Ni junctions is shown in Fig. 2 for an incident pulse fluence of 11.3 J/m 2 , which transfers ∼ 1.4 pJ to the ring (absorption crosssection ∼ 0.12 µm 2 , see Appendix F). Following an initial excitation with the ultrafast optical pulse, the electron-gas temperature rapidly rises at both junctions, reaching peak values shortly after the maximum of the incident pulse. The nickel section of the ring produces a considerable temperature difference between the two junctions: electrons at the hot junction (A) reach a peak temperature of ∼ 6000 K, which is approximately twice the temperature of the cold junction electrons (B). The peak temperature at B occurs nearly 1 ps after the maximum of A as a result of hot-electron diffusion. Over this short time interval, the absorbed energy remains almost entirely in the electron gas, while thermalization of the gold and nickel lattices, initially at 300 K, takes place over a time scale of several picoseconds. The lattice and electron temperatures become nearly identical ∼ 10 ps after irradiation, when both junctions are at ∼ 600 K, well below the melting temperature of the ring materials. Finally, the ring cools back to room temperature after a few nanoseconds due to diffusion through the surrounding glass.
The decay from maximum heating reveals three different time scales: a fast decay (∼ 1ps) produced by thermal diffusion within the electron gases of the two metals; an intermediate decay (∼ 10 ps) that can be ascribed to thermalization of the electrons into the lattice via electron-phonon scattering; and a slow decay (∼nanoseconds) arising from thermalization into the surrounding glass. This picture is consistent with the temperature dependence of the involved materials parameters (see Methods).
The temperature gradient between two junctions of metals with different Seebeck coefficients produces an electromotive force that drives a thermoelectric current along the ring circumference. For uniformly heated junctions, this effect is quantified through a potential between them S∆T , proportional to the temperature difference ∆T . The Seebeck coefficient S depends on the nature of the material between the two junctions, as well as on temperature (see Appendix E). Because of the high electron temperatures involved, the thermoelectric current density in the confined ring geometry reaches values as high as ∼ 10 13 A/m 2 . Figure 3a (left scale) illustrates the temporal evolution of the current density through junction A, which follows quite closely the temperature difference, with a maximum occurring ∼ 100 fs after the peak of the excitation pulse. The current distribution on the ring surface (Fig. 3b) reveals a dominant azimuthal component, except near the hot junction, where parasitic short-range loops are produced.
These strong currents give rise to a large magnetic induction (> 0.35 T peak value) of similar temporal profile (Fig.  3, right scale). The induction is localized inside the ring cavity (Fig. 3c), where it is rather uniform (Fig. 4, inset), with the exception of a depletion caused by the noted loop currents in the hot Au/Ni junction. The magnetic pulse has a duraction ∆t mag ∼ 1.8 ps FWHM, mainly controlled by electron diffusion and thermalization into the lattice (see Methods).
The peak magnetic induction grows monotonically with light-pulse fluence (Fig. 4), exhibiting a slightly sublinear behavior as a result of the increase in electron heat capacity with temperature [7], although the net effect involves a complex interplay between the temperature dependence of the materials thermal and electrical parameters (see Methods).
We have formulated a simple model that yields semi-analytical expressions for the peak magnetic induction (Eq. (B1)) and the magnetic pulse duration (Eq. (B2)), as discussed in Appendix B. The model is based upon the neglect of thermal diffusion in the lattice and it assumes that the optical pulse acts as a sudden impulse. This produces results in excellent agreement with full numerical simulations, as shown in Fig. 4, thus providing a simple tool that can assist in the design of rings optimized in size and composition to yield the desired levels of magnetic pulse intensity and duration. In particular, the model predicts a magnetic field duration that increases with optical pulse fluence, which essentially reflects the decrease in electron-phonon coupling and the increase in the heat capacity associated with the ever larger electron temperature that is reached with higher pumping intensities (see Appendix ??).

III. CONCLUSION AND PERSPECTIVES
Our results indicate that we can take advantage of the strong thermoelectric currents at the nanoscale in order to achieve very intense, transient, THz electromagnetic fields driven by pumping with vis-NIR light. This can be the basis of a new source of optically pumped THz radiation, whereby illuminated rings act as localized magnetic dipoles. In a different, important range of applications, illuminated bimetallic rings provide an interesting opportunity for nanoscale magnetic recording triggered by the intense magnetic field (strongly confined in the < 100 nm ring cavity) in combination with suitable magnetic grain sizes that can react to the short duration of the magnetic pulse [2]. As a practical consideration, the main limitation on the values of the achievable magnetic field comes from the breakdown current in the rings (∼ 10 12 − 10 13 A/m 2 for gold [8,9]), as well as from possible melting of the metals [10]. However, the short duration of the optical irradiation, the moderate lattice temperatures involved, and the embedding of the ring in a rigid glass matrix should push the thresholds for melting and electrical breakdown well beyond the currents and temperatures considered here.
In conclusion, we have shown that a bimetallic nanoring illuminated by ultrafast laser pulses supports transient thermoelectric currents leading to magnetic pulses of sub-picosecond duration reaching a fraction of a Tesla and localized in the < 100 nm ring cavity. Our results can facilitate the study of ultrafast, nanoscale magnetic phenomena, and they hold great potential for applications in materials characterization, the generation of terahertz radiation, and magnetic recording. ACKNOWLEDGMENT This work has been supported in part by the Engineering and Physical Sciences Research Council (U.K.), the Leverhulme Trust, the Royal Society, and the Spanish MEC (MAT2010-14885 and Consolider NanoLight.es).

Appendix A: Theoretical formalism
Our simulations of optically induced generation of magnetic fields mediated by thermoelectric currents in bimetallic rings involve solving a set of coupled equations for (1) the electromagnetic field intensity of the incident light, (2) the heat diffusion problem, (3) the resulting thermoelectric current, and (4) the magnetic field itself. In this work, we provide two levels of theory based upon (i) full numerical simulations and (ii) semi-analytical modeling. We use a commercial finite-element method (COMSOL) for the former, whereas a derivation of the latter is summarized below. But first, we discuss the relevant equations and approximations required to deal with each of the four noted subproblems.  [7] and lattice [12,13] heat capacities, ce and c l , respectively. (c,d) Electronic [14][15][16] and lattice [12,13] thermal conductivities, κe and κ l , respectively. (e) Electron-lattice coupling coefficient [7], G.
(1) Light-pulse absorption by the ring. Light absorption is calculated by solving Maxwell's equations for a Gaussian pulse of ∼ 2∆t pulse = 100 fs duration centered at a wavelength of 920 nm and incident on the ring along its axis with linear polarization as shown in Fig. 1. Because the pulse duration is much longer than the light period (3.1 fs), the absorbed power density (i.e., the position-and time-dependent power absorbed per unit volume inside the ring) is approximated as p(r, t) = p(r)f (t), assuming that it follows the pulse temporal profile f (t) = exp[−t 2 /(∆t pulse ) 2 ], multiplied by the peak absorption density p(r), which we calculate for a plane wave with the same intensity as the pulse maximum. The dielectric functions of gold (−33.56 − 1.93i), nickel (−15.47 − 26.03i), and glass (2.1) are taken from tabulated optical data [11]. (2) Thermal diffusion and temperature distribution. The absorbed power density p(r, t) acts as the source of heat for the thermal diffusion problem. We adopt a two-temperature model [7], in which the conduction-electron and atomic-lattice subsystems have their own temperatures T e (r, t) and T l (r, t), respectively, each of them evolving over very different time scales. We assume that the heating produced through p(r, t) is entirely transferred from the light to the conduction electrons, which reach local thermal equilibrium in a short time compared with the characteristic times for thermal diffusion and lattice thermalization (see below). The thermal diffusion equation is then simultaneously solved for the electronic and lattice subsystems, which are coupled through a term accounting for heat transfer from the former to the latter. We assume this term to be proportional to the local temperature difference T e − T l , via an electron-phonon coupling coefficient G. This leads to the two coupled diffusion equations [7] where c e and c l are the electron and lattice heat capacities, and κ e and κ l are the respective thermal conductivities.
We use values of the temperature-dependent coefficients C, κ, and G for Ni and Au compiled from Refs. [7,[12][13][14][15][16], as discussed in Appendix ??. For simplicity, we ignore the effect of heterojunction thermal barriers, which should contribute to create higher temperature gradients and stronger magnetic fields. (3) Thermoelectric current. From the solution of Eqs. (A1), we derive a thermoelectric electromotive force S∇T e , where S is the Seebeck coefficient. This allows us to write the thermoelectric current as where σ is the temperature-dependent electric conductivity and V represents the potential displayed in response to the thermoelectric source. We use values for the temperature-dependent coefficients S and σ for Ni and Au calculated from Refs. [7,17], as explained in Appendix E. Because T e varies smoothly over hundreds of femtoseconds, involving characteristic frequencies below the mid-infrared, we can safely assume that the metals behave as good conductors, so that the thermoelectric current is quasi-stationary and the continuity equation reduces to the vanishing of ∇ · j (this is equivalent to neglecting self-inductance of the thermoelectric current). Inserting Eq. (A2) in this expression, we find which is formally equivalent to Poisson's equation with a local response function σ and a source term −∇ · σS∇T e . Incidentally, by setting σ = 0 outside the ring, Eq. (A3) guarantees the vanishing of the normal current at the ring boundaries. We obtain V by solving Eq. (A3), and this is in turn inserted into Eq. (A2) to yield j.
(4) Magnetic induction. Finally, we use the electric current j (Eq. (A2)) to find the resulting magnetic induction from [18] where the integral is extended over the volume of the ring.

Appendix B: Semi-analytical model
A simple model that predicts the order of magnitude of the generated magnetic field, the current circulating around the ring, and the duration of the magnetic pulse can be formulated under the following approximations: (i) Sudden pulse excitation. Upon inspection of Eq. (A1a), we expect to have characteristic diffusion and thermalization times given by ∆t dif = R 2 c e /κ e and ∆t th = c e /G, respectively, where R is the average ring radius. For the high temperatures (T e > 2000 K) at which a large magnetic field is generated, we have ∆t dif > 0.5 ps and ∆t th > 2 ps (see Appendix F), in front of which we neglect ∆t pulse = 50 fs, and thus we consider the pulse to deliver an instantaneous impulse quantified by an absorbed energy density q = √ π∆t pulse p(r). This determines the maximum local electron temperature T e right after pulse irradiation by imposing q = Te T0 c e (T ) dT , where T 0 is the initial room temperature before irradiation.
(ii) Neglect of the lattice. Upon examination of the physical parameters entering Eqs. (A1) (see Appendices ?? and E), we observe that κ l κ e and c l c e , that is, the lattice takes much more heat to increase its temperate than the electron gas and it conducts heat more poorly. This explains why T l T e during the bulk of the magnetic pulse duration, so we can safely ignore the lattice and reduce Eqs. (A1) to c eṪe = ∇(κ e ∇T e ) − G T e , with the initial condition for T e (t = 0) derived above.
(iii) Average over ring cross-section. We can further assume that the temperature distribution can be well represented if we replace it by its azimuthal-angle-dependent average over the radial and vertical directions (i.e., the average over the ring cross-section). We then focus on the azimuthal component of the current j, which we can take as uniform along the ring circumference as a consequence of the vanishing of ∇ · j. Noticing that the potential V must reach the same value when ∇V is integrated along a whole ring circumference, we find from Eq. (A2) that j dϕ/σ = (1/R) S(∂T e /∂ϕ)dϕ, where we have approximated the azimuthal component of ∇ as (1/R)∂/∂ϕ. Finally, we can carry out the integral in the right-hand side of this equation from junction A to B through Au and from B to A through Ni (see Fig. 2a). This produces the expression for the z component of the magnetic induction as at the ring center, where β = (1/R) drẑ·(φ×r)/r 3 is a dimensionless geometrical factor (β ≈ V /R 3 , where V is the ring volume). Following the discussion in point (i), we estimate the pulse duration from the diffusion and thermalization times as where the brackets stand for the average along the ring circumference, which needs to be taken due to the strong dependence of ∆t dif and ∆t th on the local electron temperature. We show in Fig. 4 the pulse duration evaluated from Eq. (B2) right after irradiation.