Nonlinear Spin-Seebeck Effects

• The paper outlines how magnetic field and Dzyaloshinskii–Moriya interaction lift magnon branch compensation to generate a finite spin-Seebeck current in noncollinear antiferromagnets.
• It employs multiple formalisms, including Fokker–Planck, Boltzmann, and Kubo approaches, to model nonlinear spin transport and rectification phenomena.
• The findings demonstrate that nonlinear spin-Seebeck effects span diverse platforms, offering insights for thermally driven spintronics applications.

Nonlinear spin-Seebeck current is the thermally generated spin current that arises when the response to a thermal bias is not exhausted by a linear coefficient. In the contemporary literature, the term encompasses several distinct but related phenomena: compensation-lifted spin transport in antiferromagnets, rectification under reversal of a chiral or interfacial parameter, negative differential spin Seebeck effect, sign reversal across a field-driven phase transition, and higher-order responses such as currents proportional to $(\partial_x T)^2$. A central case is the noncollinear antiferromagnet, where a temperature gradient excites magnons and produces magnon flow from hot to cold, yet the net spin current vanishes as long as the two magnon branches remain degenerate and carry opposite spin; external magnetic field and Dzyaloshinskii–Moriya interaction (DMI) lift that compensation and produce a finite, rectifiable spin-Seebeck current (Chotorlishvili et al., 2022).

1. Compensation, branch splitting, and the emergence of spin-Seebeck current

In antiferromagnetic systems without external magnetic field and without spin-polarized charge current, two magnon modes are degenerate and carry opposite spin angular momentum. A thermal bias activates spin-wave dynamics and drives a magnon flow from the hot edge to the cold edge, but the spin carried by the two branches cancels in the net current. In the noncollinear antiferromagnetic chain studied in "Rectification of the spin Seebeck current in noncollinear antiferromagnets" (Chotorlishvili et al., 2022), the local magnon density is monitored through

$\rho = m_x^2 + m_y^2,$

and the compensated state satisfies

$$I_{\text{spin}}=0 \qquad \text{when} \qquad H_z=0,\ D=0.$$

The mode-resolved spin-pumping current is written as

$$\langle I\rangle_{\pm}=\frac{M_sV}{\gamma}\,\alpha'\,\langle \mathbf m_\pm \times \dot{\mathbf m}_\pm\rangle, \qquad \alpha'=\frac{\gamma\hbar g_r}{4\pi M_sV},$$

while the total injected current into the adjacent normal metal is the difference of the two branch contributions,

$$\langle I\rangle_{R,k} = \frac{M_sV}{\gamma}\alpha' \left\{ \langle \mathbf m_-\times \dot{\mathbf m}_-\rangle - \langle \mathbf m_+\times \dot{\mathbf m}_+\rangle \right\}.$$

The subtraction is the formal expression of branch compensation: a nonzero magnon current does not, by itself, imply a nonzero spin current.

The degeneracy is lifted by magnetic field, DMI, or both. In the same treatment, the two-mode frequencies are

$$\omega_\pm(k,D)=\omega_0(k)\pm \omega_0-\omega_D(k), \qquad \omega_D(k)=2S\mathcal D\sin(k_x a), \qquad \omega_0=\gamma \tilde H_z.$$

Here the Zeeman term splits the two branches vertically, whereas DMI produces a $k$-odd shift and hence left-right asymmetry. The Fokker–Planck analysis also introduces a branch-dependent effective temperature,

$$\beta_\pm^{\rm eff} = \beta\left(1\pm \frac{Dak}{\omega_0(k)\pm\omega_0}\right),$$

which makes explicit that DMI acts as a helical, branch-selective thermal bias (Chotorlishvili et al., 2022).

2. Rectification in noncollinear antiferromagnets

The defining nonlinear feature in the noncollinear antiferromagnetic setting is rectification. In the cited work, rectification means asymmetry under reversal of the DMI sign,

$I(-D)\neq I(D),$

which is interpreted as left-right propagation asymmetry produced by nonreciprocal magnons and the magnon Doppler effect. The exchange current $I_{ex}^z$ and DMI current $\rho = m_x^2 + m_y^2,$0 both show clear asymmetry under $\rho = m_x^2 + m_y^2,$1, and the total current $\rho = m_x^2 + m_y^2,$2 is likewise asymmetric. By contrast, the spin-pumping current $\rho = m_x^2 + m_y^2,$3 is much more symmetric, with

$\rho = m_x^2 + m_y^2,$4

for weak or moderate DMI and only slight asymmetry at strong DMI. The reported rectification scale is about $\rho = m_x^2 + m_y^2,$5 for the exchange and DMI spin currents (Chotorlishvili et al., 2022).

The symmetry logic is explicit. Without field and without DMI, the magnon branches are degenerate and opposite-spin contributions cancel. With field only, degeneracy is lifted and the net spin current becomes finite. With DMI only, the dispersion shifts horizontally but the mirror symmetry of the antiferromagnetic setting considered can still preserve spin compensation, so the spin-pumping current may remain zero. With field and DMI together, both degeneracy breaking and left-right asymmetry are present, the current is strongly enhanced, and rectification appears (Chotorlishvili et al., 2022).

This rectification problem belongs to a broader interfacial theory of noncollinear magnetism. "Interfacial Spin Seebeck effect in noncollinear magnetic systems" (Flebus et al., 2018) formulates the spin current for an arbitrary noncollinear magnetic insulator|normal-metal interface by expressing localized moments in local spin frames, diagonalizing the magnon Hamiltonian by a Bogoliubov transformation, and evaluating the interfacial exchange current to second order in $\rho = m_x^2 + m_y^2,$6. In that framework, noncollinearity removes the restriction to a single conserved spin axis, so the spin current can acquire multiple polarization components, and the ferromagnetic and bipartite-antiferromagnetic results are recovered as limiting cases.

3. Representative nonlinear regimes in other platforms

Nonlinear spin-Seebeck transport is not confined to noncollinear antiferromagnets. The literature contains several experimentally and theoretically distinct manifestations.

System	Nonlinear signature	Source
Easy-axis antiferromagnet near spin flop	SSE sign reversal across the spin-flop transition	(Reitz et al., 2020)
$\rho = m_x^2 + m_y^2,$7-wave magnet	$\rho = m_x^2 + m_y^2,$8; spin-current diode behavior	(Ezawa, 22 Feb 2026)
Normal-metal–chiral-insulator heterostructure	Negative differential SSE and rectification ratio $\rho = m_x^2 + m_y^2,$9 reaching about $$I_{\text{spin}}=0 \qquad \text{when} \qquad H_z=0,\ D=0.$$0	(Zhang et al., 25 Apr 2026)
Nanoscale insulating magnetic junction with localized spin	Asymmetric and negative differential spin Seebeck effects; diode, transistor, and switch behavior	(Ren et al., 2013)
Cryogenic ferromagnetic nanowire	Transverse SSE sign changes with detector position and/or temperature gradient; longitudinal SSE nearly linear	(Elyasi et al., 2020)

Near the antiferromagnetic spin-flop transition, the nonlinearity is tied to a mode switch rather than to DMI rectification. Below the transition, two circularly polarized modes with opposite magnetic moments contribute, and the majority thermal population produces a Néel spin current. Above the transition, the low-energy ferromagnetic-like mode becomes the relevant carrier and the spin current reverses sign. The low-temperature, long-wavelength theory of this interfacial SSE reproduces the sign flip observed in Cr$$I_{\text{spin}}=0 \qquad \text{when} \qquad H_z=0,\ D=0.$$1O$$I_{\text{spin}}=0 \qquad \text{when} \qquad H_z=0,\ D=0.$$2/Pt and Cr$$I_{\text{spin}}=0 \qquad \text{when} \qquad H_z=0,\ D=0.$$3O$$I_{\text{spin}}=0 \qquad \text{when} \qquad H_z=0,\ D=0.$$4/Ta devices (Reitz et al., 2020).

In $$I_{\text{spin}}=0 \qquad \text{when} \qquad H_z=0,\ D=0.$$5-wave magnets without spin-orbit coupling, the leading diagonal response is even in the thermal gradient,

$$I_{\text{spin}}=0 \qquad \text{when} \qquad H_z=0,\ D=0.$$6

Because $$I_{\text{spin}}=0 \qquad \text{when} \qquad H_z=0,\ D=0.$$7 is invariant under $$I_{\text{spin}}=0 \qquad \text{when} \qquad H_z=0,\ D=0.$$8, the spin current does not reverse sign when the thermal gradient is reversed; this is the basis of the spin-current-diode interpretation. The same work contrasts this second-order diagonal response with third-order transverse response in $$I_{\text{spin}}=0 \qquad \text{when} \qquad H_z=0,\ D=0.$$9-wave magnets, linear spin-Nernst response in $$\langle I\rangle_{\pm}=\frac{M_sV}{\gamma}\,\alpha'\,\langle \mathbf m_\pm \times \dot{\mathbf m}_\pm\rangle, \qquad \alpha'=\frac{\gamma\hbar g_r}{4\pi M_sV},$$0- and $$\langle I\rangle_{\pm}=\frac{M_sV}{\gamma}\,\alpha'\,\langle \mathbf m_\pm \times \dot{\mathbf m}_\pm\rangle, \qquad \alpha'=\frac{\gamma\hbar g_r}{4\pi M_sV},$$1-wave altermagnets, and absence of thermally generated spin current in the $$\langle I\rangle_{\pm}=\frac{M_sV}{\gamma}\,\alpha'\,\langle \mathbf m_\pm \times \dot{\mathbf m}_\pm\rangle, \qquad \alpha'=\frac{\gamma\hbar g_r}{4\pi M_sV},$$2-wave case (Ezawa, 22 Feb 2026).

In the normal-metal–chiral-insulator heterostructure, chiral phonons carry angular momentum that is converted into electron spin flips at the interface. The resulting spin current depends nonlinearly on thermal bias, chemical potential, and an interfacial onsite shift $$\langle I\rangle_{\pm}=\frac{M_sV}{\gamma}\,\alpha'\,\langle \mathbf m_\pm \times \dot{\mathbf m}_\pm\rangle, \qquad \alpha'=\frac{\gamma\hbar g_r}{4\pi M_sV},$$3. For $$\langle I\rangle_{\pm}=\frac{M_sV}{\gamma}\,\alpha'\,\langle \mathbf m_\pm \times \dot{\mathbf m}_\pm\rangle, \qquad \alpha'=\frac{\gamma\hbar g_r}{4\pi M_sV},$$4, the current can first increase with $$\langle I\rangle_{\pm}=\frac{M_sV}{\gamma}\,\alpha'\,\langle \mathbf m_\pm \times \dot{\mathbf m}_\pm\rangle, \qquad \alpha'=\frac{\gamma\hbar g_r}{4\pi M_sV},$$5 and then decrease, producing negative differential SSE. The current is also rectified,

$$\langle I\rangle_{\pm}=\frac{M_sV}{\gamma}\,\alpha'\,\langle \mathbf m_\pm \times \dot{\mathbf m}_\pm\rangle, \qquad \alpha'=\frac{\gamma\hbar g_r}{4\pi M_sV},$$6

and the rectification ratio

$$\langle I\rangle_{\pm}=\frac{M_sV}{\gamma}\,\alpha'\,\langle \mathbf m_\pm \times \dot{\mathbf m}_\pm\rangle, \qquad \alpha'=\frac{\gamma\hbar g_r}{4\pi M_sV},$$7

can reach about $$\langle I\rangle_{\pm}=\frac{M_sV}{\gamma}\,\alpha'\,\langle \mathbf m_\pm \times \dot{\mathbf m}_\pm\rangle, \qquad \alpha'=\frac{\gamma\hbar g_r}{4\pi M_sV},$$8 (Zhang et al., 25 Apr 2026).

The nanoscale insulating magnetic junction provides a different route to nonlinearity. There, two ferromagnetic metallic leads are coupled through a localized spin, direct electron transport is absent in the main model, and pure spin current is transmitted by exchange-assisted spin-flip scattering. For $$\langle I\rangle_{\pm}=\frac{M_sV}{\gamma}\,\alpha'\,\langle \mathbf m_\pm \times \dot{\mathbf m}_\pm\rangle, \qquad \alpha'=\frac{\gamma\hbar g_r}{4\pi M_sV},$$9,

$$\langle I\rangle_{R,k} = \frac{M_sV}{\gamma}\alpha' \left\{ \langle \mathbf m_-\times \dot{\mathbf m}_-\rangle - \langle \mathbf m_+\times \dot{\mathbf m}_+\rangle \right\}.$$0

so the linear-response tight-coupling limit gives $$\langle I\rangle_{R,k} = \frac{M_sV}{\gamma}\alpha' \left\{ \langle \mathbf m_-\times \dot{\mathbf m}_-\rangle - \langle \mathbf m_+\times \dot{\mathbf m}_+\rangle \right\}.$$1 and hence $$\langle I\rangle_{R,k} = \frac{M_sV}{\gamma}\alpha' \left\{ \langle \mathbf m_-\times \dot{\mathbf m}_-\rangle - \langle \mathbf m_+\times \dot{\mathbf m}_+\rangle \right\}.$$2. Beyond linear response, the overlap functions $$\langle I\rangle_{R,k} = \frac{M_sV}{\gamma}\alpha' \left\{ \langle \mathbf m_-\times \dot{\mathbf m}_-\rangle - \langle \mathbf m_+\times \dot{\mathbf m}_+\rangle \right\}.$$3 and Bose-like factors $$\langle I\rangle_{R,k} = \frac{M_sV}{\gamma}\alpha' \left\{ \langle \mathbf m_-\times \dot{\mathbf m}_-\rangle - \langle \mathbf m_+\times \dot{\mathbf m}_+\rangle \right\}.$$4 make the current asymmetric under thermal-bias reversal and can also generate negative differential spin Seebeck effect (Ren et al., 2013).

4. Microscopic and mesoscopic formalisms

Several theoretical frameworks are used to describe nonlinear spin-Seebeck current, and the choice of formalism follows the dominant degrees of freedom.

For symmetry-classified metallic magnets, the Boltzmann equation gives a compact hierarchy of thermal responses. In the $$\langle I\rangle_{R,k} = \frac{M_sV}{\gamma}\alpha' \left\{ \langle \mathbf m_-\times \dot{\mathbf m}_-\rangle - \langle \mathbf m_+\times \dot{\mathbf m}_+\rangle \right\}.$$5-, $$\langle I\rangle_{R,k} = \frac{M_sV}{\gamma}\alpha' \left\{ \langle \mathbf m_-\times \dot{\mathbf m}_-\rangle - \langle \mathbf m_+\times \dot{\mathbf m}_+\rangle \right\}.$$6-, and related wave cases, the $$\langle I\rangle_{R,k} = \frac{M_sV}{\gamma}\alpha' \left\{ \langle \mathbf m_-\times \dot{\mathbf m}_-\rangle - \langle \mathbf m_+\times \dot{\mathbf m}_+\rangle \right\}.$$7-th order current induced by a temperature gradient along $$\langle I\rangle_{R,k} = \frac{M_sV}{\gamma}\alpha' \left\{ \langle \mathbf m_-\times \dot{\mathbf m}_-\rangle - \langle \mathbf m_+\times \dot{\mathbf m}_+\rangle \right\}.$$8 is

$$\langle I\rangle_{R,k} = \frac{M_sV}{\gamma}\alpha' \left\{ \langle \mathbf m_-\times \dot{\mathbf m}_-\rangle - \langle \mathbf m_+\times \dot{\mathbf m}_+\rangle \right\}.$$9

This formula makes the symmetry constraint transparent: mirror symmetry can force even- or odd-order terms to vanish, so nonlinear spin-Seebeck transport is directly controlled by band-structure parity (Ezawa, 22 Feb 2026).

For metallic ferromagnetic thin films with heavy-metal and oxide interfaces, a stochastic Landau–Lifshitz–Gilbert equation is converted into a Fokker–Planck equation on the unit sphere. The resulting average spin-pumping current is

$$\omega_\pm(k,D)=\omega_0(k)\pm \omega_0-\omega_D(k), \qquad \omega_D(k)=2S\mathcal D\sin(k_x a), \qquad \omega_0=\gamma \tilde H_z.$$0

with $$\omega_\pm(k,D)=\omega_0(k)\pm \omega_0-\omega_D(k), \qquad \omega_D(k)=2S\mathcal D\sin(k_x a), \qquad \omega_0=\gamma \tilde H_z.$$1. The nonlinear dependence enters through the Langevin factor and through the effective frequencies $$\omega_\pm(k,D)=\omega_0(k)\pm \omega_0-\omega_D(k), \qquad \omega_D(k)=2S\mathcal D\sin(k_x a), \qquad \omega_0=\gamma \tilde H_z.$$2 and $$\omega_\pm(k,D)=\omega_0(k)\pm \omega_0-\omega_D(k), \qquad \omega_D(k)=2S\mathcal D\sin(k_x a), \qquad \omega_0=\gamma \tilde H_z.$$3, which incorporate external field, Rashba field, and spin Hall field. In the analytical treatment, parallel $$\omega_\pm(k,D)=\omega_0(k)\pm \omega_0-\omega_D(k), \qquad \omega_D(k)=2S\mathcal D\sin(k_x a), \qquad \omega_0=\gamma \tilde H_z.$$4 and $$\omega_\pm(k,D)=\omega_0(k)\pm \omega_0-\omega_D(k), \qquad \omega_D(k)=2S\mathcal D\sin(k_x a), \qquad \omega_0=\gamma \tilde H_z.$$5 enhance the current, antiparallel alignment suppresses it, and numerical micromagnetics show that DMI also modifies the total current (Chotorlishvili et al., 2018).

For arbitrary noncollinear magnetic interfaces, the Kubo approach of (Flebus et al., 2018) expresses the current through local spin angles $$\omega_\pm(k,D)=\omega_0(k)\pm \omega_0-\omega_D(k), \qquad \omega_D(k)=2S\mathcal D\sin(k_x a), \qquad \omega_0=\gamma \tilde H_z.$$6, interfacial exchange couplings $$\omega_\pm(k,D)=\omega_0(k)\pm \omega_0-\omega_D(k), \qquad \omega_D(k)=2S\mathcal D\sin(k_x a), \qquad \omega_0=\gamma \tilde H_z.$$7, magnon Bogoliubov matrices $$\omega_\pm(k,D)=\omega_0(k)\pm \omega_0-\omega_D(k), \qquad \omega_D(k)=2S\mathcal D\sin(k_x a), \qquad \omega_0=\gamma \tilde H_z.$$8 and $$\omega_\pm(k,D)=\omega_0(k)\pm \omega_0-\omega_D(k), \qquad \omega_D(k)=2S\mathcal D\sin(k_x a), \qquad \omega_0=\gamma \tilde H_z.$$9, and electron and magnon spectral functions. That formulation is explicitly interfacial: the current is generated by inelastic electron–magnon scattering driven by a temperature difference between the magnet and the metal.

For chiral-phonon-mediated spin transport, the appropriate method is nonequilibrium Green’s functions. In the normal-metal–chiral-insulator problem, the spin current is defined by

$k$0

and the phonon-mediated electron self-energies are solved self-consistently. The nonlinearity is not imposed phenomenologically; it emerges from the competition among thermal bias, thermally excited electron density, and effective interfacial spectral density (Zhang et al., 25 Apr 2026).

5. Bulk, interfacial, and local-detection nonlinear transport

A recurring distinction in the field is whether the nonlinear spin-Seebeck current is generated in the bulk of the magnetic medium, at an interface, or in a strongly local detector geometry.

In bulk single-crystal YIG, nonlocal spin Seebeck measurements support a decomposition of the magnon spin current into

$k$1

The first term is pure magnon spin diffusion, driven by the gradient of magnon chemical potential, and the second is the intrinsic spin Seebeck contribution, driven by the magnon-temperature gradient. Experimentally, this produces two decay lengths: a short-range, nearly temperature-independent scale of about $k$2, and a longer scale that is about $k$3 near $k$4, exceeds $k$5 at low temperature, and reaches $k$6 at $k$7. Finite-element modeling reproduces the separation between the rapidly decaying $k$8 contribution and the much more slowly decaying $k$9 contribution (Giles et al., 2017).

At cryogenic temperatures in a ferromagnetic nanowire, the distinction is instead between local transverse detection and bulk longitudinal collection. With dissipation modeled by rare-earth two-level systems, the transverse spin Seebeck effect becomes strongly nonlinear: the sign-change position moves along the wire as temperature is varied, and on detector sites in the right half of the wire the current can show a peak and then reverse sign as $$\beta_\pm^{\rm eff} = \beta\left(1\pm \frac{Dak}{\omega_0(k)\pm\omega_0}\right),$$0 increases. By contrast, the longitudinal SSE remains nearly linear in $$\beta_\pm^{\rm eff} = \beta\left(1\pm \frac{Dak}{\omega_0(k)\pm\omega_0}\right),$$1 even when $$\beta_\pm^{\rm eff} = \beta\left(1\pm \frac{Dak}{\omega_0(k)\pm\omega_0}\right),$$2 and far outside the strict linear-response regime. This shows that detector locality and boundary magnon accumulation can determine whether strong nonlinearities are observed at all (Elyasi et al., 2020).

In nonlocal metallic spin valves, Joule heating of the injector provides another local route to nonlinear thermal spin transport. In Py/Cu and Py/Ag lateral spin valves, the nonlocal spin signal is symmetric in current polarity at low current but becomes asymmetric above a critical current of about $$\beta_\pm^{\rm eff} = \beta\left(1\pm \frac{Dak}{\omega_0(k)\pm\omega_0}\right),$$3 in a Py/Cu sample. The interpretation is that electrical spin injection is supplemented by a thermally generated spin current from the spin-dependent Seebeck effect in the Py injector. This thermal contribution is even in current polarity because it scales with Joule heating, and above $$\beta_\pm^{\rm eff} = \beta\left(1\pm \frac{Dak}{\omega_0(k)\pm\omega_0}\right),$$4 it accounts for about $$\beta_\pm^{\rm eff} = \beta\left(1\pm \frac{Dak}{\omega_0(k)\pm\omega_0}\right),$$5 of the total spin injection (Erekhinsky et al., 2018).

6. Terminological scope and recurrent misconceptions

One persistent misconception is that any thermally driven magnon flow is automatically a spin-Seebeck current. The noncollinear-antiferromagnet result shows the opposite: magnon current can be nonzero while the net spin current is zero because the two degenerate magnon branches carry opposite spin. A second misconception is that DMI alone necessarily yields a finite spin-pumping current; in the antiferromagnetic setting of (Chotorlishvili et al., 2022), DMI alone can preserve branch compensation, whereas field plus DMI produces the strong enhancement and rectification.

The terminology of the field is also broader than the strict definition of thermally generated spin current. In quantum-dot transport, "spin-current Seebeck effect" denotes the generation of an ordinary electric voltage by a spin bias under open-circuit conditions,

$$\beta_\pm^{\rm eff} = \beta\left(1\pm \frac{Dak}{\omega_0(k)\pm\omega_0}\right),$$6

with the response strongly enhanced by intradot Coulomb interaction; the reported maximum $$\beta_\pm^{\rm eff} = \beta\left(1\pm \frac{Dak}{\omega_0(k)\pm\omega_0}\right),$$7 can be enhanced by a factor of $$\beta_\pm^{\rm eff} = \beta\left(1\pm \frac{Dak}{\omega_0(k)\pm\omega_0}\right),$$8 (Yang et al., 2013). In graphene, "non-linear spin Seebeck effect" can denote spin-current-induced charge voltage in a paramagnet with energy-dependent conductivity, with

$$\beta_\pm^{\rm eff} = \beta\left(1\pm \frac{Dak}{\omega_0(k)\pm\omega_0}\right),$$9

and the nonlinear spin signal appearing in the second-order coefficient $I(-D)\neq I(D),$0 (Vera-Marun et al., 2011). These are spin-to-charge conversion effects rather than direct measurements of thermally generated spin current.

A related but distinct room-temperature example is the nonlinear Seebeck effect in Ni$I(-D)\neq I(D),$1Fe$I(-D)\neq I(D),$2|Pt bilayers. There the measured quantity is a second-harmonic nonlinear Seebeck voltage, not a spin current, but the proposed mechanism is explicitly spin-current mediated: a temperature gradient in Pt generates spin current through the spin Nernst effect, the interfacial spin accumulation modulates the Seebeck coefficient of NiFe, and the voltage scales as

$I(-D)\neq I(D),$3

The sign reversal with layer-order inversion and the inverse-length scaling distinguish this nonlinear thermopower from the conventional linear Seebeck effect (Hirata et al., 4 Jul 2025).

Collectively, these results show that nonlinear spin-Seebeck current is not a single mechanism but a class of nonequilibrium spin-caloritronic responses. The operative ingredients vary by platform—mode compensation and its lifting, mirror-symmetry breaking, interfacial spectral asymmetry, density-of-states overlap, detector locality, or bulk magnon-temperature gradients—but in each case the defining feature is the failure of a purely linear, odd-in-bias description of thermally generated spin transport.