Low-damping sub-10-nm thin films of lutetium iron garnet grown by molecular-beam epitaxy

Insulating ferrimagnets are of interest for spintronic applications because they can possess very small damping parameters, as low as $10−5$ in the bulk. ¹ They also provide the potential for improving the efficiency of magnetic manipulation using spin-orbit torques from heavy metals ^2,3 and topological insulators, ^4,5 because ferrimagnetic insulators will not shunt an applied charge current away from the material generating the spin-orbit torque. Making practical devices from ferrimagnetic insulators will require techniques capable of growing very thin films (a few tens of nm and below) while maintaining low damping. Much of the previous research in this field has focused on yttrium iron garnet (Y₃Fe₅O₁₂, YIG) grown by pulsed-laser deposition (PLD) or off-axis sputtering, ^6–10 but YIG is just one in a family of rare earth iron garnets with potentially useful properties. ¹¹ Here, we examine the magnetic and structural properties of thin, (111)-oriented films of lutetium iron garnet (Lu₃Fe₅O₁₂, LuIG) grown by an alternative method, molecular-beam epitaxy (MBE). ¹² We find that MBE is capable of providing sub-10-nm films with very low values of damping, rivaling, or surpassing other deposition techniques. We are able to grow LuIG films down to 2.8 nm, or 4 layers along the interplanar spacing d₁₁₁ (0.71 nm), ^11,13 while retaining high crystalline quality. We report in- and out-of-plane (OOP) ferromagnetic resonance (FMR) measurements as a function of film thickness, demonstrating reduced two-magnon scattering compared to previous work. We achieve effective damping coefficients as low as $11.1(9)×10−4$ for 5.3 nm LuIG films and $32(3)×10−4$ for 2.8 nm films, which can be compared to the best previous report for very thin YIG, $38×10−4$ for a 4 nm film. ⁶ 

As an iron garnet, LuIG has ferrimagnetic properties similar to YIG. The magnetic moments in both materials arise from their $Fe3+$ ions, which interact via super-exchange through oxygen atoms. ^11,14 In bulk samples, LuIG has a slightly higher room-temperature saturation magnetization (1815 Oe) than YIG (1760 Oe). ^11,14,15 The bulk lattice parameters for LuIG (12.283 Å) and YIG (12.376 Å) differ by 0.75%. ^16,17 Both materials can be grown on isostructural gadolinium gallium garnet (Gd₃Ga₅O₁₂, GGG) substrates, which have a cubic lattice parameter of 12.383 Å. The resulting mismatch causes biaxial tensile strain with a maximum value of 0.81% and 0.07% for LuIG and YIG, respectively. High-quality YIG films have been grown previously using off-axis sputter deposition ^10,18–21 and pulsed-laser deposition (PLD). ^6,22–28 The best reported damping values for thin YIG films grown by PLD to date include $2.3×10−4$ for a 20 nm film, ⁶ $3.2×10−4$ for a 10 nm film treated with a post-growth etching procedure, ²⁹ and $0.7×10−4$ for a 20 nm film treated with a post-growth high-temperature anneal. ³⁰ For off-axis sputtering, the best reported values include $6.1×10−4$ for a 16 nm film, ²¹ $12.4×10−4$ for a 10.2 nm film, ¹⁹ and $0.9×10−4$ for a 22 nm film with a post-growth high-temperature anneal. ²⁰ Previous measurements of films thinner than 10 nm recorded significant two-magnon scattering, ^6,19 and much larger damping parameters of $38×10−4$ for a 4 nm film and $16×10−4$ for a 7 nm film. ⁶ 

Here, we report the growth of epitaxial LuIG films with thicknesses from 2.8 to 40 nm by reactive MBE on (111) GGG substrates. (We study LuIG, rather than YIG, primarily because Lu is available within our MBE chamber.) Our substrates are prepared by annealing at 1300 °C for 3 h in an air furnace to produce well-defined unit-cell steps and smooth terraces (see supplementary material). During growth, we simultaneously co-supply Lu and Fe with an accuracy of ±5%, to achieve the stoichiometric atomic ratio of Lu:Fe = 3:5. We use distilled ozone (O₃) at a background pressure of $1.0×10−6$ Torr as the oxidant. The growth temperature is 950–970 °C, achieved by radiatively heating the backside of the GGG substrates, which are coated with 400 nm of Pt to enhance thermal absorption. The quality of crystal growth is monitored using in-situ reflection high-energy electron diffraction (RHEED) along both the $[11¯0]$ and $[112¯]$ in-plane (IP) azimuthal directions. The RHEED intensity oscillations (Fig. 1(a)) indicate layer-by-layer growth, ³¹ with an oscillation period corresponding to the d₄₄₄ spacing, which is a quarter of a single LuIG layer (⁠$d111=0.71$ nm) along the (111)-orientation. We also observe sharp RHEED features and clear Kikuchi lines during growth, as seen in Figs. 1(b) and 1(c) for a 10 nm film, demonstrating that our films are of high crystalline quality. These features are not observed if the flux drifts more than ±5% or if the growth temperature is less than 900 °C. We do not perform any post-growth annealing on our films.

We quantify the strain state and verify the crystalline quality with four-circle X-ray diffraction (XRD) measurements. The normalized rocking curves for films with different thicknesses are shown in Fig. 2(a). All of the samples except possibly the 2.8 nm film have full-width at half-maximum (FWHM) values that are less than 0.004°, limited by the GGG substrate. This indicates that the LuIG films are commensurately strained, and are at the maximal strain state of 0.81% set by the lattice mismatch with the substrate. The rocking curve measurements on the 2.8 nm film lack sufficient signal-to-noise for an accurate quantitative analysis, but the results from the thicker films suggest that the strain state is also commensurate for this film. The surfaces of the films are characterized by atomic force microscopy. Figure 2(b) shows the 2.8 nm film, with a measured surface roughness of 0.26 nm (RMS) over a 5 μm × 5 μm scan area. This indicates that the surface quality is substrate limited, which we observe for all thicknesses. Figure 2(c) shows the $θ/2θ$ XRD patterns of the LuIG thin films for all thicknesses grown. The visible Laue oscillations confirm thickness measurements we make with the RHEED intensity oscillations and flux calibrations. Low-angle X-ray reflectivity (XRR) determines the film thicknesses as 2.84(1), 5.33(2), 9.94(2), 20.16(3), and 40.37(10) nm, which we nominally report as 2.8, 5.3, 10, 20, and 40 nm.

The magnetic properties of the MBE-grown LuIG films are characterized by measuring the frequency and thickness dependence of ferromagnetic resonance (FMR). The samples are placed, LuIG-side down, on a broadband coplanar waveguide so that the Oersted field of the waveguide excites FMR at GHz frequencies. ³² We measure the FMR spectra at fixed frequency by sweeping the applied magnetic field, oriented either in-plane (IP) parallel to the coplanar waveguide or out-of-plane (OOP). For the IP measurements, we position the film so that the applied magnetic field is always along the $[112¯]$ crystal orientation. The measured signal corresponds to the derivative absorption, which we detect via the voltage from a detector diode. We achieve optimal sensitivity using lock-in amplification by modulating both the input power and the applied field. All of the FMR measurements are performed at room temperature. Further details of the FMR apparatus are described in the supplementary material.

Figure 2(d) shows the IP-FMR response at 5 GHz for LuIG samples with different thicknesses. Two trends are apparent as the film thickness is reduced: (i) the resonance position shifts to higher fields and (ii) the linewidth increases substantially. Below we show that both of these effects can be explained by two-magnon scattering. ^33–35 We focus first on the behavior of the resonance fields. We have measured the IP-FMR resonances for each film thickness at frequencies from 1 to 10 GHz. The evolution as a function of frequency is shown in Fig. 3(a) and as a function of thickness in Fig. 3(b).

In the presence of two-magnon scattering, the IP resonance field $Hr∥$ predicted by the Kittel equation in the thin-film limit takes the form ^33,36

with f the excitation frequency, t the film thickness, $ΔHr$ a renormalization shift associated with two-magnon scattering, and γ the gyromagnetic ratio. We measured $|γ|/2π=2.77(2)$ MHz/Oe based on the frequency dependence of the OOP resonance field $Hr⊥$ (see supplementary material). The effective anisotropy field $4πMeff$ is expected to depend on the film thickness, because it contains contributions from both bulk demagnetization and surface anisotropy

Here, M_s is the saturation magnetization and K_s is the surface anisotropy energy. The renormalization shift produced by two-magnon scattering can be related to the surface anisotropy as ^33,36

where r is a parameter characterizing the strength of two-magnon scattering.

We performed a global least-squares fit of Eqs. (1)–(3) to all the data in Fig. 3 using three fitting parameters r, $4πMs$⁠, and K_s. As shown by the lines in Fig. 3, we find excellent fits assuming that all three parameters are independent of film thickness, obtaining the values $r=4.9(2)×10−4$ Oe⁻¹, $4πMs=1609(1)$ Oe, and $Ks=−8.52(8)×10−3$ erg/cm². We also attempted to fit the data without the two-magnon contribution (i.e., with the constraint r = 0 Oe⁻¹), but we found significant discrepancies for the 2.8 film, especially at low frequencies (see supplementary material). The non-zero value of r implies that the two-magnon mechanism is active. For our 2.8 nm film, the renormalization shift is $ΔHr=110$ Oe, similar to that found in a 2.7 nm NiFe film. ³⁶ The value of $4πMs$ determined by the fit is significantly lower than the bulk LuIG value of 1815 Oe. ¹⁵ This reduction is qualitatively consistent with the tensile strain in our films from the GGG substrate. The tensile strain is expected to enhance the antiferromagnetic super-exchange interaction between the two inequivalent $Fe3+$ lattices in the LuIG and therefore reduces the overall saturation magnetization. ^11,14 Another contribution to the small value of M_s could come from tetrahedral Fe $3+$ vacancies, particularly at the film surface. ^23,37 By reducing the M_s value, these can also increase the surface anisotropy field (⁠$2Ks/Mst$⁠), and therefore increase the damping through two-magnon scattering. The negative sign that we find for K_s indicates that the surface anisotropy reduces the effective demagnetization field $4πMeff$ compared to the bulk value. The magnitude of K_s is relatively weak, however (e.g., more than two orders of magnitude smaller than K_s for annealed CoFeB, ³⁸ but close to the value predicted for (111)-oriented YIG, ³⁷ $Ks=−5.2×10−3$ erg/cm²). With our values for $4πMs$ and K_s, only for extremely thin LuIG films, <0.8 nm, the magnetic anisotropy might be turned perpendicular to the sample plane. For any thickness above this, $4πMeff$ favors in-plane magnetization.

Next we consider the FWHM linewidths (⁠$ΔH$⁠) of the IP FMR resonances for our LuIG films as a function of thickness and FMR frequency. The linewidths of our samples are sufficiently narrow that small inhomogeneities in the films can result in overlapping but distinguishable resonances, as has often been seen previously in measurements on thin garnet films. ^6,25,39 To make an accurate determination of the intrinsic linewidths, we fit each measured curve to the sum of multiple (2 in this analysis) Lorentzian derivative curves with their widths constrained to be identical (see supplementary material for details). This procedure produces values for the linewidth that are consistent with the results for films that can be cleaved into samples sufficiently small to isolate a single resonance (see supplementary material).

Figure 4(a) shows the measured frequency dependence of the linewidth for each of our films. We observe a linear dependence on frequency up to ∼8 GHz. At higher frequencies, the linewidths deviate from linearity, most obviously for the 2.8 and 5.3 nm films. This high-frequency curvature is qualitatively consistent with the effect of two-magnon scattering, as observed previously in PLD-grown YIG films. ⁶ Using the expression ³² 

we can define an effective Gilbert damping parameter, α, for each value of film thickness based on linear fits to the data below 8 GHz (Fig. 4(b)). The zero-field linewidth $ΔH0$ represents the extrinsic contributions to the linewidth in this model and increases in proportion to the two-magnon scattering strength. ³⁴ The line shown in Fig. 4(b) is a fit to a phenomenological form

with $αG=0.9(6)×10−4, Aα2M=125(45)×10−4$ nm², and $Bα2M=36(11)×10−4$ nm.

Our damping values are among the best reported for any garnet film and extend the viable thickness of low-damping thin films well below 10 nm. We measure $α=11.1(9)×10−4$ for 5.3 nm LuIG films and $32(3)×10−4$ for 2.8 nm films. We speculate that our MBE growth procedure minimizes the amount of surface roughness and other defects even for very thin LuIG films, compared to other deposition techniques, and thereby provides a reduced level of two-magnon scattering. Similar MBE growth procedures may also allow the production of sub-10-nm films made from YIG and other garnets, assisting in the development of a wide variety of spintronic devices incorporating these materials. In future experiments, we will investigate controlling the O₃ pressure to determine if O$2+$ vacancies at the film surface might contribute to the small value of M_s we measure, and if a reduced vacancy concentration might reduce the damping even further.

See supplementary material for detailed information on the step terraces in the GGG substrate, FMR measurement system, gyromagnetic ratio determination, analysis of the resonance shift from two-magnon scattering, and analysis of multiple resonances in the FMR data.

We acknowledge F. Guo for helpful discussion on two-magnon theory, and a reviewer for suggesting the importance of $Fe3+$ vacancies. This work was supported by the National Science Foundation (DMR-1406333) with partial support from the Cornell Center for Materials Research (CCMR), part of the NSF MRSEC program (DMR-1120296). We made use of the Cornell Nanoscale Facility, a member of the National Nanotechnology Coordinated Infrastructure (NNCI), which is supported by the NSF (ECCS-1542081) and also the CCMR Shared Facilities. The work of H.P. and D.G.S. is supported by the Air Force Office of Scientific Research under Award No. FA9550-16-1-0192.