# Ultrafast manipulation of the magnetization

## Microwaves and terahertz pulses

_{0}

^{2}0 c/2μ0, implying that a 100 μJ, halfcycle THz pulse (0.5 ps) focused to 1 mm2 will produce a peak magnetic field B

_{0}= 1.3 T, i.e., more than 100 times that of a strip-line, and with a THz excitation bandwidth. This offers the possibility of probing the ultimate limit of magnetization dynamics, which is at least a factor 1000 faster than conventional field-induced spin switching. A fur ther possibility for rapidly per turbing the magnetic moments in a dynamic XMCD experiment is X-ray detected ferromagnetic resonance, using continuous-wave or pulsed microwave (GHz) radiation [7]. When resonant with the Zeeman-split energy levels of the magnetic ions, the microwaves excite damped magnetic precession, the details of which are sensitive to dynamic couplings and magnetization relaxation. If several magnetic species are simultaneously present in the sample, the elemental selectivity of XMCD allows the dynamics of each to be studied individually. The high peak brightness and excellent time resolution of the SwissFEL would be ver y beneficial for this technique, avoiding the present restriction to samples with very low damping.

## Laser-induced phenomena

_{el}), the electronic spin order (T

_{mag}) and the lattice vibrations (T

_{lat}) (see Infobox), has been used to describe the demagnetization process. It is assumed that the laser pump pulse initially delivers energy to the electron reservoir Tel and that each reservoir individually remains in equilibrium. But due to their relatively weak intercoupling, the three temperatures may differ significantly, giving rise to strong non-equilibrium effects which have not yet been investigated. Microscopic models have difficulty in explaining how the laser excitation of the conduction electrons can cause such a rapid transfer of angular momentum away from the spin system or indeed what is its destination. Among the proposals put forward are: hiding the angular momentum in electronic orbits, or transferring it to the lattice via special hot-spots in the electron band structure with exceptional spin-orbit coupling or via an enhancement of the spin-orbit interaction by the presence of phonons [9]. A phenomenological treatment of the entire de- and remagnetization process using an atomic analog of the LLG equations, with the magnetization M replaced by the atomic spin S, has been published by Kazantseva et al. [10]. These authors introduced an effective field acting on the atomic moments which includes a stochastic, fluctuating component, which they then related to the LLG damping constant α using the fluctuation-dissipation theorem (see Infobox). From their atomistic numerical simulations (see Fig. I.7), Kazantseva et al. [10] find that the magnetization can be non-zero in spite of a spin temperature T

_{mag}which exceeds the Curie temperature of Ni (T

_{C}= 631 K), leading them to question the concept of equilibration of the spin system and hence of a spin temperature. The authors find that the coupling constant which governs the post-pulse recovery of the magnetization is the same as that responsible for the ultrafast demagnetization. They explain the much slower recovery, and the fact that the recovery is slower for a more complete demagnetization, with the concept of a magnetic entropic barrier (see Chapter IV) – i.e., if the magnetization vanishes, it takes time for the system to reorganize itself.

### The Landau-Lifshitz Gilbert Equation

To describe the time-evolution of the magnetization subjected to a magnetic field H, one uses the Landau- Lifshitz Gilber t equation, which has the form:where is the gyromagnetic ratio (=1.76 × 10

^{11}rad s

^{-1}T

^{-1}, for g =2), and α is the damping constant. The resulting motion of the magnetization is that of a damped precession about the effective field H (see Fig. I.i3). The first term causes a precession with angular frequency ω=γH, and the second term exponentially damps the precession at the rate αω. In a one-tesla field, the precession period is 36 ps, and the damping time 1/αω may be a factor 100 longer.

## Spin-torque: ultrafast switching by spin currents

Beyond the conventional switching by applied external fields, magnetization manipulation can also be achieved by using spin transfer from spin-polarized currents that flow in a magnetic structure. This leads to ultra-fast reversal of nano-pillar elements as well as current-induced domain wall motion. Here the magnetization dynamic timescales are not limited by the gyromagnetic ratio and can be potentially much faster. Including a spin-polarized current (u is proportional to the current density and the spin-polarization of the current), the extended Landau-Lifschitz Gilber t equation now reads: with the last two terms accounting for the adiabatic and non-adiabatic spin-torque. The strength of the effect is given by u and the non-adiabaticity parameter β. Spin currents can be generated by spin-injection, spin pumping and, on a femtosecond timescale, by exciting spin-polarized charge carriers with a fs laser [23]. Furthermore, heating the electron system with a short laser pulse will generate large temperature gradients, and the resulting spin currents and such spin-torqueinduced ultra-fast magnetization dynamics can be ideally probed using the SwissFEL## Langevin dynamics and the fluctuation-dissipation theorem

In their numerical treatment of ultrafast magnezation recovery after an optical pulse, Kazantseva et al. [10] used the formalism of Langevin dynamics to relate the amplitude of a fluctuating magnetic field experienced by a local moment to the strength of the viscous damping it undergoes. The general relationship between fluctuations and damping is expressed in the fluctuation-dissipation theorem (FDT), which we derive here for the simple case of the one-dimensional Brownian motion of a par ticle [24]. Consider a particle with mass M and velocity v(t) which interacts with the local environment via both a viscous force, characterized by the mobility μ, and a randomly fluctuating force f (t). The resulting (Langevin) equation of motion is:or, with the definitions γ = 1/Mμ and A(t) = f(t)/μ:

This equation has the steady-state solution (for γ t>>1):

We require that A(t) have a zero mean value , a vanishing autocorrelation time and a time-independent variance The (steady-state) variance of the velocity is then:

Making use of the equipar tition theorem:

we find the following condition:

or:

This relation between the viscous and random forces is a special case of the more general FDT.

## Optomagnetism

A large effort is presently being made to use intense, fs pulses of circularly-polarized light to write magnetic information in thin film media. The goal is to explore the limits of high-speed magnetic writing. The production of an effective magnetic field by a circularly- polarized light wave is called the inverse Faraday effect. This is the basis of optomagnetism, i.e. the manipulation of the magnetization by laser light. A circularly-polarized light wave can produce an effective magnetic field as follows [25]: Beginning from the expression for the energy density of a light wave,one can compute an effective magnetic field as

An expression due to Onsager relates the off-diagonal elements of the dielectric tensor and the magnetization:

The resulting effective field (see Fig. I.i4):

can be as large as 1 T. A demonstration of sub-ps magnetic writing has been made by Stanciu et al. [26] using 40 fs pulses in GdFeCo (see Fig. I.i5), but fundamental questions remain regarding how the ultrafast transfer of angular momentum occurs.