Long-distance spin-transport across the Morin phase transition up to room temperature in ultra-low damping single crystals of the antiferromagnet α-Fe2O3

Antiferromagnetic materials can host spin-waves with polarizations ranging from circular to linear depending on their magnetic anisotropies. Until now, only easy-axis anisotropy antiferromagnets with circularly polarized spin-waves were reported to carry spin-information over long distances of micrometers. In this article, we report long-distance spin-transport in the easy-plane canted antiferromagnetic phase of hematite and at room temperature, where the linearly polarized magnons are not intuitively expected to carry spin. We demonstrate that the spin-transport signal decreases continuously through the easy-axis to easy-plane Morin transition, and persists in the easy-plane phase through current induced pairs of linearly polarized magnons with dephasing lengths in the micrometer range. We explain the long transport distance as a result of the low magnetic damping, which we measure to be ≤ 10−5 as in the best ferromagnets. All of this together demonstrates that long-distance transport can be achieved across a range of anisotropies and temperatures, up to room temperature, highlighting the promising potential of this insulating antiferromagnet for magnon-based devices.


I. SUPPLEMENTARY DISCUSSION
In this section we describe the phenomenological model of magnon transport as a function of the magnetic anisotropies of hematite controlled by the temperature T and the external magnetic field H.

A. Model
The magnetic state is characterized by the Néel vector n(T, H) whose dynamics is calculated from the dynamical equation Here c is the limiting velocity of the magnons, γ is the gyromagnetic ratio, α G isthe Gilbert damping, H ex is the exchange field, which keeps the magnetic sublattice moments antiparallel, H curr = εθ H j ×ẑ/(2ed AF M s ) is directed along the spin polarization p (|p| = 1) of the current in the Pt electrode, j is the current density, is the Planck constant, d AF is the penetration depth of spin current into hematite, 0 < ε ≤ 1 is the spin-polarization efficiency, θ H is the spin Hall angle, e is the electron charge, and M s = |n|. The expression for the magnetic energy density w AF (n; T, H) in the presence of the constant external magnetic field H reads (for the details of derivation, see Ref. [1]): where H DMI > 0 is the homogeneous DMI field responsible for a small spin canting (and finite magnetization) in the easy-plane phase, H 6an > 0 is the in-plane anisotropy associated with the rhombohedral symmetry of hematite. The out-of-plane anisotropy H 2 (T ) depends on the temperature 1,2 . It is positive in the easy-axis phase and changes sign at the Morin temperature (H 2 (T M ) = 0 ). The coordinate system is related with the crystallographic axes with Z aligned along easy axis below the Morin temperature and X is parallel to one of the in-plane easy axes above T M . Note, that the sample-related coordinates used in the experimental setup ( Fig.1a and 2a of the main text) are rotated by the angle ψ = 33 • around the Y axis, so that x = Z cos ψ + X sin ψ, z = −X cos ψ + Z sin ψ. (1) assuming that δn(t, k) ∝ exp (−iωt + ik · r).
This approach is appropriate for magons with k vectors far from the Brillouin zone edge which give the main contribution to the observed spin transport.
The spin polarization of an eigenmode is a vector parallel to the dynamical magnetization 3 From the orthogonality condition δn ⊥ n (0) it follows that m dyn n (0) . We consider only modes with stationary magnetization, as only these modes contribute to spin transport signal.
As follows from Eq. (3), magnetization (and spin) depends on the polarization of the magnon mode. We introduce the polarization 0 ≤ s ≤ 1 as the ellipticity of the mode, as will be specified below (see Eq. (5)). The maximal polarization corresponds to circularly polarized modes with an ellipticity s = 1. The minimal polarization corresponds to linearly polarized modes with an ellipticity s = 0 .

B. Magnon modes in absence of current
In this section, we calculate the magnon spectra in absence of the external spin current and dissipation. Figure  In region I the linearized equations (1) for magnon modes take the form: where δn 1,2 and ω 2 1,2 are the eigen-vectors and eigen-values of the matrix γ 2 H ex M s (∂ 2 w AF /∂n j ∂n k ) | n 0 , ω H = γH · n (0) . As follows from Eqs. (4) the eigen-modes in presence of the magnetic field are spin-polarized with field-dependent ellipticity where are eigen-frequencies of the spin-polarized modes with k = 0. Frequencies of the modes with nonzero k ore obtained by substitution ω 2 1,2 → ω 2 1,2 + c 2 k 2 . In the present geometry, all eigenmodes in region I are spin-polarized due to the influence of the magnetic field, both below (in the easy-axis phase) and above (in the easy-plane) the Morin transition temperature. The polarization degree (ellipticity) is proportional to the component of the Néel vector along the field direction and vanishes in region II. This also means that the magnon gas can be spin-polarized even in the absence of the external spin-current. An exception is the case of H = 0 below the Morin temperature, where the eigenfrequencies are degenerate and the eigenmodes are circularly polarized with s = 1.
Ellipticity of magnon modes (shown with the colour code in Fig. S1) achieves the maximal value close but slightly below the phase transition line H crit (T ). However, ellipticity substantially diminishes in the region below the Morin point where the AF is formally still uniaxial, but the value of uniaxial anisotropy is small and comparable with the in-plane anisotropy. In this case a very small magnetic field induces an orthorhombic anisotropy, suppresses elliptical polarization of magnons and makes spin transport less efficient thus reducing the amplitude of the signal.This shows up as the peak in the simulated temperature dependence of the voltage signal which appears below the Morin temperature (grey curve in Fig.1c of the main text).
In region II ω H = 0 and the magnon eigenmodes in the absence of the spin-current are linearly polarized and carry no spin.

C. Magnons in the presence of a spin current
To study the effect of the external spin current on magnons we linearise Eq. (1) assuming that H curr n (0) . In region I, the corresponding equations become: From Eqs. (7) it follows that the spin current modifies the effective damping coefficient of the spin-polarized modes: Opposite signs correspond to the modes polarized along/opposite to the spin current.
According to the fluctuation-dissipation theorem, the modification of the effective damping given by Eq.(8) can be viewed as a modification of the effective temperature for each magnon mode 4 As a result, the spin current creates a nonequilibrium spin accumulation µ (thermodynamically conjugate variable to magnon spin) which assuming small fluctuations can be calculated, using a standard approach (see, e.g., 5 ): where f ( ε) = [exp (ε/(k B T )) − 1] −1 is the Bose-Einstein equilibrium distribution function for each of the modes, is the Planck constant, k B is the Boltzmann constant.
In region II, where ω H = 0, and ω 1 = ω 2 , we take into account the spacial dependence of the spin current. The orthogonal components δn 1 (k 1 ) and δn 2 (k 2 ) are coupled through the Fourier component H curr (k 1 − k 2 ) of the current, as follows from the equations: From Eq. (11) it follows that the spin current not only creates a nonequilibrium distribution of magnons, but also modifies the structure of the magnon modes. In particular, instead of linearly polarized modes , the modes in the presence of a spin current can be circularly polarized along n (0) with s = 1 . Such modes can be presented as a linear superposition of equilibrium modes δn 1 (k 1 ) and δn 2 (k 2 ) with the same frequency, satisfying the relation ω 2 1 + c 2 k 2 1 = ω 2 2 + c 2 k 2 2 . The effective damping and temperature of these modes are renormalised similar to Eqs. (8), (9), and the renormalisation is proportional to the Fourier component To estimate the Fourier spectrum H curr (k 1 − k 2 ) we assume that within the yz plane the current-induced spin accumulation is homogeneously distributed within the rectangular region −w/2 < y < w/2, − < z < 0, where w (∝ 300 nm) is the width of Pt electrode and (∝ 1 nm) is spin penetration depth. This gives a wide distribution of ∆k z ∝ 3 nm −1 in the z direction and a relatively narrow distribution ∆k y ∝ 10 µm −1 in the direction of the spin transport measurement.
The circularly polarized modes with different k vectors decay in space due to the dephasing effect. The characteristic length of the dephasing L can be estimated from the dispersion. We consider the mode with k 1 = (0, k y , 0) and k 2 = (0, k y , k z ), so that the magnon spin dephases in the z direction while the magnons propagate in the y direction.
Taking into account that the group velocity v gr = c 2 k/ω, so that v z k y = v y k z , we obtain for the propagation length in the y direction: D. Spin-transport data for different inter-stripe distances In this section we present the temperature dependence for the spin-transport data obtained for an inter-stripe distance of 7µm. The decrease of the decay length with temperature leads to disappearance of a significant spin-transport signal above 240 K for this distance.
Furthermore, we show the field dependence of the spin-transport signal at room temperature for inter-stripe distances of 800 nm and 1 µm, which look similar as the data for 500 nm shown in the main text.

D. Spin-transport data for different inter-stripe distances
In this section we present the temperature dependence for the spin-transport data obtained for an interstripe distance of 7 μm. The decrease of the decay length with temperature leads to disappearance of a significant spin-transport signal above 240 K for this distance. Furthermore, we show the field dependence of the spin-transport signal at room temperature for inter-stripe distances of 800 nm and 1 μm, which look similar as the data for 500 nm shown in the main text.