Exceptional sign changes of the nonlocal spin Seebeck effect in antiferromagnetic hematite

Low-power spintronic devices based on the propagation of pure magnonic spin currents in antiferromagnetic insulator materials offer several distinct advantages over ferromagnetic components including higher-frequency magnons and a stability against disturbing external magnetic fields. In this work, we make use of the insulating antiferromagnetic phase of iron oxide, the mineral hematite $\alpha \text{−}{\mathrm{Fe}}_2{\mathrm{O}}_3$ to investigate the long-distance transport of thermally generated magnonic spin currents. We report on the excitation of magnons generated by the spin Seebeck effect, transported both parallel and perpendicular to the antiferromagnetic easy-axis under an applied magnetic field. Making use of an atomistic hematite toy model, we calculate the transport characteristics from the deviation of the antiferromagnetic ordering from equilibrium under an applied field. We resolve the role of the magnetic order parameters in the transport, and experimentally we find significant thermal spin transport without the need for a net magnetization.

Figure 1

(a) Schematic magnon dispersion as a function of an applied field for a uniaxial antiferromagnet. The application of a magnetic field splits the degeneracy of the two lowest magnon modes. At the spin-flop field $H_{sf}$, the frequency of one mode reaches zero and the two circular modes change character. (b) Schematic of the two easy-axis modes. The right circularly polarized mode ${\omega }_{\alpha }$ and the left circularly polarized mode ${\omega }_{\beta }$.

Figure 2

(a) Geometry used to investigate the nonlocal spin Seebeck effect (NL-SSE) in hematite (blue). A passing charge current heats the hematite below one Pt wire, generating a thermal gradient $\mathbf{\nabla }T$. The easy-axis is inclined at ${33}^{\circ }$ to the surface plane and a magnetic field $\mathit{H}$ is applied parallel in the in-plane component of the Néel vector. (b) Local SSE as a function of magnetic field at $T=240\mathrm{K}$ (magenta) and $T=200\mathrm{K}$ (black). The dashed line indicates the spin-flop field at the respective temperature. (c) NL-SSE at $T=240\mathrm{K}$ (magenta) and $T=200\mathrm{K}$ (black) as a function of the applied magnetic field. The dashed lines indicate the spin-flop field at the respective temperature. (d) NL-SSE as a function of the applied magnetic field parallel to the in-plane component of the easy-axis for temperatures above and below the Morin transition temperature ($T_M=257\mathrm{K}$). (e) NL-SSE as a function of field and temperature. Open circles indicate the spin-flop field from Ref. [28] and triangles indicate the inversion field from panel (d). The dashed line indicates $T_M$. The magnitude of $\mathit{H}$ stated is that along $\mathit{x}$.

Figure 3

(a) Nonlocal schematic for wires parallel to $\mathit{x}$. The magnetic field is parallel to the easy-axis in-plane projection and perpendicular to the thermal gradient $\mathbf{\nabla }T$. (b) Nonlocal antiferromagnetic spin Seebeck effect as a function of field for temperatures below $T_M$. The magnitude of $\mathit{H}$ stated is that along $\mathit{x}$. (c) Current dependence of the nonlocal signal normalized to the input power. The inset shows the same data but on a larger y-axis due to the divergent behavior at low input currents (d, e) Phase diagram of the nonlocal spin Seebeck signal as a function of field and temperature for two current densities, $2\times {10}^{11}\mathrm{A}{\mathrm{m}}^{-2}$ and $7\times {10}^{11}\mathrm{A}{\mathrm{m}}^{-2}$ respectively. The circles represent the spin-flop field taken from Ref. [28].

Figure 4

(a) Geometry used for calculations with a size $N_x\times N_y\times N_z=16\times 16\times 12288$. A temperature step of $25\mathrm{K}$ was used to excite the magnetic ordering and we probe the perturbation from equilibrium of the Néel vector and magnetization some distance from this excitation. The easy-axis is aligned along the transport direction and a magnetic field $\mathit{H}$ is applied at ${33}^{\circ }$ to the easy-axis. (b) Perturbation of the Néel vector from equilibrium (${\mathit{n}}_0$) as a function of the applied field, projected along the magnetic field, where $\mathrm{\Delta }{n}_{\mathrm{p}}$ is defined in the main text. (b) Perturbation of the magnetization from equilibrium (${\mathit{m}}_0$) as a function of the applied field, projected along the magnetic field, where $\mathrm{\Delta }{m}_{\mathrm{p}}$ is defined in the main text. The dashed lines indicates the spin-flop field of the system.

Figure 5

(a) $x, y$, and $z$ components of the Néel vector under an applied magnetic field at ${33}^{\circ }$ to the easy-axis for our choice of parameters. We find the spin-flop to occur at ${\mu }_0\mathit{H}=6.4\mathrm{T}$, similar to the experimental observation. (b) $x, y$, and $z$ components of the magnetization under an applied magnetic field for our choice of parameters. Above the spin-flop field, a large component of $\mathit{m}$ appears parallel to the field.

Figure 6

(a) Local spin Seebeck effect measured for the geometry in Fig. 2 as a function of the applied magnetic field for two current densities. (b) nonlocal spin Seebeck effect measured for the geometry in Fig. 2 as a function of the applied magnetic field for two current densities. Error bars represent the standard deviation of the data point.

Figure 7

Nonlocal spin Seebeck signal parallel to the in-plane projection of the easy-axis in hematite for a magnetic field applied parallel to the transport direction across a distance of (a) 500 nm or (b) $7\mu \mathrm{m}$. The magnitude of $\mathit{H}$ stated is that along $\mathit{x}$. The general shape and trends are identical between the two distances. The error bars represent the standard deviation of the data point.

Figure 8

(a) Nonlocal geometry for investigating the nonlocal spin Seebeck effect in hematite for transport parallel to the in-plane projection of the easy-axis. The magnetic field is applied at an angle $\beta$ as indicated where $\beta =0^{\circ }$ occurs for $\mathit{H}$ parallel to $\mathit{x}$. (b) Local (red, right axis) and nonlocal (black closed and open symbols, left axis) spin Seebeck signals at $200\mathrm{K}$ under a magnetic field of ${\mu }_0\mathit{H}=2\mathrm{T}$ rotated through an angle $\beta$. The solid lines represent fits to a $sin\beta$ function. (c) Local (red, right axis) and nonlocal (black closed and open symbols, left axis) spin Seebeck signals at $200\mathrm{K}$ under a magnetic field of ${\mu }_0\mathit{H}=8\mathrm{T}$ rotated through an angle $\beta$. The solid lines represent fits to a $sin\beta$ function. (d) Nonlocal spin Seebeck signal for a fixed value of $\beta$ for a wire separation of 500 nm at $200\mathrm{K}$.

Figure 9

(a) Nonlocal geometry for investigating the nonlocal spin Seebeck effect in hematite for transport perpendicular to the in-plane projection of the easy-axis. The magnetic field is applied at an angle $\beta$ as indicated where $\beta =0^{\circ }$ occurs for $\mathit{H}$ parallel to $\mathit{x}$. (b) Nonlocal spin Seebeck signal for a fixed value of $\beta$ for a wire separation of 500 nm at $200\mathrm{K}$ and a current density of $2\times {10}^{11}\mathrm{A}{\mathrm{m}}^{-2}$.

Figure 10

(a) Schematic geometry of Pt wires relative to the easy-axis direction along $\mathit{x}$. (b) Local spin Seebeck signal for a voltage dropped along $\mathit{x}$ as a function of the applied magnetic field, also along $\mathit{x}$ for several temperatures.