Giant localised spin-Peltier effect due to ultrafast domainwalls motion in antiferromagnetic metals

Spin thermo-electric phenomena have attracted wide attention recently, e.g. the Spin Peltier effect (SPE) heat generation by magnonic spin currents. Here we find that the Spin Peltier ef-fect also manifests as a heat wave accompanying fast moving magnetic textures. High speed andextreme magnetic excitation localisation are paramount for efficient transfer of energy from thespin-degrees of freedom to electrons and lattice. While satisfying both conditions is subject to se-vere restrictions in ferromagnets, we find that domain walls in antiferomagnets can overcome theselimitations due to their potential ultrahigh mobility and ultra-small widths originating from the rel-ativistic contraction. To illustrate our findings, we show that electric current driven domain wallmotion in the antiferromagnetic metal Mn2Au can carry a localised heat wave with the maximumamplitude up to 1 K . Since domain walls are well localised nanoscale magnetic objects, this effecthas the potential for nanoscale heating sensing and functionalities.

Here, we demonstrate the possibility to use electric driven AFM DW motion to generate significant localised heating wave, with a transient electronic temperature raise at least three orders of magnitude larger than in the ferromagnetic case. Notably, by developing a kinetic model where both local and nonlocal electron, phonon and spin relaxation are included, we are able to identify relevant timescales and extend the existing framework for the AFM DW dynamics onto the case of non-equilibrium ultrafast dynamics.
We show that the magnetic energy conversion takes place in the ultrafast regime where the DW motion at magnonic velocities implies subpicosecond energy transfer to the electronic system. Subsequently, those hot electrons release their excess of energy to the lattice via the electron-phonon coupling, a thermalization process that takes several picoseconds. Finally, lateral heat diffusion transports thermal energy away from the DW position in the hundred picosecond scale.
Concerning the use of AFM in nanoheating technologies, the standard way of applying cycling magnetic fields would hardly work due to the small coupling owing to their vanishing net magnetization. However, relatively low electric currents are able to drive the DWs in AFMs up to magnonic velocities which lie in the elusive range of THz. At those ultrafast spin dynamics, two effects emerge which need to be considered; i) Lorentz contraction of the characteristic lengths due to the relativistic nature of AFM spin dynamics, and ii) ultrafast magnetic energy conversion into heat. While the former has been investigated in literature for some classes of AFMs 20 , the latter effect remains largely unknown.
In this work, we reveal the fundamental role of the ultra-high speeds and Lorentzian contraction of DW dynamics in the magnetic energy conversion into dissipation of a moving AFM magnetic texture. In some sense the heat production of electrically driven DW motion is analogous to the Joule heating, when passing an electric current through a resistive material. In our proposal, the role of the moving electrons is played by the DW, which implies that the mechanisms of energy conversion into heat occur at the site of the DW. This would enable a highly localized heat wave source, provided that the DW moves faster than typical diffusion processes, with the added benefit of that the heat source is movable.
In terms of viable materials for this approach, antiferromagnetic metals would appear to be favored over counterpart insulators since in the latter the magnetic energy would be absorbed into the magnon bath rather than transferred to the electronic degrees of freedom 21 . Moreover, magnon relaxation in insulators is a slow process, which could be comparable to the thermal diffusion. In metals however, spin degrees of freedom efficiently coupled to the electronic system 22 . This results for example in femtosecond (10 −15 s) time scale dynamics of spins, electrons and lattice 23 , opening the door to Petahertz (10 15 s −1 ) spintronics 24 . This strong coupling between the electronic and spin degrees of freedom provides an ideal benchmark for the possibility of ultrafast energy transfer from spin to the electron system by pure ultrafast domain wall dynamics in antiferromagnetic metals.

Results
Heat generation due to a AFM DW moving at ultrafast speeds.-The response of a magnet to an external stimulus strongly depends on the way dissipation takes place, which in turn controls the magnetic response in processes such as DW motion, switching and spin transport. The dynamics of the angular momentum dissipation in magnets is well described by the Gilbert relaxation term (see Methods), whose rate, is proportional to the so-called Gilbert damping, α. This is connected to the magnetic energy dissipation described by the Rayleigh dissipation functional,Q DW = dV ηṠ 2 (x), where η = µ sl α/γ, γ is the gyromagnetic ratio and µ sl is the sublattice magnetisation. We use the Rayleigh dissipation functional to estimate the parameter dependence of the initial temperature rise while the DW is moving.
Within the Lagrangian formalism for a stationary moving (along x-direction) 1D domain wall in a layered antiferromagnet (or ferromagnet), the Rayleigh dissipation function (per atomic spin) can be derived as where q = (x−v DW t)/∆ DW , v DW is the DW velocity and ∆ DW is the DW width. For a stationary moving DW in the absence of thermal diffusion, the temperature profile accompanying the DW propagation can be estimated as where C is the electron (phonon) bath heat capacity. We note that the temperature rise scales with the ratio v DW /∆ DW , which is advantageous for AFM DW due to the Lorentz contraction of ∆ DW down to the atomic limit, and the possibility to achieve magnonic velocities.
In the following we explore the non-equilibrium (non steady) dynamics of DW in the metallic AFM, Mn 2 Au, driven by spin-orbit fields in a track and its impact on the track temperature. Efficient DW motion can be achieved in certain AFM metals by injecting electric currents into them. In particular, crystals with locally broken inversion symmetry at the magnetic sites A and B form inversion partners, the inverse spin galvanic effect produces a staggered local spin accumulation with opposite polarities.
The effect of the current is then to produce local staggered spin-orbit (SO) field which is perpendicular to the spin-polarised current direction and is linearly proportional to its magnitude. The torque generated on each AFM sublattice has, therefore, the same form as in ferromagnets. Together with the AFM metal characterised by its instantaneous position and velocity of its centre, (q, v DW ) (see Fig. 1a). In AFMs, the Lorentz contraction means that the width ∆ DW of the DW depends on v DW and its limited by the rest. An electric current passing through Mn 2 Au creates a staggered spin-orbit torque H SO which drives the DW at a velocity, v DW = (γ/α)H SO ∆ DW . Due to energy conservation, a stationary moving DW dissipates energy into the medium at the same rate as Zeeman energy lowers due to the domain switching.
The dynamics of the redistribution of this excess of energy into the different subsystems is the main result of this work.
Atomic spin dynamics simulations (ASD) permit us to calculate each of theṠ i (t), and consequently q(t) and v DW (t) and ∆ DW are obtained. Those time-dependent quantities are then fed into Eq. 1 to calculate the instantaneous localQ DW (x i , t). This quantity enters into the equation of motion for the electron and lattice temperature dynamics, which is described by the two temperature model described in Methods section. Figure 1b depicts the transient dynamics of the local electron and phonon temperature profiles due to a fast moving DW. The electron temperature shows a peak temperature lagging slightly behind the DW centre. This is due to the direct coupling between the electron and spins, (g s−e ) 22 . At the same time the phonon temperature shows a much smoother profile, owning to the indirect coupling to the heat source (moving DW) via the electron system, (g e−ph ). The heat wave is well localised around the centre of the DW. The excess of energy in the electronic system is rapidly transferred to the lattice via the electron-phonon coupling at characteristic timescales of the order ∼ 1 ps. At the same time, lateral heat transport is also present, flow of energy from hot to cold regions. We should note that additional channels of energy conversion also exists. For instance, magnon creation, which in turn can transport energy away from the heat source. In our simulations we do not see significant spin wave creation, probably due to the conditions we are assuming here: low temperature and l15 ps linear ramp time of driving current as opposed to a square pulse. Furthermore, we neglected the Joule heating contribution in our model, although its contribution may be larger than those related to the DW motion. However, the Joule heating only provides a homogeneous background. We hope that these effects would be possible to distinguish by proper calibration in the real experimental set up.
In our simulations, we start by injecting an electric current with the time profile illustrated in We discuss now the total heat dissipated in our system per cycle (Figure 3a). This excitation protocol produces a phonon heat accumulation at the track centre. In terms of the phonon (electron) temperature the value of 1 K is reached already in 300 ps (Figure 3b) with 4 field cycles with a full width at half maximum of 1 micrometer (Figure 3a). The rapid accumulation is possible due to the high-speed character of the AFM DWs. At this timescales the heat transfer to the outside media is expected to be small so that a giant magnetocaloric effect is induced.

Discussion
The above described mechanism opens the door to control heating at the nanoscale in a fast manner.
It is worth discussing the differences and similarities between the heating process by AFM DW and that of ferromagnetic nanoparticles performing coherent magnetisation rotation by external oscillating magnetic fields. The latter is a standard way to heat, for example, tumor cells in magnetic hyperthermia treatment. Energetic considerations show that the magnetic energy density release is different to zero for irreversible processes only, and is equal to the hysteresis loop area. ∆ε = m(H)HdH, where m is the magnetisation density. In the absence of other heat losses, one can estimate the maximum temperature rise as ∆T ∼ ∆ε/C where C is the specific heat density. Although useful, this argument can produce an impression that a relatively large temperature rise can be achieved by only increasing the applied magnetic field. The real situation is, however, more complex since dynamical considerations need to be taken into account. Specifically, the rotational speed is in the nanoseconds range, i.e. the maximum possible heat will correspond to fields with the magnitude of the coercive field applied at GHz frequencies. This timescale is much slower than the AFM dynamics considered here. Faster field cycling implies minor hysteresis loops leading to a huge decrease of the heating output. Additionally, at this timescale, the interface heat transfer (to the outside media) is also more efficient than in our case. For comparison, simple estimations show that a small magnetite nanoparticles of 10 nm diameter, under the best conditions of major hysteresis loop would heat up to 10 mK per field cycle. This estimation is two order of magnitude lower than our calculations for AFM DW motion.
We note that for commonly used ferromagnetic materials, heating by moving DWs would not be so efficient as in AFM. For example, for standard parameters of permalloy, we estimate that the DW motion can carry electron temperature pulses of a maximum of 1 mK.
Thermal diffusion also plays an important role in the delocalisation of the temperature rise. The heat diffusion rate is defined by the parameter η = v DW ∆ DW /D where D = k e /C e is the electron diffusivity coefficient. Considering the typical metal value D = 10 −4 m/s 2 , the above parameter η 1 for permalloy and η > 1 for Mn 2 Au. Thus the negligible heat wave accompanying the permalloy DW will be completely delocalised, whereas for Mn 2 Au it is localised at the DW.
Importantly, since eventually heat will spread all over the sample, the particular dynamics of the relaxation of the localised temperature rise is paramount for devising experiments and devices able to exploit this new concept we present here. The effect hinges to the field of ultrafast spin caloritronics fostered by the recent demonstration of subpicosecond spin Seebeck effect 26 . Along this line, it remains a challenge to drive AFM DWs into magnonic velocities, and to develop experimental techniques in order to detect DW dynamics at these ultrafast timescales. Notably, we believe that nanoscale confined heating at the domain wall position can be used to track the DW position and velocity by measuring the temperature increase with, for example, scanning thermal miscroscopy. Recent reports indicate that by using the anomalous Nerst effect, thermal nanoscale detection of the DW motion (although in the timescale much smaller than our) is possible 27 .
Interestingly, the effect is not restricted to moving DWs in AFMs but is a universal characteristic of magnetic textures moving at high velocities. Therefore, the concept can be extended to other textures, such as skyrmions or vortices. As example, DWs in cylindrical nanowires and nanotubes lack the Walker breakdown phenomenon and reach velocities similar to those considered here 28 . Systems with perpendicular anisotropy and Dzyaloshinki-Moriya interactions 29 may also be good candidates to observe the predicted effect. However, the AFM has additional advantage of very small DW width, due to Lorentz contraction at high velocities, which can be reached using SO torques with reasonable current intensities.

Methods
Atomic spin structure of Mn 2 Au. We use a realistic unit cell spin structure for the Mn 2 Au, see Ref. 30 for more details. The total energy is described with the following atomistic classical spin Hamiltonian, where S i is a classical vector,|S i | = 1. The exchange coupling between spins at sites i and j, J ij are where C e , C ph are the electron and phonon specific heats, respectively. The coupling between electron and lattice systems is defined by the electron-phonon coupling constant, g e−ph . Lateral heat transport is defined by k e ∂ 2 T /∂x 2 , where k e is the electronic thermal conductivity and τ d is the heat difusion time.
Exact values used in our model can be found in Table 1.