The inverse thermal spin-orbit torque and its relation to DMI

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

Frank Freimuth, Stefan Blügel and Yuriy Mokrousov Peter Grünberg Institut and Institute for Advanced Simulation, Forschungszentrum Jülich and JARA, 52425 Jülich, Germany
###### Abstract

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 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.

###### pacs:
72.25.Ba, 72.25.Mk, 71.70.Ej, 75.70.Tj

## 1 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 so-called 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 ground-state 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  [23]. In this case the energy current obtained from the Kubo formalism contains besides the heat current also the material-dependent part of the Poynting vector, where 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 in terms of the magnetization direction and its gradients contains a term linear in the gradients of magnetization, the so-called Dzyaloshinskii-Moriya interation (DMI) [26, 27]:

 (1)

where is the position and the index runs over the three cartesian directions, i.e., . The DMI coefficients 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 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 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 and , respectively. In section 5 we present ab-initio calculations of TSOT and ITSOT in Mn/W(001) magnetic bilayers.

## 2 Ground-state energy current associated with the Dzyaloshinskii-Moriya interaction

To be concrete, we consider a flat cycloidal spin spiral propagating along the direction. The magnetization direction is given by

 ^nc(r)=^nc(x)=⎛⎜⎝sin(qx)0cos(qx)⎞⎟⎠, (2)

where is the spin-spiral wavenumber, i.e., the inverse wavelength of the spin spiral multiplied by . The free energy contribution given in (1) simplifies for the spin spiral of (2) as follows:

 FDMI(r) =FDMI(x)=Dx(^nc(x))⋅[^nc(x)×∂^nc(x)∂x]= (3) =qDx(^nc(x))⋅^ey=qDxy(^nc(x)),

where is the unit vector pointing in direction and we defined . Whether is nonzero or not depends on crystal symmetry. The tensor is axial and of second rank like the SOT torkance tensor [22]. Additionally, it is even under magnetization reversal, i.e., . Therefore, has the same symmetry properties as the even SOT torkance [16]. According to (3) the cycloidal spiral of (2) is affected by DMI if 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 direction), where also the component of the even SOT torkance tensor is nonzero [16, 22, 31, 32].

We consider now a Neel-type domain wall that moves with velocity in direction. The magnetization direction at time , which we denote by , is illustrated in Figure LABEL:figure1a. can be interpreted as a modification of ((2)), where the -vector depends on position:

 ^n0(x)=⎛⎜⎝sin(q(x)x)0cos(q(x)x)⎞⎟⎠. (4)

Since the domain wall moves with velocity , the magnetization direction at position and time is given by

 ^n(x,t)=^n0(x−wt). (5)

In Figure LABEL:figure1 we discuss the magnetization direction at position at the three times , and . At time the domain wall is far away from . Therefore, the magnetization is collinear at and . At time the domain wall starts to arrive at . Consequently, the magnetization gradient becomes nonzero and thus . Due to the motion of the domain wall the DMI contribution 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 with respect to time is given by

 ∂FDMI(x,t)∂t==Dx(^n0(x−wt))⋅[^n0(x−wt)×∂2^n0(x−wt)∂x∂t]++∂Dx(^n0(x−wt))∂t⋅[^n0(x−wt)×∂^n0(x−wt)∂x]==Dx(^n0(x−wt))⋅[^n0(x−wt)×∂2^n0(x−wt)∂x∂t]++∂Dx(^n0(x−wt))∂x⋅[^n0(x−wt)×∂^n0(x−wt)∂t]==∂∂x{Dx(^n0(x−wt))⋅[^n0(x−wt)×∂^n0(x−wt)∂t]}==−∂∂xJDMIx, (6)

where in the last line is the component of the DMI energy current density

 \mathscrbfJDMI= (7) = −\mathscrbfD(^n)(^n×∂^n∂t).

By considering additionally spirals propagating in and direction we find that the general form of (6) is the continuity equation

 ∂FDMI∂t+∇⋅\mathscrbfJDMI=0 (8)

of the DMI energy current . According to (7) and (8) the energy current 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 in addition to the ITSOT heat current that we wish to determine. Thus, we need to subtract from the energy current in order to extract the ITSOT heat current.

It is reassuring to verify that the material-dependent part 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 and magnetic field is given by

 Forb(r,t)=−Morb(r,t)⋅B(r,t). (9)

We assume that the magnetic field is of the form

 B(r,t)=B0(x−wt)^ez, (10)

i.e., the magnetic field at time can be obtained from the magnetic field at time by shifting it by , as illustrated in Figure LABEL:figure2. Additionally, we assume that the orbital magnetization is of the same form, i.e., . Consequently, also . can be expressed as in terms of the vector potential

 A(r,t)=^ey∫x−wt0B0(x′)dx′. (11)

Due to the motion of the profile of the energy density in (9) changes as a function of time. The partial derivative of with respect to time is

 ∂Forb∂t =−∂Morb∂t⋅B−Morb⋅∂B∂t (12) =w∂Morb∂x⋅[∇×A]+Morb⋅[∇×E] =w∂Morb∂x⋅^ez∂Ay∂x+Morb⋅[∇×E] =−∂Morb∂x⋅^ez∂Ay∂t+Morb⋅[∇×E] =−E⋅[∇×Morb]+Morb⋅[∇×E] =∇⋅[E×Morb],

where we used the Maxwell equation and valid in Weyl’s temporal gauge with scalar potential set to zero. Thus,

 ∂Forb∂t+∇⋅\mathscrbfJorb=0 (13)

with

 \mathscrbfJorb=−E×Morb, (14)

as expected.

In the following we discuss several additional formal analogies and similarities between DMI, classical electrodynamics and orbital magnetization. We introduce the tensors and with elements

 Cij(r)=^ei⋅[^n(r)×∂^n(r)∂rj] (15)

and

 ¯Cij(r)=∂^ni(r)∂rj (16)

to quantify the noncollinearity of . and are related through the matrix

 \mathscrbfK(^n)=⎛⎜⎝0−^n3^n2^n30−^n1−^n2^n10⎞⎟⎠ (17)

as . The free energy can be expressed in terms of and as follows:

 FDMI(r)=∑jDj(r)⋅[^n(r)×∂^n(r)∂rj] (18) =∑ijDji(r)^ei⋅[^n(r)×∂^n(r)∂rj] =∑ijDji(r)Cij(r)=Tr[\mathscrbfD(r)\mathscrbfC(r)]= =Tr[\mathscrbfD(r)\mathscrbfK(^n(r))¯\mathscrbfC(r)]=Tr[¯\mathscrbfD(r)¯\mathscrbfC(r)],

where we defined . Similarly, in (7) can be expressed in terms of as

 \mathscrbfJDMI= −\mathscrbfD(^n×∂^n∂t)=−¯\mathscrbfD∂^n∂t. (19)

The energy density in (9) involves the curl of the vector potential , while the material-dependent part of the Poynting vector, i.e., , involves the time-derivative of . Similarly, the spatial derivatives enter in (18) via the tensor while the temporal derivative enters in (19). Thus, in the theory of DMI the magnetization direction plays the role of an effective vector potential.

The curl of orbital magnetization constitutes a bound current 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 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.

## 3 Inverse thermal spin-orbit torque (ITSOT)

In ferromagnets with broken inversion symmetry and SOI, a gradient in temperature leads to a torque on the magnetization, the so-called thermal spin-orbit torque (TSOT) [22, 21]:

 τ=−β∇T. (20)

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 to the Hamiltonian  [16]

 δH=sin(ωt)ω[^n×∂^n∂t]⋅T, (21)

where is the torque operator. is the exchange field, i.e., the difference between the potentials of minority and majority electrons. is the spin magnetic moment operator, is the Bohr magneton and is the vector of Pauli spin matrices. The energy current driven by magnetization dynamics is thus given by

 \mathscrbfJE=−\mathscrbfB(^n)[^n×∂^n∂t], (22)

where the tensor with elements

 (23)

describes the Kubo linear response of the energy current operator

 JE=12V[(H−μ)v+v(H−μ)] (24)

to magnetization dynamics. is the chemical potential, is the velocity operator and the retarded energy-current torque correlation-function is given by

 GRJEi,Tj(ℏω,^n)=−i∞∫0dteiωt⟨[JEi(t),Tj(0)]−⟩. (25)

In (23) we take the limit frequency , 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 in the GHz range.

Within the independent particle approximation (23) becomes , with

 (26)

where and are the retarded and advanced single-particle Green functions, respectively. 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

 Bintij=2ℏN∑kn∑m≠nfknIm⟨ψkn|Tj|ψkm⟩⟨ψkm|JEi|ψkn⟩(Ekm−Ekn)2=1NV∑knfkn[Aknji−(Ekn−μ)Bknji], (27)

where

 Aknij=ℏ∑m≠nIm[⟨ψkn|Ti|ψkm⟩⟨ψkm|vj|ψkn⟩Ekm−Ekn] (28)

and

 Bknij=−2ℏ∑m≠nIm[⟨ψkn|Ti|ψkm⟩⟨ψkm|vj|ψkn⟩(Ekm−Ekn)2] (29)

and are the Bloch wavefunctions with corresponding band energies , , and is the number of points.

As discussed in section 2 we subtract ((7)) from in order to obtain the heat current :

 \mathscrbfJQ=\mathscrbfJE−\mathscrbfJDMI=−~β[^n×∂^n∂t], (30)

with

 ~β=\mathscrbfB−\mathscrbfD. (31)

Inserting the Berry-curvature expression of DMI [22, 28]

 Dij=1NV∑kn{fknAknji+1βln[1+e−β(Ekn−μ)]Bknji}, (32)

we obtain for the intrinsic contribution

 ~βintij= Bintij−Dij= (33) = − [fknAknji+1βln[1+e−β(Ekn−μ)]Bknji]} = −1NV∑knBknji{fkn(Ekn−μ)+

where . Using

 fkn(Ekn−μ)+1βln[1+e−β(Ekn−μ)]==−∫μ−∞dEf′(Ekn+μ−E)(Ekn−E)==−∫μ−∞dE∫∞−∞dE′f′(E′+μ−E)(E′−E)δ(E′−Ekn)==−∫∞−∞dE′f′(E′)(E′−μ)Θ(E′−Ekn), (34)

where is the Heaviside unit step function, we can rewrite (33) as

 ~βintij(^n)=−1eV∫∞−∞dEf′(E)(E−μ)tintji(^n,E). (35)

Here,

 tintij(^n,E)=−eN∑knΘ(E−Ekn)Bknij (36)

is the intrinsic SOT torkance tensor [16, 22] at zero temperature as a function of Fermi energy and is the elementary positive charge.

The intrinsic TSOT and ITSOT are even in magnetization, i.e., . (26) contains an additional contribution which is odd in magnetization, i.e., , and which is given by

 ~βoddij(^n)=1eV∫∞−∞dEf′(E)(E−μ)toddji(^n,E), (37)

where is the odd contribution to the SOT torkance tensor as a function of Fermi energy [16]. The total coefficient, i.e., the sum of all contributions, is related to the total torkance for magnetization in direction by

 ~βij(^n)=−1eV∫∞−∞dEf′(E)(E−μ)tji(−^n,E), (38)

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]

 βij(^n)=1e∫∞−∞dEf′(E)(E−μ)Ttij(^n,E). (39)

Comparison of (38) and (39) yields

 β(^n)=−VT[~β(−^n)]T (40)

and thus

 (−\mathscrbfJQτ/V)=(Tλ(^n)~β(^n)T−Λ(^n))⎛⎝∇TT^n×∂^n∂t⎞⎠, (41)

where is the thermal conductivity tensor and describes Gilbert damping and gyromagnetic ratio [18]. As expected, the response matrix

 \mathscrbfA(^n)=(Tλ(^n)~β(^n)T−Λ(^n)) (42)

satisfies the Onsager symmetry .

(38) and (30) are the central result of this section. Together, these two equations provide the recipe to compute the heat current driven by magnetization dynamics . We discuss applications in section 5.

## 4 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 expression of can also be obtained elegantly and easily by invoking the third law of thermodynamics: For the ITSOT must vanish, , because otherwise we could pump heat at zero temperature and thereby violate Nernst’s theorem. Hence, according to (31). In other words, at the energy current density in (22) is identical to the DMI energy current density because the heat current is zero. Thus, at we obtain from (27)

 Dij=Bintij=1NV∑knfkn[Aknji−(Ekn−μ)Bknji], (43)

which agrees with (32) at .

Similarly, we can derive the expression of orbital magnetization from the energy current discussed in (14): For 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 does not contain any heat current and is therefore identical to . We introduce the tensor to describe the linear response of the energy current to an applied electric field , i.e., . We describe the effect of the electric field by the vector potential and take the limit later. The Hamiltonian density describing the interaction between electric current density and vector potential is , from which we obtain the time-dependent perturbation

 δH=−sin(ωt)ωeE⋅v. (44)

Introducing the retarded energy-current velocity correlation-function

 GRJEi,vj(ℏω)=−i∞∫0dteiωt⟨[JEi(t),vj(0)]−⟩ (45)

we can write the elements of the tensor as

 Rij=elimω→0ImGRJEi,vj(ℏω)ℏω. (46)

This allows us to determine as , where the intrinsic Berry-curvature contribution to the response tensor is given by

 Rintij= −2eℏN∑knfkn∑m≠nIm⟨ukn|JEi|ukm⟩⟨ukm|vj|ukn⟩(Ekm−Ekn)2 (47) =

with

 Mknij=eℏ∑m≠nIm⟨ukn|vi|ukm⟩⟨ukm|vj|ukn⟩Ekn−Ekm (48)

and

 Nknij=2eℏ∑m≠nIm⟨ukn|vi|ukm⟩⟨ukm|vj|ukn⟩(Ekm−Ekn)2. (49)

From we obtain

 Morb=−12^ekϵkijRintij. (50)

It is straightforward to verify that given by (50) agrees to the 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 at . We can extend these derivations to 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]

 αxy=1e∫∞−∞dEf′(E)E−μTσxy(E) (51)

in the case of the anomalous Nernst effect, where is the zero-temperature anomalous Hall conductivity as a function of Fermi energy and the anomalous Nernst current due to a temperature gradient in direction is . 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 Mott-like 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

 JQy=TαxyEx. (52)

Employing the general identity (34) (but in contrast to section 3 we now use it backwards) we obtain

 ~βintij= −1NV∑knBknji{fkn(Ekn−μ)+ (53)

from (35) and, similarly, (52) can be written as

 JQy =−1NV∑knNknyx{fkn(Ekn−μ)+ (54)

The finite- expressions of and 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 . This leads to (32) for the DMI spiralization tensor and to

 Morbz=1NV∑knfkn{Mknyx+1βNknyxln[1+e−β(Ekn−μ)]} (55)

for the orbital magnetization. (55) agrees to the finite- expressions of derived elsewhere [33, 23].

## 5 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 of and :

 tij(^n,¯\mathscrbfC) =∑kt(1,0)ijk^nk+∑klt(0,1)ijkl¯Ckl+ (56) +∑klmt(1,1)ijklm^nk¯Clm+∑kt(2,0)ijkl^nk^nl+⋯.

The coefficients , , ,…in this expansion can be extracted from magnetically collinear calculations. Analogous expansions of the TSOT and ITSOT coefficients are of the same form. Here, we consider only and , which give rise to the following contribution to the torque :

 τ =toddxx(^ez)^n×(E×^ez)+ (57) +tevenyx(^ez)^n×[^n×(E×^ez)],

where we used that for magnetization direction along it follows from symmetry considerations that , , and . 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 direction set along