Deterministic control of an antiferromagnetic spin arrangement using ultrafast optical excitation

A central prospect of antiferromagnetic spintronics is to exploit magnetic properties that are unavailable with ferromagnets. However, this poses the challenge of accessing such properties for readout and control. To this end, light-induced manipulation of the transient ground state, e.g. by changing the magnetic anisotropy potential, opens promising pathways towards ultrafast deterministic control of antiferromagnetism. Here, we use this approach to trigger a $\it{coherent}$ rotation of the entire long-range antiferromagnetic spin arrangement about a crystalline axis in $GdRh_2Si_2$ and demonstrate $\it{deterministic}$ control of this rotation upon ultrafast optical excitation. Our observations can be explained by a displacive excitation of the Gd spins$'$ local anisotropy potential by the optical excitation, allowing for a full description of this transient magnetic anisotropy potential.


Introduction
Antiferromagnets have attracted great interest in recent years due to their potential to push forward the field of spintronics 1 . A central advantage of antiferromagnetic (AF) over ferromagnetic (FM) spintronics arises from their self-cancelling spin arrangement with zero net moment, rendering them largely insensitive to external fields. Such stability holds great potential for devices (e.g. for digital storage density and longevity), but also poses significant challenges to implement magnetic functionality, requiring new approaches to interact with magnetic order. To this end, a key promise of antiferromagnetic spintronics is to exploit and control magnetic properties that are unique to antiferromagnets, such as the ordering wave vector or changes to the spin arrangement itself [2][3][4] . Another potential benefit of AF spintronics stems from their enhanced magnetic resonance frequencies, promising efficient control of the magnetic state by ultrashort pulses. While in thermal equilibrium a number of approaches for control of AF spintronic systems have been demonstrated 2 , more recently also significant progress has been made in manipulating magnetism in antiferromagnetic and ferrimagnetic systems using ultrashort THz [5][6][7][8] or visible 9,10 light pulses . However, many of these approaches relied on very specific sample geometries or material properties, e.g. noncentrosymmetric lattice symmetries 2,11 .
Another possible route for control of antiferromagnetism using optical stimulation is to utilize the local anisotropy of the AF ordered ions to steer the magnetic state of a system. For example, different configurations that are sufficiently close in energy may be susceptible to thermally driven effects on the local (single-ion) magnetic anisotropy. A sudden rise in heat induced by a femtosecond laser excitation can abruptly change this anisotropy, to which the long-range AF order would then respond by realigning according to a new easy axis. In this work, we demonstrate deterministic control of this effect in the prototypical A-type antiferromagnet GdRh2Si2. The system is an exemplary case of a magnetically soft X-Y antiferromagnet 12 , due to the low anisotropy of the closed Gd 4f shell, which has no orbital angular momentum (L=0). Its weak magnetic anisotropy makes it ideal for our study, as it allows efficient modification of the anisotropy potential of the Gd ions through ultrafast photoexcitation. Following the excitation, we observe a coherent and deterministic rotation of the entire AF arrangement of Gd spins about a crystalline axis, without loss of the longrange coherence of the AF order. Further analysis of the coherent rotations of the AF structure also allows us to fully determine the local magnetic anisotropy potential.

Equilibrium Antiferromagnetic Behaviour
Before investigating the response to laser excitation, we begin by characterizing the equilibrium behaviour of the AF order in GdRh2Si2. We study its magnetic structure, which consists of ferromagnetic Gd layers (ab planes) stacked antiparallel to each other along the [001] direction 13 (the tetragonal axis, Fig. 1a), using resonant X-ray diffraction (RXD) 14,15 .
Tuning the incoming X-rays to the Gd M5 edge (3 5/2 → 4 transitions, see supplementary Figure 1) we gain sensitivity exclusively to the long-range AF order of the Gd 4f spins by studying the (001) magnetic reflection, which is forbidden for diffraction from the crystal lattice. Since we diffract from ordered spins, the scatterers are not isotropic and the orientation of the spin arrangement relative to the light polarization vector can cause the intensity to depend on the azimuthal angle Ψ (Fig. 1a). Detecting the scattered intensity as function of Ψ therefore allows us to completely determine the orientation of the Gd 4f spin moments , which are aligned along an easy axis within the ab plane, at an angle relative to the [100] direction (Fig. 1d). The normalized diffraction intensity of the (001) reflection at various temperatures is shown in Fig. 1b. It exhibits a remarkable shift of the cosine-shaped function with temperature, indicating a change of the magnetic easy axis. Our data can be well described by a magnetic structure factor that depends on and as: 16,17 ( , Ψ) ∝ | ( ) cos(Ψ − ( ))| 2 = or .
Here and are functions of either temperature ( ) or pump-probe delay time ( , see following). Using Eq. 1 we extract the temperature-dependent behaviour of ( ) and ( ) from the data in Fig. 1b and find that ( ) fits well to a mean-field behaviour for S=7/2 (see methods), and that ( ) changes significantly in the range 30 K -70 K (Fig. 1c). This corresponds to a gradual rotation of the aforementioned easy axis in the direction from ~[110] towards [010] upon heating (see Fig. 1d). We emphasize that this a collective rotation of the entire AF spin structure about the [001] axis (i.e. the material remains AF with vanishing net moment).
To understand this effect, we consider the possible contributions to magnetic anisotropy in GdRh2Si2. Because the Gd 4f shell is spherical ( = 0), its anisotropy terms are typically very weak, and only two contributions can occur: from on-site exchange coupling to the conduction electrons which do experience magnetic anisotropy (and mediate the RKKY interaction), and from dipolar interactions 18,19 . The dipolar contribution is not expected to contribute to the rotation effect (it itself depends on the spins' orientation), leaving the conduction electrons.
Therefore, to gain insight on the observed rotation, the electron density at the Fermi level was calculated for several temperatures. The effect of temperature was simulated as a spin disorder using the disordered local moment approximation (DLM, see methods).

Antiferromagnetic Behaviour upon Ultrafast Excitation
Having established the equilibrium behaviour of the magnetic structure, we now optically excite the system in order to test if we can induce a similar behaviour on ultrafast time scales, as sketched in Fig. 2a. We studied the temporal evolution of the AF structure at a base temperature of 11 K, where ≈ 0 and ≈ 58°. We excited the system with 1.55 eV laser pulses, which arrive at delay times before the femtosecond X-ray probe pulse. The (001) intensity ( , Ψ) is found to either increase or decrease depending on Ψ, but no change is observed in the shape or position of the diffraction peak (Figs. 2b,c). This allows us to track the evolution of ( , Ψ) by measuring the time-dependent intensity at the maximum of the diffraction peak only. The evolution of ( , Ψ) with time is strongly fluence dependent, exhibiting both an oscillatory feature and a suppression in magnitude (Fig. 2d). Similar to the equilibrium case, the temporal evolution of ( ) and ( ) can be disentangled using Eq. 1 by measuring ( , Ψ) for several values of Ψ. We measured pump-probe delay scans at four values of Ψ, as indicated in Fig. 2a, and used these results to fit Eq. 1 for each value of (see supplementary note 1). We find that Eq. 1 provides a good description of the transient behaviours of ( ) and ( ) (Fig. 3). ( ) exhibits suppression and recovery dynamics, reminiscent of the ultrafast demagnetization dynamics found in various magnetic systems 21 , while ( ) exhibits a combination of a continuous rotation and a superimposed coherent oscillation, corresponding to a collective rotation of the entire AF structure upon photoexcitation (Fig. 2a).
We now discuss the dynamics of and in more detail. A hierarchy of the quantities discussed in the next sections is shown in Fig 4a for guidance.  The demagnetization of ( ) (Fig. 3a) is well-described by two exponential decay channels (see methods), "fast" and "slow", with corresponding time constants and demagnetization amplitudes . The fast channel occurs on a subpicosecond time scale (300 -600 fs, see example in the inset of Fig. 3a), while the slow channel is ~100 times slower (20 -40 ps) and accounts for most of the demagnetization amplitude ( > , Fig. 4c). Similar biexponential demagnetization has been reported for Lanthanide metals, both for AF 22,23 and FM 22,24,25 cases. In all cases the high speed subpicosecond channel accounts for a small fraction of total demagnetization, while the slower channel (i.e. a few ps to tens of ps) accounts for the rest. The appearance of two time scales has been interpreted as one demagnetization process that is initially fast due to the presence of hot electrons which have not yet thermalized with the lattice, and then becomes slower once the electrons and phonons have thermalized 26,27 . The time scales we observe (Fig 4b) are in very good agreement with those found for both demagnetization channels of ferromagnetic Gd metal 24,28 and other Gd-based materials 9,25 . Demagnetization amplitudes exhibit a linear behavior up to a critical fluence (Fig. 4c). In this linear range, the time scales slow down, associated with the rise in susceptibility near the ordering temperature 29 . For the slow channel, this occurs at / ≈ 1.
We now discuss the transient rotation dynamics of the AF structure, observed through ( ) (Fig. 3b). The main feature is a gradual reorientation of the AF structure towards higher values, analogous to the rotation upon heating (dashed lines in Fig. 3b). An additional coherent oscillation of the AF structure is observed superimposed on the gradual rotation. We stress that these behaviours are collective and coherent motions of the entire AF spin structure, and in analogy to the behaviour upon heating, represent a variation in the ions' local magnetic anisotropy. The total change in can be described as Here, Φ and Ω represent the reorientation and the oscillation, respectively (indicated in Fig.   3b). Unlike in the demagnetization data, no subpicosecond dynamics are observed. The gradual reorientation Φ( ) occurs on timescales of tens of ps (Fig. 4d), similar to the values of . At late delays the reorientation begins to recover on a slower timescale , such that the whole process can be described as The amplitude Φ 0 and time constant both grow with fluence. Such a dependence is a direct example of a controllable variation in an AF spin arrangement. To demonstrate deterministic control, the inset of Fig. 3b presents the fluence dependence of ( ) − (0) at a delay of 1 (this delay was chosen such that all oscillations are clearly damped). A linear fluence dependence is observed in a wide fluence range, reaching complete rotation of the AF structure towards the [010] direction (Δ > 30 °) for the highest fluences.
Further insight into the underlying time-dependent magnetic anisotropy potential can be gained from the coherent oscillations Ω( ). This term exhibits fluence-dependent frequencies of 6-12 of GHz (Fig. 4d), and is damped on time scales well-below the recovery of Φ( ) . It is well described in the form a Displacive excitations are typically described by a phase = 0, accounting for a step-like shift of the potential minimum, compared to the oscillation response time 30 . Here these response times are of similar order (but rotation times are still shorter than half the oscillation period, as required for displacive excitations, see Fig.  4d). This more gradual turn-on of the oscillations leads to a finite oscillation phase (see methods).
Our method provides a complete description of the transient magnetic anisotropy, which exhibits a linear relation ∝ Φ 0 in Fig. 4f, demonstrating that both quantities (easy-axis angle and anisotropy strength) change together upon excitation. Assuming the transient system to be in a thermally equilibrated state, our transient values of can be interpreted as a measure of the equilibrium anisotropy potential. This assumption is reasonable, considering that the reorientation occurs over tens of ps, which is substantially longer than typical e-ph thermalization time scales in metals of only a few ps 32 . Thus, our results demonstrate not only a method to control the AF spin structure using light, but also to probe the anisotropy potential of an antiferromagnet through a displacive excitation, which can be understood as an antiferromagnetic Γ-point magnon 33 .
In this work, we have demonstrated the optical manipulation of the local (single-ion) magnetic anisotropy via its temperature dependence. Beyond such thermal effects, strong variations in this energy landscape could also be achieved by other means, such as directly tuning magnetic interactions, varying the crystal field, distorting the local environment with an excited phonon mode, and more. Such approaches could also offer faster response times to the excitation.
Ultimately the purpose is to embed such functionalities in spintronic devices. For example, in the context of magnetic memory such effects could facilitate ultrafast all-optical control of the pinning direction in write-head spin valves, or enable ultrafast steering of spin current directions in exchange-biased layers. Understanding controllable magnetic properties is challenging with antiferromagnets because of their inaccessibility through conventional means. The insight gained on ( ) through our approach was enabled by the relatively slow GHz frequencies characteristic of our model material system. Antiferromagnets that are candidates for applications typically exhibit THz frequencies 34 , which renders similar effects more appealing for future devices, but also more challenging to identify. The effects we present may also be applicable beyond the scope of antiferromagnets, because of the Gd ions' combination of high spin moment and low anisotropy. For example, the easy axis of Gd metal is also known to rotate upon warming by up to 60° 35 , suggesting that optically-induced reorientation towards a transient easy axis could also occur. This and other similarities between our observations and other Gd-based magnets, suggest that optical control of AF order may be possible for other Gd systems. Since Gd is commonly employed in applications requiring its high 4f moment, this property may prove as useful for the design of deterministically controllable spintronic devices.

Sample Preparation
Samples were cleaved single crystals of GdRh2Si2, grown as described in 13 . Due to the layered crystal structure, the cleaved sample face normal is precisely parallel to the tetragonal [001] axis. The crystals used were approximately 1 mm 3 in size, with faces much larger than the beam sizes. Equilibrium RXD experiments Experiments were conducted at beamline X11MA of the Swiss Light Source 36 , using the RESOXS end station 37 and at beamline ID32 at the ESRF 38 .
Following Ref. 39 , in Fig. 1c the temperature dependence of ( ) was fit to a mean field approximation of the form The detection scheme allowed for single-photon counting, such that collected intensities represent photon counts per second. As such, the error bars in Fig. 2b,c,d are taken as Δ = √ . Reciprocal space scans such as those in Fig. 2b and 2c were collected for every fluence at several delays, and no broadening or shifting of the peaks was observed. Therefore only the peak heights were collected in time traces.
The time resolved experiments were conducted at four azimuths at 45° intervals: Ψ = 15°, 60°, 105° and 150°. The probed areas on the surface were chosen to ensure a single-domain state: two dimensional scans of the sample surface were taken before and after excitation.
Chosen areas were large and homogenous, and exhibited a uniform response to excitation.
The areas chosen for different azimuths were also consistent with each other, in agreement with the azimuthal trend they should exhibit according to Eq. 1. This was the case for all except the last azimuth, Ψ = 15°. For that dataset a large degree of domain mixing was apparent in the data, such that a mixture of 105° and 15° domains (90° apart) contributes to the observed intensity. The non-mixed signal for Ψ = 15° was extracted as described in supplementary methods 2. Contamination between the 60° and 150° azimuths is also possible, but due to their large difference in intensity (see Fig. 2b,c) any appreciable effect would only occur for Ψ = 150°. To account for this, a small fraction (2-4%) of the Ψ = 60° signal is subtracted from the 150° signal before fitting Eq. 1.
The demagnetizations in the curves in Fig. 3a were fit to an equation of the form The temporal resolution is estimated at 120 fs. To account for this, fits to Eq. (M3) were conducted with a convolved Gaussian response function of 120 fs width.
Lastly, fits to Eq. 4 are done with a constant damping time of = 150 ps, as this does not affect the results appreciably. Fitted Values of range between 116° and 97°.
Calculation of absorbed fluence: All fluences reported are total absorbed fluences. In order to determine the optical constants of GdRh2Si2, we conducted reflectivity measurements using 1.55 eV light. From this we estimate the complex index of refraction at this photon energy as = 0 + = (3.34 ± 0.10) + (3.19 ± 0.19) . Using this, recorded incident fluences were corrected for reflection and refraction to produce the total absorbed fluence. Note that a value of 1 mJ/cm 2 corresponds to an excitation density of 420 J/cm 3 within the first layer of the material. From this index of refraction we estimate a penetration depth of 20.6 nm at the Bragg angle .

Anisotropy potential
The anisotropy values K presented in Fig. 4f were calculated from the observed oscillations.
We first convert them to eigenfrequencies by correcting for the damping as 0 2 = 2 + −2 . We then estimate the anisotropy field by following the derivation in Ref. 42 (similarly to Ref. 31 ) as 0 = 2 , with = /ℎ and = 2 for Gd, in which is Bohr's magneton and ℎ is Planck's constant. Finally, for a simple estimate of the constant , we use the relation = / , in which is the total Gd 4f moment ( ,) and is the Landé factor.
First-principles calculations The calculations were performed using a first-principles Green's function method within the multiple scattering theory 43,44 and within the density function theory in a local density approximation 45 . To correctly describe strongly localized Gd f-states, a first-principles Hubbard model with U=3eV was used 46 .
The effect of temperature was simulated as a spin disorder using the disordered local moment approximation (DLM) 47,48 , implemented using a coherent-potential approximation 49,50 . In the DLM approach, the absence of spin disorder corresponds to the ground state at T=0, which is the antiferromagnetic order in the case of GdRh2Si2. An increase of temperature leads to disordering of magnetic moment orientations. A fully disordered moment configuration corresponds to the paramagnetic state high above TN.
The easy axis direction was calculated as follows. For a given 4f spin disorder level, the spins were aligned (anti)parallel to a given direction in the ab plane. The energy of each spin direction was calculated, and the minimal energy is taken as the easy axis angle for the given spin disorder.