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The inverse thermal spin-orbit torque and the relation of the Dzyaloshinskii-Moriya interaction to ground-state energy currents

Using the Kubo linear-response formalism we derive expressions to calculate the heat current generated by magnetization dynamics in magnets with broken inversion symmetry and spin-orbit interaction (SOI). The effect of producing heat currents by magnetization dynamics constitutes the Onsager reciprocal of the thermal spin-orbit torque (TSOT), i.e., the generation of torques on the magnetization due to temperature gradients. We find that the energy current driven by magnetization dynamics contains a contribution from the Dzyaloshinskii-Moriya interaction (DMI), which needs to be subtracted from the Kubo linear response of the energy current in order to extract the heat current. We show that the expressions of the DMI coefficient can be derived elegantly from the DMI energy current. Guided by formal analogies between the Berry phase theory of DMI on the one hand and the modern theory of orbital magnetization on the other hand we are led to an interpretation of the latter in terms of energy currents as well. Based on \textit{ab-initio} calculations we investigate the heat current driven by magnetization dynamics in Mn/W(001) magnetic bilayers. We predict that fast domain walls drive strong ITSOT heat currents.


Introduction
The interaction of heat current with electron spins is at the heart of spin caloritronics [1]. It leads to thermal spin-transfer torques (STTs) on the magnetization in spin valves, magnetic tunnel junctions, and domain walls when a temperature gradient is applied [2,3,4,5,6,7,8]. While the thermal STT does not require spin-orbit interaction (SOI), it only exists in noncollinear magnets. In spin valves and magnetic tunnel junctions this noncollinearity arises when the magnetizations of the free and fixed layers are not parallel, while in domain walls it arises from the continuous rotation of magnetization across the wall.
In the presence of SOI electric currents and heat currents can generate torques also in collinear magnets: In ferromagnets with broken inversion symmetry the socalled spin-orbit torque (SOT) acts on the magnetization when an electric current is applied ( Figure 1a) [9,10,11,12,13,14,15,16,17] . The inverse spin-orbit torque (ISOT) consists in the production of an electric current due to magnetization dynamics ( Figure 1b) [18,19,20]. The application of a temperature gradient results in the thermal spin-orbit torque (TSOT) (Figure 1c) [21]. TSOT and SOT are related by a Mott-like expression [22].
In this work we discuss the inverse effect of TSOT, i.e., the generation of heat current due to magnetization dynamics in ferromagnets with broken inversion symmetry and SOI (Figure 1d). We refer to this effect as inverse thermal spin-orbit torque (ITSOT). While the SOT is given directly by the linear response of the torque to an applied electric field [16], expressions for the ITSOT are more difficult to derive because the energy current obtained from the Kubo formalism contains also a groundstate contribution that does not contribute to the heat current. Analogous difficulties are known from the case of the inverse anomalous Nernst effect, i.e., the generation of a heat current transverse to an applied electric field E [23]. In this case the energy current obtained from the Kubo formalism contains besides the heat current also the material-dependent part −E × M orb of the Poynting vector, where M orb is the orbital magnetization. This energy magnetization does not contribute to the heat current and needs to be subtracted from the Kubo linear response [23,24,25].
When inversion symmetry is broken in magnets with SOI the expansion of the free energy F in terms of the magnetization directionn(r) and its gradients contains a term linear in the gradients of magnetization, the so-called Dzyaloshinskii-Moriya interation (DMI) [26,27]: where r is the position and the index j runs over the three cartesian directions, i.e., r 1 = x, r 2 = y, r 3 = z. The DMI coefficients D j can be expressed in terms of mixed Berry phases [22,28]. DMI does not only affect the magnetic structure by energetically favoring spirals of a certain handedness but also enters spin caloritronics effects [29,30]. Here, we will show that DMI gives rise to the ground-state energy current J DMI j = −D j · n × ∂n ∂t when magnetization precesses. This DMI energy current needs to be subtracted from the linear response of the energy current in order to obtain the ITSOT heat current.
This work is structured as follows. In section 2 we show that magnetization dynamics drives a ground-state energy current associated with DMI and we highlight its formal similarities with the material-dependent part of the Poynting vector. In section 3 we develop the theory of ITSOT. We derive the energy current based on the Kubo linear-response formalism and subtract DMI in order to extract the heat current. In section 4 we show that the expressions of DMI and orbital magnetization can also be derived elegantly by equating the energy currents obtained from linear response theory to DMI and −E × M orb , respectively. In section 5 we present ab-initio calculations of TSOT and ITSOT in Mn/W(001) magnetic bilayers.

Ground-state energy current associated with the Dzyaloshinskii-Moriya interaction
To be concrete, we consider a flat cycloidal spin spiral propagating along the x direction. The magnetization direction is given bŷ where q is the spin-spiral wavenumber, i.e., the inverse wavelength of the spin spiral multiplied by 2π. The free energy contribution F DMI (r) given in (1) simplifies for the spin spiral of (2) as follows: whereê y is the unit vector pointing in y direction and we defined D ij (n) = D i (n) ·ê j . Whether D xy is nonzero or not depends on crystal symmetry. The tensor D ij (n) is axial and of second rank like the SOT torkance tensor [22]. Additionally, it is even under magnetization reversal, i.e., D ij (n) = D ij (−n). Therefore, D ij (n) has the same symmetry properties as the even SOT torkance [16]. According to (3) the cycloidal spiral of (2) is affected by DMI if D xy is nonzero. This is the case e.g. for magnetic bilayers such as Mn/W(001) and Co/Pt(111) (the interface normal points in z direction), where also the component t yx of the even SOT torkance tensor is nonzero [16,22,31,32].
We consider now a Neel-type domain wall that moves with velocity w < 0 in x direction. The magnetization direction at time t 0 = 0, which we denote byn 0 (x), is illustrated in Figure 2a.n 0 (x) can be interpreted as a modification ofn c (x) ((2)), where the q-vector depends on position: Since the domain wall moves with velocity w, the magnetization directionn(x, t) at position x and time t is given byn In Figure 2 we discuss the magnetization direction at position x 0 at the three times t 0 = 0, t 1 > t 0 and t 2 > t 1 . At time t 0 = 0 the domain wall is far away from x 0 . Therefore, the magnetization is collinear at x 0 and F DMI (x 0 , t 0 ) = 0. At time t 1 the domain wall starts to arrive at x 0 . Consequently, the magnetization gradient ∂n(x 0 , t 1 )/∂x 0 becomes nonzero and thus F DMI (x 0 , t 1 ) = 0. Due to the motion of the Illustration of a Neel-type domain wall that moves into the negative x direction. Arrows represent the magnetization directionn(x, t).n(x 0 , t) is highlighted by oval boxes. (a)n(x, t 0 ) =n 0 (x) is locally collinear at x 0 and therefore F DMI (x 0 , t 0 ) = 0. (b)n(x, t 1 ) =n 0 (x − wt 1 ) starts to become noncollinear at x 0 and therefore F DMI (x 0 , t 1 ) = 0. (c)n(x, t 2 ) is strongly noncollinear at x 0 . domain wall the DMI contribution F DMI (x, t) to the free energy is time dependent: How much DMI free energy is stored at a given position in the magnetic structure is determined by the local gradient of magnetization, which moves together with the magnetic structure. The partial derivative of F DMI (x, t) with respect to time is given by where J DMI x in the last line is the x component of the DMI energy current density By considering additionally spirals propagating in y and z direction we find that the general form of (6) is the continuity equation of the DMI energy current DMI . According to (7) and (8) the energy current DMI is driven by magnetization dynamics and its sources and sinks signal the respective decrease and increase of DMI energy density. When we compute the energy current driven by magnetization dynamics in section 3 we therefore need to be aware that this energy current contains DMI in addition to the ITSOT heat current that we wish to determine. Thus, we need to subtract DMI from the energy current in order to extract the ITSOT heat current. It is reassuring to verify that the material-dependent part orb = −E × M orb of the Poynting vector, which needs to be subtracted from the energy current to obtain the heat current in the case of the inverse anomalous Nernst effect [23], can be identified by arguments analogous to the above. We sketch this in the following. The energy density due to the interaction between orbital magnetization M orb and magnetic field B is given by We assume that the magnetic field is of the form i.e., the magnetic field at time t can be obtained from the magnetic field at time t 0 = 0 by shifting it by wt, as illustrated in Figure 3. Additionally, we assume that the orbital magnetization is of the same form, i.e., M orb (r, Due to the motion of the profile of B(r, t) the energy density in (9) changes as a function of time. The partial derivative of F orb (r, t) with respect to time is where we used the Maxwell equation ∇ × E + ∂B ∂t = 0 and E = − ∂A ∂t valid in Weyl's temporal gauge with scalar potential set to zero. Thus, as expected.
In the following we discuss several additional formal analogies and similarities between DMI, classical electrodynamics and orbital magnetization. We introduce the tensors (r) and¯ (r) with elements to quantify the noncollinearity ofn(r). and¯ are related through the matrix as = ¯ . The free energy F DMI (r) can be expressed in terms of and as follows: Illustration of a magnetic field ramp that moves into the negative x direction. Arrows represent the magnetic field B 0 (x − wt)ê z at position x and time t. B 0 (x) describes a ramp that increases linearly with x. The magnetic field at position where we defined¯ = . Similarly, DMI in (7) can be expressed in terms of¯ as The energy density F orb = −M orb ·(∇×A) in (9) involves the curl of the vector potential A, while the material-dependent part of the Poynting vector, i.e., orb = −E×M orb = ∂A ∂t ×M orb , involves the time-derivative of A. Similarly, the spatial derivatives ∂n/∂r i enter F DMI in (18) via the tensor¯ while the temporal derivative ∂n/∂t enters DMI in (19). Thus, in the theory of DMI the magnetization directionn plays the role of an effective vector potential. The curl of orbital magnetization constitutes a bound current J mag = ∇ × M orb that does not contribute to electronic transport. Hence it needs to be subtracted from the linear response electric current driven by gradients in temperature or chemical potential in order to obtain the measurable electric current [23]. Similarly, the spatial derivatives τ bound ] that result from the presence of gradients in temperature or chemical potential constitute torques that are not measurable and need to be subtracted from the total linear response to temperature or chemical potential gradients in order to obtain the measurable torque [22]. Table 1 summarizes the formal analogies and similarities between the orbital magnetization and DMI. Table 1. Formal analogies between the theories of orbital magnetization (OM) and Dzyaloshinskii-Moriya interaction (DMI). The vector potential A is assumed to satisfy Weyl's temporal gauge, hence the scalar potential is set to zero.

OM DMI
'vector potential' An

Inverse thermal spin-orbit torque (ITSOT)
In ferromagnets with broken inversion symmetry and SOI, a gradient in temperature T leads to a torque τ on the magnetization, the so-called thermal spin-orbit torque (TSOT) [22,21]: The inverse thermal spin-orbit torque (ITSOT) consists in the generation of heat current by magnetization dynamics in ferromagnets with broken inversion symmetry and SOI. The effect of magnetization dynamics can be described by the time-dependent perturbation δH to the Hamiltonian H [16] where T (r) = m×nΩ xc (r) is the torque operator. Ω xc (r) = 1 2µ B V eff minority (r) − V eff majority (r) is the exchange field, i.e., the difference between the potentials of minority and majority electrons. m = −µ B σ is the spin magnetic moment operator, µ B is the Bohr magneton and σ = (σ x , σ y , σ z ) T is the vector of Pauli spin matrices. The energy current E driven by magnetization dynamics is thus given by where the tensor with elements describes the Kubo linear response of the energy current operator to magnetization dynamics. µ is the chemical potential, v is the velocity operator and the retarded energy-current torque correlation-function is given by In (23) we take the limit frequency ω → 0, which is justified when the frequency is small compared to the inverse lifetime of electronic states, which is satisfied for magnetic bilayers at room temperature and frequency ω/(2π) in the GHz range.
Within the independent particle approximation (23) becomes B ij = B where G R (E) and G A (E) are the retarded and advanced single-particle Green functions, respectively. f (E) is the Fermi function. contains scattering-independent intrinsic contributions and, in the presence of disorder, additional disorder-driven contributions. The intrinsic Berry-curvature contribution is given by where and and |ψ kn are the Bloch wavefunctions with corresponding band energies E kn , f kn = f (E kn ), and N is the number of k points. As discussed in section 2 we subtract DMI ( (7)) from E in order to obtain the heat current Q : Inserting the Berry-curvature expression of DMI [22,28] we obtain for the intrinsic contributioñ where β = (k B T ) −1 . Using where Θ is the Heaviside unit step function, we can rewrite (33) as Here, is the intrinsic SOT torkance tensor [16,22] at zero temperature as a function of Fermi energy E and e = |e| is the elementary positive charge. The intrinsic TSOT and ITSOT are even in magnetization, i.e.,β int ij (n) =β int ij (−n). (26) contains an additional contribution which is odd in magnetization, i.e.,β odd ij (n) = −β odd ij (−n), and which is given bỹ where t odd ji (n, E) is the odd contribution to the SOT torkance tensor as a function of Fermi energy [16]. The totalβ ij (n) coefficient, i.e., the sum of all contributions, is related to the total torkance t ji (−n, E) for magnetization in −n direction bỹ which contains (35) and (37) as special cases.
It is instructive to verify that the ITSOT described by (38) is the Onsager-reciprocal of the TSOT ((20)), where [22] Comparison of (38) and (39) yields and thus  where λ is the thermal conductivity tensor and Λ describes Gilbert damping and gyromagnetic ratio [18]. As expected, the response matrix satisfies the Onsager symmetry (n) = [ (−n)] T . (38) and (30) are the central result of this section. Together, these two equations provide the recipe to compute the heat current Q driven by magnetization dynamics ∂n/∂t. We discuss applications in section 5.

Using the ground-state energy currents to derive expressions for DMI and orbital magnetization
The expression (32) for the DMI-spiralization tensor was derived both from semiclassics [28] and static quantum mechanical perturbation theory [22]. Alternatively, the T = 0 expression of can also be obtained elegantly and easily by invoking the third law of thermodynamics: For T → 0 the ITSOT must vanish,β → 0, because otherwise we could pump heat at zero temperature and thereby violate Nernst's theorem. Hence, → according to (31). In other words, at T = 0 the energy current density E in (22) is identical to the DMI energy current density DMI = − n × ∂n ∂t because the heat current is zero. Thus, at T = 0 we obtain from (27) which agrees with (32) at T = 0. Similarly, we can derive the T = 0 expression of orbital magnetization from the energy current orb = −E × M orb discussed in (14): For T → 0 the inverse anomalous Nernst effect (i.e., the generation of a transverse heat current by an applied electric field) has to vanish according to the third law of thermodynamics. Hence, the energy current driven by an applied electric field at T = 0 does not contain any heat current and is therefore identical to orb . We introduce the tensor to describe the linear response of the energy current to an applied electric field E, i.e., = E. We describe the effect of the electric field by the vector potential A = −E sin(ωt)/ω and take the limit ω → 0 later. The Hamiltonian density describing the interaction between electric current density J and vector potential is −J · A, from which we obtain the time-dependent perturbation Introducing the retarded energy-current velocity correlation-function we can write the elements of the tensor as This allows us to determine orb as orb = int E, where the intrinsic Berry-curvature contribution to the response tensor is given by with and It is straightforward to verify that M orb given by (50) agrees to the T = 0 expressions for orbital magnetization derived from quantum mechanical perturbation theory [33], from semiclassics [23], and within the Wannier representation [34,35].
Combining the third law of thermodynamics with the continuity equations (8) and (13) provides thus an elegant way to derive expressions for and M orb at T = 0. We can extend these derivations to T > 0 if we postulate that the linear response to thermal gradients is described by Mott-like expressions. In the case of the TSOT this Mott-like expression is (39), while it is [23,36,37] in the case of the anomalous Nernst effect, where σ xy (E) is the zero-temperature anomalous Hall conductivity as a function of Fermi energy E and the anomalous Nernst current due to a temperature gradient in y direction is j x = −α xy ∂T /∂y. While (39) and (51) were, respectively, derived in the previous section and in [23], we now instead consider it an axiom that within the range of validity of the independent particle approximation the linear response to thermal gradients is always described by Mottlike expressions. Thereby, the derivation in the present section becomes independent from the derivation in the preceding section. Applying the Onsager reciprocity principle to (39) and (51) we find that the ITSOT and the inverse anomalous Nernst effect are, respectively, described by (38) and by Employing the general identity (34) (but in contrast to section 3 we now use it backwards) we obtainβ from (35) and, similarly, (52) can be written as The finite-T expressions of and M orb are now easily obtained, respectively, by subtracting the ITSOT heat current given by (53) from the energy current (27) and by subtracting the heat current (54) from J y = R int yx E x . This leads to (32) for the DMI spiralization tensor and to for the orbital magnetization. (55) agrees to the finite-T expressions of M orb z derived elsewhere [33,23].

Ab-initio calculations
We investigate TSOT and ITSOT in a Mn/W(001) magnetic bilayer composed of one monolayer of Mn deposited on 9 layers of W(001). The ground state of this system is magnetically noncollinear and can be described by the cycloidal spin spiral (2) [31]. Based on phenomenological grounds [38,19] we can expand torkance as well as TSOT and ITSOT coefficients locally at a given point in space in terms ofn and¯ : ijkl , which give rise to the following contribution to the torque τ : where we used that for magnetization directionn along z it follows from symmetry considerations that t xx = t yy , t xy = −t yx , t even xx = 0 and t odd yx = 0. The SOT in this system has already been discussed by us [16]. In order to obtain TSOT and ITSOT, we calculate the torkance for the magnetically collinear ferromagnetic state with magnetization directionn set along z as a function of Fermi energy and use (39) and (38) to determine the TSOT and ITSOT coefficients β andβ, respectively. Computational details of the density-functional theory calculation of the electronic structure as well as technical details of the torkance calculation are given in [16]. The torkance calculation is performed with the help of Wannier functions [39,40] and a quasiparticle broadening of Γ = 25 meV is applied.
Due to symmetry it suffices to discuss the TSOT coefficients β even yx and β odd xx , which are shown in Figure 4 as a function of temperature. For small temperatures we find β ij ∝ T as expected from which is obtained from (39) using the Sommerfeld expansion. Slightly above 100K both β even yx and β odd xx stop following the linear behavior of the low temperature expansion (58): After reaching a maximum both β even yx and β odd xx decrease and finally change sign. At T = 300K the thermal torkances are β even yx = 5.24 × 10 −35 Jm/K and β odd xx = −3.21 × 10 −36 Jm/K. Thermal torkances of comparable magnitude have been determined in calculations on FePt/Pt magnetic bilayers [21].
Using (40) and the volume of the unit cell of V = 1.58 × 10 −28 m 3 to convert the TSOT coefficients into ITSOT coefficients, we obtainβ even yx = −99.49µJ/m 2 and β odd xx = −6.09µJ/m 2 at T = 300K. When the magnetization precesses around the z axis in ferromagnetic resonance (this situation is sketched in Figure 1d) with frequency ω and cone angle θ according tô n(t) = [sin(θ) cos(ωt), sin(θ) sin(ωt), cos(θ)] T , the following ITSOT heat current is obtained from (30) in the limit of small θ: J Q x = ωθ β odd xx cos(ωt) −β even yx sin(ωt) J Q y = ωθ β even yx cos(ωt) +β odd xx sin(ωt) , where we made use ofβ xx =β yy =β odd xx and −β xy =β yx =β even yx , which follows from symmetry considerations. Using the ITSOT coefficientsβ even yx andβ odd xx determined above at T = 300K we can determine the amplitudes of J Q x and J Q y . Assuming a cone angle of 1 • and a frequency of ω = 2π·5GHz we find that the amplitude of the oscillating heat current density J Q x is ωθ βeven The heat current density J Q y has the same amplitude. We can use the thermal conductivity of bulk W of λ xx =174 W/(Km) [41] at T =300 K to estimate the temperature gradient needed to drive a heat current of this magnitude: (55kW/m 2 )/λ xx =316 K/m. The thickness of the Mn/W(001) film is 1.58 nm. The amplitude of the heat current per length flowing in x direction is thus 55 kW/m 2 ·1.58 nm≈ 87µW/m. These estimates suggest that J Q is measurable in ferromagnetic resonance experiments.
According to (60) the heat current can be made larger by increasing the cone angle. However, in ferromagnetic resonance experiments the cone angle θ is small. Therefore, we estimate the heat current driven by a flat cycloidal spin spiral that moves with velocity w in x direction. Its magnetization direction is given bŷ (62) Withn c (r, t) × ∂n c (r, t)/∂t = wqê y we get J Q x = −β even xy wq (63) from (30), i.e., a constant-in-time heat current in x direction. Usingβ even xy = 99.49µJ/m 2 determined above and a spin-spiral wavelength of 2.3nm [31] we obtain a heat current density of J Q x =-270kW/m 2 for a spin spiral moving with a speed of w=1ms −1 . This estimate suggests that fast domain walls moving at a speed of the order of 100ms −1 drive significant heat currents that correspond to temperature gradients of the order of 0.1K/(µm).

Summary
Magnetization dynamics drives heat currents in magnets with broken inversion symmetry and SOI. This effect is the inverse of the thermal spin-orbit torque. We use the Kubo linear-response formalism to derive equations suitable to calculate the inverse thermal spin-orbit torque (ITSOT) from first principles. We find that a groundstate energy current associated with the Dzyaloshinskii-Moriya interaction (DMI) is driven by magnetization dynamics and needs to be subtracted from the linear response of the energy current in order to extract the heat current. We show that the groundstate energy currents obtained from the Kubo linear-response formalism can also be used to derive expressions for DMI and for orbital magnetization. The ITSOT extends the picture of phenomena associated with the coupling of spin to electrical currents and heat currents in magnets with broken inversion symmetry and SOI. Based on abinitio calculations we estimate the heat currents driven by magnetization precession and moving spin-spirals in Mn/W(001) magnetic bilayers. Our estimates suggest that fast domain walls in magnetic bilayers drive significant heat currents.