Magnetization dynamics: From the Landau–Lifschitz equation to spintronics Open Access

Most of our digital electronic devices are based on complementary metal oxide semiconductor (CMOS) technology.¹ It allows for the integration of billions of transistors on a single chip. The quiescent current and the required energy per operation are small enough that a mobile phone can be powered by a battery. The reason is that the field effect transistors are controlled by electric fields. Therefore, in steady state (besides leakage currents), no current needs to flow between the gate and the channel of the transistor. The energy needed to switch a logic gate is given by the energy stored in the gate capacitance. CMOS technology scaled in a very favorable way as smaller transistors have a smaller gate capacitance and can operate at a lower supply voltage. Both reduce the energy per operation and increase the switching speed. This technology is very successful for combinational and sequential logic, finite-state machines, processors, and random access memory devices. However, nonvolatile memory is difficult to realize as CMOS devices. FLASH is the most common nonvolatile CMOS memory technology. It stores the data in an insulated metal island on top of the channel of a field effect transistor. This floating gate is charged and discharged by carrier injection for writing. The reading is performed by measuring the resistance of the channel. Although reading FLASH memory cells is fast (as the difference in resistance between 1 and 0 states is large), writing is generally slow:² It takes around 100  $μs$ to write a bit. In addition, the endurance, which is the number of write cycles before a memory cell is damaged, is around $104$⁠. Therefore, a better solid-state storage device is desirable.

Among other storage technologies such as phase change memory³ and ferroelectric random access memory,⁴ ferromagnetic random access memory⁵ (MRAM) is very promising.

Figure 1 illustrates an MRAM cell. Each cell consists of an exchange biased “fixed” and a “free” magnetic layer. The information is stored in the magnetization direction of the free layer. The two layers are separated by a spacer layer, which is either a non-magnetic metal or a tunnel barrier. To read the state of a memory cell, the resistance can be measured: the giant magnetoresistance (or tunnel magnetoresistance) effect^6–9 leads to a different resistance for the parallel and antiparallel orientation of the magnetizatins of the free- and fixed-layers.

The writing of such a device can be achieved in four different ways:

• The oldest technique is to apply a magnetic field and use the Zeeman interaction between the field and the spins to cause switching. Below one nanosecond, the actual switching mechanism is governed by the Landau–Lifschitz–Gilbert equation (LLG),^10,11 which is the equation of motion of magnetism. Jo Stöhr and Hans Christoph Siegmann pioneered experiments using the subpicosecond magnetic field pulses generated by the SLAC linear accelerator to switch magnetic domains,¹² demonstrating the validity of the LLG equation for femtoseconds.

• If we pass a current through a spin valve, it gets spin polarized by the fixed layer. This spin polarized current is injected into the free layer and interacts with its magnetization. If the current is sufficiently large, the magnetization of the free layer can be switched. This “spin transfer torque” (STT) is being used in commercial MRAM devices.

• Similar to the spin transfer torque, a spin current can also be generated by the spin-orbit torque via the spin-Hall effect.¹³ 

• The magnetization of a ferromagnet can also be switched by a femtosecond laser pulse.¹⁴ The mechanism was unknown for a long time. Time-resolved x-ray spectroscopy played an essential role in understanding the switching mechanism¹⁵ and how the same effect can be implemented in a realistic device by electrical pumping.¹⁶ 

To understand the effect of a spin current on the magnetization of a ferromagnet, we first need to introduce the equation of motion for the magnetization, the Landau–Lifschitz–Gilbert equation.^10,11 It is the basis of micromagnetic simulations.

The LLG equation was developed to describe the motion of magnetic domains as well as ferromagnetic resonance on the timescale of $>10 ps$⁠. Jo Stöhr and Hans Christoph Siegmann have demonstrated its validity even for the subpicosecond time scales.^12,18 Figure 2 shows the experimental setup. An initially uniformly magnetized ferromagnetic film was exposed to the electron beam of the SLAC linear accelerator. A single electron pulse of $140 fs$ pulse length was sent through the center of the sample. The highly relativistic electron pulse (kinetic energy: 20 GeV) generates a circular magnetic field distribution shown in Fig. 2(b). It shows all possible angles with the initial magnetization of the sample. In addition to the magnetic field $B→$⁠, there is also a radial electric field component $E→$⁠. The magnetic field pulse reaches a maximum of 60 T and decays with larger distance r from the center of the beam. After exposure, the magnetization of the sample was investigated using scanning electron microscopy with polarization analysis (SEMPA).¹⁹ 

Figure 3 shows the magnetization after exposure. The result is similar to the result from a previous experiment with picosecond electron pulses:¹⁸ If the magnetization is parallel or anti-parallel to the applied magnetic field, no switching can be observed. This is true even at the magnetic field of $>50 T$⁠. The lowest field required for switching is achieved if $M→$ is perpendicular to $B→$⁠: here, switching occurs at the largest distance from the center of the electron beam. This becomes obvious if we look at the LLG equation (1): the precessional term reaches a maximum at perpendicular alignment between the field and the magnetization. In this geometry, the magnetization gets lifted out of the plane by the initial field pulse. The demagnetizing field in turn causes the switching on the picosecond timescale. The pattern in Fig. 3 can be fully reconstructed using the LLG equation, the shape anisotropy, and an electric field-induced anisotropy.¹² This experiment demonstrated that the LLG equation is valid, even at extreme field strengths of $>50 T$ and on a timescale of 140 fs.

The LLG equation is quite complex as the magnetic field $H(x,t)$ is the total field that acts on the magnetization. It consists of dipolar-, exchange-, and external fields. Therefore, emerging phenomena can be observed, such as the formation of magnetic vortices.^20,21 Figure 4 shows the behavior of a magnetic vortex after applying an in-plane magnetic field pulse: magnetic islands (patterned by focused ion beam milling on a waveguide) are exposed to in-plane magnetic field pulses. The response of the magnetization was detected by x-ray photoemission electron microscopy²² (XPEEM) using the time structure of the synchrotron to detect the dynamics. The top row of Fig. 4 shows the static domain structure of the magnetic islands. The balance between the exchange- and dipolar energy leads to the flux closure domain states visible in the figure. In the center, where the domain walls cross, a magnetic vortex is formed. Here, the magnetization is perpendicular to the plane. The lower part of Fig. 4 shows the vortex core trajectory after the field pulse. The core moves on an elliptical orbit. Surprisingly, the sense of rotation can differ although the domain structures are equal. This experiment shows that the motion of the vortex (and with it the entire domain structure) is determined by the orientation of the vortex core. This was not expected because the vortex core is much smaller than its orbit.²⁰ The magnetic vortex is an example of an emerging phenomenon: it acts like a quasiparticle and shows its own dynamics. Although it is fully described by the LLG equation, we do not expect the vortex dynamics before we perform experiments or simulations. Later, it has been demonstrated that the orientation of a magnetic vortex core can be reversed by the creation of a vortex—antivortex pair and anihilation of the original vortex with the antivortex.²³ 

An essential property of the LLG equation is that the magnitude $|M→|$ is not altered by the precessional and damping term: In Eq. (1), both terms are perpendicular to $M→̇$⁠. In contrast, when the solid is heated, the magnitude of the magnetization is reduced. Beaurepaire et al.²⁴ have demonstrated that a ferromagnet can be demagnetized by a laser pulse within around 100 fs. This timescale was surprising, as the LLG equation is governed by the transfer and conservation of angular momentum; damping in ferromagnetic resonance experiments typically requires hundreds of picoseconds.²⁵ 

Koopmans et al.²⁶ provide insight into ultrafast demagnetization, revealing that the damping parameter $α$ of the Landau–Lifshitz–Gilbert equation (LLG) is dimensionless and does not directly describe a timescale. Instead, it is the combination of this parameter with the precessional frequency that determines the timescale for demagnetization. This means that even if the precession is driven by an extremely strong field (e.g., exchange interaction), the resulting damping can still occur on the femtosecond timescale. It took another decade to understand the mechanism of the angular momentum transfer. One of the reasons was that two effects play an important role: local spin flips²⁷ and the generation of spin currents.²⁸ 

From ultrafast demagnetization two new fields emerged: manipulation of the magnetization (in ferrimagnets) directly by optical excitations^29,30 (all-optical switching) and generation of femtosecond spin current pulses and its potential to manipulate the magnetization by spin injection.^31–33 Ultrafast x-ray magnetic circular dichroism shed light on the all-optical switching mechanism in ferrimagnets: As the electron gas is heated up, the two sublattices of the ferrimagnet demagnetize at a different rate, causing the formation of a transient ferromagnetic state.^34,35 The interaction of these magnetic subsystems, in turn, leads to the reversal of the magnetization. This discovery led to all-optical switching in multilayer structures composed of ferromagnetic layers.^36–38 It is unlikely that a laser-driven switching mechanism will be used in a memory device, as visible light is difficult to focus onto individual elements in a high-density storage medium. Recently, Yang et al.¹⁶ have successfully demonstrated that picosecond current pulses can rapidly heat magnetic nanostructures to induce switching, mirroring all-optical toggle switching mechanisms. This experiment highlights the potential for fundamental discoveries made using femtosecond lasers and x-ray pulses to be translated into practical applications in memory devices controlled by current pulses.

If we inject a spin-polarized current into a ferromagnet, the injected spins will interact with the magnetization. An early experiment indirectly demonstrating this interaction was performed by Weber et al.³⁹ They injected a beam of spin-polarized electrons into a free-standing thin ferromagnetic film. The spin polarization of the transmitted beam was measured by a Mott spin polarimeter. They observed that the spin polarization of the transmitted beam was altered in two ways: (i) it precessed around the magnetization of the film at a rate of $33°/nm$ in iron. (ii) In addition, there exists a damping mechanism, which causes the spin polarization to relax into the magnetization direction.

If we neglect the transfer of angular momentum to the lattice, we expect the change in spin polarization to act back on the magnetization.

The possibility of magnetization manipulation by spin injection has been independently predicted by Berger⁴⁰ and Slonczewski.⁴¹ The first experimental observation of this process was possible in magnetic nano-pillars grown by electrochemical deposition.⁴² In this experiment, switching was only possible in the presence of an additional magnetic field. Later, complete spin-injection-driven switching could be achieved with state-of-the-art lithographic patterning.⁴³ 

The distribution of the transit times for the injected electrons in combination with the precessional motion within the exchange field leads to dephasing of the precessional motion.⁴¹ On average, the transversal (with respect to the magnetization direction) component of the injected spin polarization averages out to zero. Angular momentum conservation dictates that this transversal component is passed on to the magnetization of the ferromagnet.

The first functioning spin-transfer-torque MRAM cell has been demonstrated by Katine et al.⁴³ Here, a spin valve was switched by passing a current through the device. Switching was achieved by generating a spin current in the fixed layer, which is then injected into the free layer. Both layers were magnetized in the sample plane and therefore had a large easy-plane shape anisotropy. This geometry required a high current density on the order of $108 A/cm2$ to switch. The reasons for this are the low spin polarization of injected current, the strong easy-plane anisotropy⁴⁸ as well as the almost parallel alignment of the free and fixed layer. According to Eq. (3), this reduces the initial spin torque in the free layer. In fact, if the alignment is perfect, no switching should occur.

Thus far, all of the experiments on spin transfer torque have used the magneto-resistance effect to measure the motion of the magnetization. Without an imaging technique, it is not possible to know if the magnetization within the free layer remains uniform during the switching process. In general, imaging spin-transfer effects in metallic structures is difficult. The main difficulties are the following:

• The magnetic field generated by the charge current interacts with the magnetic structure as well. At a diameter of $>100$ nm, the Oersted field of the charge current density starts to dominate the spin dynamics.

• To carry the charge current that generates the spin current, thick copper contacts are needed. These are opaque to optical detection methods.

• A temporal resolution of $<100$ ps is needed to observe the spin current-induced switching process.

With the development of modern x-ray microscopes at $3rd$ generation synchrotron sources, these requirements can be met. Scanning transmission x-ray microscopy⁴⁹ (STXM) has been shown to be well suited for these experiments. In a STXM, the synchrotron radiation is focused onto the sample using a Fresnel zone plate. A point detector is mounted behind the sample and the sample is scanned laterally through the x-ray focus, while the transmitted intensity is measured. The STXM is well adapted to time-resolved experiments, as the detector is typically a photo diode. The high electrical bandwidth makes it very flexible to adapt the detection scheme to the time structure of the excitation pulses, as well as to the time structure of the source.⁵⁰ For these reasons, STXM is successful in imaging magnetodynamics on the nanometer scale.^51–55 The spatial resolution of 30 nm in combination with a time resolution of 100 ps and the chemical, as well as the magnetic contrast, provides access to the switching dynamics in the spin valves. The setup is shown in Fig. 5. In this experiment,⁵³ the spin valve has been patterned on a silicon nitride membrane. Using x-ray magnetic circular dichroism, the magnetization directions in the plane of the sample can be probed. A current pulse sequence is applied to the sample, which alternates between pulses to switch the sample between the parallel (“set”) and the antiparallel (“reset”) states while the switching dynamics is detected using the x-ray magnetic circular dichroism effect. A special photon counting system detects the photons from individual x-ray pulses of the synchrotron.⁵⁰ 

Figure 6 shows the result of the time resolved measurements. The images in panels (a)–(i) are taken at the times indicated in the upper part of the figure. Before the “set” pulse, the free layer (here called the “nanomagnetic element”) is uniformly magnetized. The switching process starts in panel (b). The magnetization changes from the uniform state to a vortex-like configuration. The vortex core enters and propagates through the nanomagnetic element, leaving behind a trail of switched magnetization. The switching by the “reset” pulse happens in the same way by initiating a C-like state, the formation and translation of a magnetic vortex through the structure.

Micromagnetic simulations have shown that the switching mechanism is caused by both the spin torque effect and the Oersted magnetic field of the charge current.^52,56 Initially, the magnetization was assumed to be uniform along the easy axis of the magnet. Therefore, the spin torque is zero as long as the spin polarization is exactly antiparallel to the magnetization [see Eq. (3)]. The Oersted field is circular around the center of the nanomagnetic element and causes the magnetization to be bent into a C-like state. This nonuniform configuration, in turn, is partially non-parallel to the injected spin polarization. Simulations show that the damping-like part of the spin torque enhances the bending of the C-state into a vortex. It also causes the propagation of the vortex across the nanomagnet.

This experiment shows that the Oersted field initiates the switching by bending the magnetization of the free layer. In this way, the spin torque can act on the free layer, and negative damping causes the formation and propagation of a vortex across the magnetic storage element. In the following years, more efficient techniques for spin torque switching have been found, as described in detail by Pinarbasi et al.⁵⁷ With the development of magnetic tunnel junctions (MTJ) the switching current as well as the resistance difference between the parallel and antiparallel states could be reduced.^58–60 Another approach to achieve a strong initial torque is to align the free- and fixed-layer magnetizations perpendicular to each other.^61,62 As MTJs are used in these experiments, the NEXI torque is increased.⁴⁶ Therefore, the magnetization of the free layer can be reversed by precession. These devices show a subnanosecond switching speed.^62,63 However, the information is stored in a magnetic layer that is magnetized along the in-plane direction. Thus, it is hard to switch while it requires to be large enough for long-term thermal stability:⁴⁸ Thermal stability is given by the in-plane shape anisotropy, whereas the current required for switching depends on both the in-plane and easy-plane anisotropy. Modern MRAM cells, therefore, use storage elements that are magnetized perpendicular to the plane. Today, 1 GB memory devices are commercially available.

Although MRAM devices are currently only used in niche applications where good endurance and high speed are needed, it is likely that MRAM will find wider-spread applications in memory-centric computation and machine learning applications: the strict separation between the microprocessor, the memory, and storage device is not beneficial, as it requires the transfer of large amounts of data. A technology that unifies memory and storage and could be integrated into the processing hardware would be desirable.

Magnetism is essential for our information society, as even today large amounts of data are stored on hard disks. CMOS technology offers FLASH as an alternative, all-solid-state storage technology, which is very successful in consumer devices. Still, FLASH is limited in writing performance and endurance. Here, magnetic random access memory is an alternative approach. It offers the writing speed of dynamic RAM and does not wear out when writing to it. Therefore, magnetism may play a more important role in the future. Jo Stöhr made essential contributions to the understanding of spin dynamics and spin torque switching. Thanks to his work, we understand that the LLG equation is valid even for the subpicosecond timescale. The complexity of magnetization dynamics due to the dipolar and exchange fields leads to emerging phenomena. An example is the magnetic vortex, which acts like a quasiparticle and shows its own dynamics. Thanks to x-ray microscopy, it is now possible to directly observe magnetic vortices and the role they play in spin torque switching.

Experiments in ultrafast magnetism employ femtosecond lasers for excitation²⁴ and synchrotron- and free-electron laser radiation for imaging.^49,64,65 In this way, all-optical switching and its microscopic origin have been discovered.^14,15 These techniques give unprecedented insight into solid-state dynamics on the femto- and attosecond time scales. In contrast, all technological applications involve electrical currents and voltages for reading and writing. A way to bridge this gap has been shown by Yang et al.:¹⁶ they demonstrate an electrically switchable memory cell, which uses the same mechanism as found in all-optical switching experiments. Modern electronics is capable of generating picosecond current pulses. The heat pulse generated in this way can be used to manipulate ferrimagnetic structures in the same way as observed in all-optical switching experiments. The ultrafast electrically driven switching experiments^16,66 link the knowledge we gained from femtosecond lasers and x-ray sources to a possible device.

The author has no conflicts to disclose.

Yves Acremann: Conceptualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.