Room-temperature helimagnetism in FeGe thin films

Chiral magnets are promising materials for the realisation of high-density and low-power spintronic memory devices. For these future applications, a key requirement is the synthesis of appropriate materials in the form of thin films ordering well above room temperature. Driven by the Dzyaloshinskii-Moriya interaction, the cubic compound FeGe exhibits helimagnetism with a relatively high transition temperature of 278 K in bulk crystals. We demonstrate that this temperature can be enhanced significantly in thin films. Using x-ray scattering and ferromagnetic resonance techniques, we provide unambiguous experimental evidence for long-wavelength helimagnetic order at room temperature and magnetic properties similar to the bulk material. We obtain α intr = 0.0036 ± 0.0003 at 310 K for the intrinsic damping parameter. We probe the dynamics of the system by means of muon-spin rotation, indicating that the ground state is reached via a freezing out of slow dynamics. Our work paves the way towards the fabrication of thin films of chiral magnets that host certain spin whirls, so-called skyrmions, at room temperature and potentially offer integrability into modern electronics.


S1. SAMPLE PREPARATION
The Fe-Ge binary alloy system has a complex equilibrium phase diagram, whereby a large variety of phases exist with different stoichiometries, structural, and magnetic properties.
We performed thin growth by magnetron sputtering using a stoichiometric (Fe:Ge=1:1) target. Except for the desired B20 phase, the most probable impurity phases to expect for our growth conditions are the B35 and the monoclinic phase. Both are antiferromagnetic, providing good magnetic contrast with the B20 phase.
FeGe thin film samples were grown in a home-built, two-chamber UHV magnetron sputtering system with a base pressure of 5 × 10 −9 mbar. The MgO substrates were degreased prior to loading as described in the main text, followed by a high-temperature anneal in UHV for up to 8 h. The FeGe films are sputtered with an Ar partial pressure of 6 × 10 −3 mbar, using a DC power of 40 W. The substrate temperature was kept at 400 • C for the growth of the FeGe H samples. A plot of the dependence of the magnetic transition temperature as a function of growth temperature is shown in Fig. S4. The thickness is monitored by an in-situ quartz crystal microbalance, and ex-situ using x-ray reflectivity (XRR), as shown in  Intensity (arb. units)

(°) q
FIG. S1. X-ray reflectivity data for a typical FeGe film. The film was grown at 400 • C on a 1"diameter MgO(001) wafer. The numerous Kiessig fringes are indicative of well-defined interfaces.
The data were fitted using the Parratt algorithm (red line). The thickness of this particular film is 65.2 nm (nominally 70 nm), with a root-mean-square roughness of 1-2 nm. This film was used for transverse field muon-spin rotation measurements.

S2. STRUCTURAL PROPERTIES
Based on the high-resolution out-of-plane x-ray diffraction (XRD) measurements (example shown in Fig. 1 in the main text), we are able to extract the FeGe(002) d-spacing of that film, resulting in a lattice constant of 4.6480Å. Given that the bulk FeGe lattice constant is 4.70Å, it can be concluded that the compressive strain is ∼1.1%.
Apart from the determination of the out-of-plane lattice parameters by XRD, we studied the perfection of the interfaces, and the thickness of the epitaxial films, using x-ray reflectivity. XRR scans were carried out on a D5000 diffractometer using Cu Kα 1 radiation. The reflectivity data was fitted using the Parratt32 algorithm 1 . An example of typical XRR data for these films is shown in Fig. S1.
In order to determine the crystalline relationship between film and substrate, we per-

S3. MAGNETIC PROPERTIES
The magnetic properties of the films were determined using a superconducting quantum interference device (SQUID) magnetometer with a vibrating sample magnetometer module (Quantum Design). Magnetisation measurements were performed as a function of temperature, for the field applied in-plane and out-of-plane, in a temperature range between 10 and 300 K.  The susceptibility 1/µ 0 · dM/dH, derived from the data shown in (a), is presented in Fig.   S3b.
The magnetic transition temperature T c , obtained from determining the minimum in the

S4. FMR RESONANCES
For the interpretation of the FMR data we use the free energy functional for a chiral magnetic film F = F 0 + F dip with F 0 = d rF 0 and where the unit vectorn( r, t) represents the orientation of the local magnetisation, A is the exchange stiffness constant, K is the magnetic anisotropy, M s is the saturation magnetisation, and H is the applied magnetic field. The second term is the Dzyaloshinskii-Moriya interaction parametrised by the helix pitch vector Q. The remaining term F dip describes the dipolar interaction of the magnetisation field at finite momenta k, withn( k) = d rn( r)e −i k r and the volume V .
We consider the magnetic field applied perpendicular to the film along the z-axis, H = Hẑ. The ground state is obtained by minimizing the energy functional. There is a phase transition between the conical phase and the field-polarised phase at the critical field H c2,z = The magnon excitation spectrum is derived in the standard linear spin-wave approximation following Refs. 5 and 6. The spin-wave excitation is parametrised in terms of a complex wave function ψ,n In the following, we discuss the result for the field-polarised phase, H > H c2,z , where the magnetisation is field-polarised andn eq =ẑ. We limit ourselves however to a discussion of magnon excitations homogeneous within the plane of the film, i.e., with vanishing in-plane wavevector, k ⊥ = 0. The stationary wave equation then reduces to (for k ⊥ = 0) In addition, it follows from Eq. (3) that the wave function must obey the boundary condition For a film with surfaces located at z = 0 and z = d with the film thickness d, the normalised eigenfunctions consistent with the boundary condition are given by with the integer quantum number p = 0, 1, 2, 3, .... They specify the so-called perpendicular standing spin wave (PSSW) modes 4 of a film with Dzyaloshinskii-Moriya interaction. The corresponding discrete energy spectrum is given by Note that the PSSW modes differ from a conventional ferromagnet 7 due to the oscillating factor e −iQz as was pointed out in Ref. 8.
In Fig. 3 of the main text, we have assumed that at large fields the two modes seen experimentally correspond to p = 0 and 1. With these assumptions, the resulting fit yields µ 0 H c2,z ≈ 0.05 T and Dπ 2 /d 2 gµ B µ 0 H c2,z ≈ 3 with g = 1.9. We have refrained from fitting the data at low fields to a theory for the conical phase. The small value for the critical field indicates that the film at zero field is already close to the phase transition between the conical and the paramagnetic phase hampering a quantitative comparison with theory.