# Spin transfer torques and spin-dependent transport in a metallic F/AF/N tunneling junction

## Abstract

We study spin-dependent electron transport through a ferromagnetic-antiferromagnetic-normal metal tunneling junction subject to a voltage or temperature bias, in the absence of spin-orbit coupling. We derive microscopic formulas for various types of spin torque acting on the antiferromagnet as well as for charge and spin currents flowing through the junction. The obtained results are applicable in the limit of slow magnetization dynamics. We identify a parameter regime in which an unconventional damping-like torque can become comparable in magnitude to the equivalent of the conventional Slonczewski’s torque generalized to antiferromagnets. Moreover, we show that the antiferromagnetic sublattice structure opens up a channel of electron transport which does not have a ferromagnetic analogue and that this mechanism leads to a pronounced field-like torque. Both charge conductance and spin current transmission through the junction depend on the relative orientation of the ferromagnetic and the antiferromagnetic vectors (order parameters). The obtained formulas for charge and spin currents allow us to identify the microscopic mechanisms responsible for this angular dependence and to assess the efficiency of an antiferromagnetic metal acting as a spin current polarizer.

## I Introduction

The last few years have witnessed a growing interest in the use of antiferromagnets as active elements in spintronic devices.Gomonay and Loktev (2014); Jungwirth et al. (2016, 2018) Antiferromagnets are an attractive platform for novel magnetic recording devices due to their large typical resonance frequency in the THz regime, robustness against magnetic perturbations and the absence of stray magnetic fields. Recent experiments have further revealed that spin transport is strongly affected by antiferromagnetic order. Specifically, precision measurements of magnetoresistance, spin current absorption, and its transmission have proven to be powerful tools for studying antiferromagnetic order in thin film multilayer structures. Mewes et al. (2010); Park et al. (2011); Merodio et al. (2014a, b); Kriegner et al. (2016); Kravets et al. (2017); Wang et al. (2014, 2015); Moriyama et al. (2015); Prakash et al. (2016); Qiu et al. (2016); Wang et al. (2017) Manipulation and switching of the antiferromagnetic order parameter, the Néel vector, are possible via current-induced spin torques. In particular, the effectiveness of relativistic (Néel) spin-orbit torques, first proposed theoretically in Ref. Železný et al., 2014, has been demonstrated in several recent experimental works. Wadley et al. (2016); Roy et al. (2016); Grzybowski et al. (2017); Bodnar et al. (2018) The relativistic Néel spin-orbit torque, however, requires significant spin-orbit coupling and a rather particular crystalline structure. It is therefore of interest to understand generic properties of antiferromagnetic metals that persist even in the non-relativistic limit, i.e., for negligible spin-orbit coupling. This is the main aim of this paper, where a minimal microscopic model will be employed to analytically study dynamics and transport in antiferromagnetic nanostructures. Our study complements existing theoretical approaches,Núñez et al. (2006); Haney and MacDonald (2008); Xu et al. (2008); Hals et al. (2011); Gomonay et al. (2012); Cheng et al. (2014); Saidaoui et al. (2014); Železný et al. (2014); Yamane et al. (2016); Saidaoui et al. (2017); Manchon (2017) which have been predominantly phenomenologicalNúñez et al. (2006); Haney and MacDonald (2008); Hals et al. (2011); Gomonay et al. (2012); Cheng et al. (2014); Yamane et al. (2016) or relied on extensive numerical computations.Xu et al. (2008); Železný et al. (2014); Saidaoui et al. (2014, 2017) Our approach is in a similar spirit to the work of Stiles and ZangwillStiles and Zangwill (2002) on ferromagnetic spin transfer torques and aims at revealing the anatomy of antiferromagnetic spin-transfer torques.

We consider an antiferromagnetic metal (AF), tunneling coupled to two leads (Fig. 1). The leads are made of ferromagnetic metal (F), and normal metal (N), respectively. We aim at exploring basic transport characteristics of F/AF/N junctions that do not require specific material properties. Spin-orbit coupling is therefore neglected. The tunneling barriers are chosen as spin-conserving and the antiferromagnetic order is established on a bipartite lattice. When a bias voltage or temperature difference is applied between the leads, charge and spin currents flow and generate spin transfer torques acting on the localized spins at the two sublattices in the AF. The main role of the ferromagnetic lead in this setup is to generate a finite spin-polarization. The normal metal lead, in turn, provides an explicit channel of relaxation for the electrons on the AF, a crucial ingredient for a number of phenomena discussed below.

We derive microscopic formulas for the four symmetry-allowed spin transfer torques and identify physical mechanisms responsible for each of them. We find that in addition to Slonczewski’s damping-like torque, familiar from ferromagnetic multilayers, two unconventional types of torque can become relevant in certain parameter regimes. We also obtain analytical expressions for the dependence of the charge and spin currents on the angle between the ferromagnetic and antiferromagnetic order parameters. The spin current transmission depends strongly on this angle, indicating that spin-valve applications are plausible for antiferromagnetic metals. The predictions on the torques and currents may be experimentally tested against each other since they are all given in terms of a common set of control parameters.

Before summarizing our findings in more detail, it is instructive to first review the phenomenology of magnetization dynamicsGomonay and Loktev (2014) in bipartite antiferromagnets. This phenomenology is based on the observation that the strong exchange interaction responsible for the antiferromagnetic order between and implies that the normalized total spin angular momentum is much smaller than the normalized Néel order parameter where . Ignoring quantities of order , one obtains and . Then the symmetry-permitted terms of spin torque at the leading order in in the dynamical equations of and are given by

(1) | |||||

(2) |

where is a unit vector along the polarization vector of the injected spin current. The subscripts and stand for field-like and anti-damping-like respectively. represents the conventional Slonczewski’s spin transfer torque and acts like an external magnetic field. The so-called Néel spin-orbit torque, which can be effective in systems with spin-orbit coupling and is therefore outside the scope of this paper, would enter the equations as a contribution to . The remaining has not been discussed very much so far in the literature. All four torques are allowed by symmetries and could therefore be introduced on purely phenomenological grounds. In this work, we will go one step further and discuss their relative strengths and microscopic origin for the case of the F/AF/N junction. In general, the dynamical equations for and do not only include torques, but additional terms accounting for damping and noise. While these can be discussed within the formalism described below, they are beyond the scope of the present work.

It is also instructive to interpret the various types of torque in the two-ferromagnet picture, in which the overlap between electronic orbitals located at and sublattice sites is negligibly small. In this limit, one may consider AF as a pair of oppositely oriented ferromagents as far as electron transport is concerned. For a ferromagnet, it is well known that quantum mechanical dephasing Stiles and Zangwill (2002) leads to a strong Slonczewski’s damping-like torque Slonczewski (1996, 2002) and a negligibly small field-like torque. Assuming that the spin torque acting on each individual ”ferromagnetic” moment is of the Slonczewski type then corresponds to where is the total spin current flowing into AF. We note that these estimates are based on a model in which the injected spin current does not appreciably change the state of AF. It therefore implicitly assumes the presence of a relaxation mechanism faster than the rate of tunneling at the F/AF interface so that it quickly wipes out any influence of the injected current. In this sense, the above estimates apply to a regime of weak F/AF coupling.

Our calculations generalize the two-ferromagnetic result by including the intersublattice overlap along with N as the source of the relaxation that dissipates the injected current. We show that the intersublattice overlap opens up an additional channel of electron transport in which the dephasing can be avoided and the transverse spin is conserved. This results in a novel contribution to the field-like torque proportional to the square of the overlap amplitude. The inclusion of N turns out to be crucial here as is inversely proportional to the relaxation rate. The finite relaxation rate also allows us to explore the regime of strong tunneling at F/AF interface. We find that the antiferromagnetic state modified by the ferromagnetic current generates nonvanishing and , of which the latter may reach a magnitude comparable to that of .

Similarly based on the two-ferromagnet picture, each sublattice contributes to the conductance a ferromagnetic angular dependent term, Landauer (1970); Inoue and Maekawa (1996) proportional to respectively. This angular dependence cancels in the total conductance, however, because of the antiferromagentic order . Phenomenologically this is a consequence of the symmetry between the and sublattices and the charge current is predicted to be a function of in the leading order cylindrical harmonics expansion. Still staying within the picture of two superimposed ferromagnets, the spin current flowing at AF/N () is expected to be given approximately by since Slonczewski’s spin torque arises from absorption of the transverse components of electron spin by the localized moments via dephasing Stiles and Zangwill (2002) and effectively projects out the spin current polarization onto the Néel vector. In contrast, the longitudinal spin is conserved in the absence of spin-orbit interaction. In short, an antiferromagnetic metal should act as an ideal spin-valve according to the two-ferromagnetic picture. We compute the charge and spin currents at both F/AF and AF/N interfaces and explicitly confirm the dependence of the charge current and the spin-valve-like behaviour of the spin current. Again it is essential to include the two leads since the dependent charge current is second order in F/AF tunneling while the spin transmission requires spin currents at the two interfaces.

The rest of the paper is organized as follows. In Sec. II, after introducing the model Hamiltonian, we give an overview of the physical properties of F/AF/N junction. We explain meanings of all the parameters appearing in the final results of torques and currents. Sections III and IV contain the main results for the spin torques and the charge and spin currents in the stationary state. We conclude with discussions of our results and their connection to the previous studies in view of experiments and applications in Sec. V. In Appendix A and B, we describe our theoretical approach and show details of the calculations. In order to facilitate the comparison with previous studies, Appendix C develops a scattering theory approach and Appendix D discusses F/F/N and AF/AF/N junctions within our framework.

## Ii The model

We consider a system depicted in Fig. 1 where an antiferromagnetic metal (AF) is connected to a ferromagnetic left lead (F) and a normal metallic right lead (N) by respective tunneling barriers. The model Hamiltonian consists of five distinct parts;

(3) |

The first two terms describe conduction electrons in F and N. Only the electrons in F have spin-dependent energy eigenvalues. Labels and are used exclusively for the orbital degrees of freedom of ferromagnetic and normal metallic electrons. represents AF and is given by

(4) | |||||

Here, and annihilate the energy eigenstates with the eigenvalues residing in the and sublattice respectively, in the absence of intersublattice overlap of the atomic orbitals and also of the exchange interaction with the localized spins . The overlap amplitudes are assumed to be diagonal in this basis mainly for the ease of implementing the sublattice symmetry ( denotes the complex conjugate of ). We note that this form is generic in the case where denotes the crystalline momentum. Yamane et al. (2016) Specifically for a checker board structure, and correspond to the Fourier transforms of the nearest-neighbor and next-nearest-neighbor hopping amplitudes respectively. is the exchange split of the antiferromagnetic electrons. are the Pauli matrices in spin space and we set . The Hilbert space for a given is four dimensional; two for the spin and two for the sublattice. Two by two matrices in the spin space and the sublattice space are distinguished by subscripts and except for the Pauli matrices for which we use in the sublattice space. and represent spin-conserving tunneling processes to and from F and N respectively;

(5) | |||||

(6) |

The tunneling matrices are vectors in the sublattice space, defined by

(7) | |||||

(8) |

The spin rotation matrix reads

(9) |

encoding the difference in the reference frame between the ferromagnet and the antiferromagnet, as indicated in Fig. 2.

They have been chosen such that in the ferromagnetic frame.

We remark on the dynamics of . The Hamiltonian should be augmented by terms which do not involve the electrons, e.g. the antiferromagnetic exchange and crystalline anisotropy. However, the details of these terms will not affect our computation as long as the resulting magnetization dynamics is sufficiently slow. This condition will be quantified in the followings. We also assume throughout that any deviation from the collinearity relation is due to time-dependent dynamical part of . This allows us to specify a common spin quantization axis for the two sublattices. In reality, there can be some effective fields (torques to be derived) generated through the tunneling to F that will induce a nonzero total spin in equilibrium or stationary state. It implies that there will be an additional requirement that the antiferromagnetic exchange be much stronger than the spin torques, which is expected to be safely satisfied in most circumstances of interest.

### ii.1 Band structure

Figure 3 shows how the electron wave functions and the energy spectrum change as we turn on the intersublattice overlap and tunneling to the leads . In the two-ferromagnet limit , the strong exchange interaction splits the energy of up and down spins within each sublattice. Since in the equilibrium, the spins of upper and lower energy states are swapped between and sublattices and the bands are doubly degenerate in spin. Thus we call them top and bottom bands, even though the energy gap is essentially the exchange spin splitting. The introduction of does not qualitatively change the band structure as it preserves the sublattice symmetry and conservation of the component of spin. As we shall see, however, the nonzero overlap between the and wave functions opens up new channels of transport and alters some observables qualitatively. The energy gap is taken to be the largest relevant energy scale of the problem. When the coupling to F is taken into account, the degeneracy between the sublattices is lifted and spin ceases to be a good quantum number in general (Fig. 3). The energy split originating from the tunneling is denoted by , where

(10) | |||||

(11) |

The corresponding split bands are labelled by . This splitting plays an important role in interpreting component of spin torque. Although the band structures of F and N are also modified by the influence of AF, these modifications are neglected by the assumption that the leads are much greater in spatial dimension and electron density of states than AF.

### ii.2 Relaxation rates

In considering transport of electrons, the lifetime of the electronic eigenstates, whose inverse we call relaxation rate, plays a crucial role alongside with the band structure. In our model, this occurs for the electrons in AF only through a tunneling into either of the two leads as we have not included other sources of scatterings such as disorder, electron-electron collisions, or phonons. First of all, the relaxation rate associated with the tunneling into N is independent of spin and sublattice by assumption; we denote it by . The relaxation into F, in contrast, depends on both spin and sublattice, and is also a function of . As derived in Sec. A, the bands and have the same ferromagnetic relaxation rate while the other two and decay at a different rate . The origin of the difference between is the spin-dependent tunneling into F. We define the isotropic and anisotropic parts of the ferromagnetic tunneling rates, and respectively, as

(12) |

The quantity which also governs the angle-dependence of the energy split of the bands has been introduced in Eq. (11). The relaxation rates defined in Eq. (12) are independent of . Microscopic expressions for in terms of the tunneling matrix elements are given by

(13) | |||||

(14) | |||||

(15) |

They can be estimated if one assumes that . This leads to

(16) | |||

(17) |

where are the respective densities of states at the Fermi energies of F and N. In contrast to and , is determined by states with a wide range of energies in F. Therefore, the dependence of on characteristic energy scales is more difficult to estimate. As an example, for a ferromagnet with quadratic dispersion one finds that the dimensionless product scales with , where is the bandwidth. Therefore, one may expect the inequality to hold.

We can now be more precise about the approximation we have made, namely the slowness of the dynamics of . Its characteristic frequency must satisfy , so that the electron dynamics, characterized by the typical dwell time , is much faster than the magnetization dynamics.

### ii.3 Nonequilibrium stationary state

In the remainder of the paper, we consider dynamics of AF driven by fixed electron distribution functions for F and N respectively. The externally applied bias voltage or temperature difference induces charge and spin currents in AF. Consequently a nonequilibrium spin accumulation develops in AF, which generates spin transfer torques via the exchange interaction. The nonequilibrium state is characterized by the lesser Green’s function where are any of and the expectation value is taken over the nonequilibrium probability distribution of the quantum mechanical states. We focus on a stationary state in which all the macroscopic observables such as currents and torques are independent of time. The stationary state is fully determined by and the instantaneous magnitude and orientation of the localized spins in AF. The calculation of is relegated to Appendix A. Once is known, the torques and currents are readily computed as explained in the next sections.

It is helpful to compare our setup with the weak tunneling regime discussed in Sec. I. The latter concerns a situation where AF is in a prescribed equilibrium state and a weak contact with either F or N is introduced adiabatically. As long as AF stays close to the equilibrium state , the tunneling charge currents that flow across the interfaces AF/F () and AF/N () are given by

(18) | |||||

(19) |

The factor takes account of the spin degrees of freedom. The summation over corresponds to the top and bottom bands. Similarly, one can compute the tunneling spin current leaving F ( as

(20) |

Note that here the spin polarization is along the ferromagnetic axis and the current does not depend on . In the normalization used in this manuscript, the tunneling spin current equals twice the Slonczewski’s spin transfer torque in the weak tunneling regime,Chudnovskiy et al. (2008) setting a reference time scale for the magnetization dynamics. Under a bias voltage , one can estimate with the antiferromagnetic density of states . Thus the aforementioned slowness condition is self-consistent as long as . We also remark that the spin current measured in AF is given by , where the polarization is along the Néel vector . The spin current at AF/N is zero in this weak tunneling limit. Therefore, one can interpret and roughly as charge and spin currents per antiferromagnetic electronic state. Note that the weak tunneling regime cannot be a stationary state unless a strong relaxation mechanism maintains AF in the state . We use a full nonequilibrium electron distribution of AF self-consistently determined by the tunneling processes to compute the currents. The results reduce to (18) and (20) for where N acts as the relaxation source against the currents from F. Yet the currents and torques in the strong tunneling regime are also expressed in terms of those same relaxation rates as we shall see.

Our formalism is formally valid for arbitrary values of and as long as the interfaces are in the tunneling regime;

(21) |

Here are the number of conduction channels for F and N and assumed to be large integers. These conditions can be interpreted that the conductivity of each channel must be small, i.e. they are nonmetallic tunneling contacts. Nevertheless, the total current can be large as there can be many channels. Although the band gap can be arbitrary, we focus on the regime for which physics can be discussed in the language of the antiferromagnetic band structure. The general results can be found in the Appendix.

Finally, it is worth repeating that our model system does not include spin-orbit interactions. The inclusion of electron-electron-interactions or interactions with phonons is also beyond the scope of this work. Relaxation is therefore modeled entirely through the coupling to the leads.

## Iii Four types of spin torque

From the lesser Green’s function Eq. (100), one can readily compute the spin torques. The Heisenberg equations of motion for the averaged spin for are given by

(22) | |||||

(23) |

In the antiferromagnetic frame, are pointing in the and directions respectively so that have only and components. The slowness of the magnetization dynamics implies one can replace the electron operators by their expectation values. Rearranging (22) and (23) and discarding terms proportional to on the right-hand-sides yield (1) and (2) with the coefficients identified to be

(24) | |||||

(25) | |||||

(26) | |||||

(27) |

We have noted

(28) | |||||

(29) |

One can observe that torques appearing in the equation for are expectation values of the total electron spin while those driving come from the staggered spin () expectation values.

Based on this observation, we call staggered torques and non-staggered torques. When the torques are expressed as effective magnetic fields, the field direction coincides with the direction of the electron spin accumulation. Confusingly, if an effective field is staggered, the direction of the corresponding torque is non-staggered since the torque is a product of the field and the local magnetization, which is itself staggered. Our designation of staggered and non-staggered refers to the effective fields, not the torques. The situation is summarized in Fig. 4. Note that in the antiferromagnetic frame, has and components only (Fig. 2). Therefore, the field-like torques are related to the injected transverse spin (i.e. spin perpendicular to ) and the damping-like torques require the spin expectation value that is orthogonal to the polarization of the injected spin current. From these considerations alone, one can anticipate that the field-like torques can arise from mechanisms that conserve the transverse spin inside the antiferromagnet while the non-conservation of transverse spin is essential in generating the damping-like torques. Carrying out the traces in Eqs. (24) - (27) is a straightforward matter as presented in Appendix B. Below we discuss each of the four components in detail.

### iii.1 Slonczewski’s spin transfer torque

We start from the contribution which is most familiar in the ferromagnetic dynamics, which turns out to be in our notation;

(30) | |||||

where the spectral functions are given by

(31) |

In our terminology, it can also be called staggered anti-damping-like torque. Note that at the leading order in , the distinction between the bands has disappeared from the final expression. To identify it with the spin transfer torque, we take the limit and obtain ()

(32) | |||||

The right-hand-side is precisely the spin current per sublattice in the leading order tunneling approximation (20) with assumption .Chudnovskiy et al. (2008) Even though physical interpretation of this formula has been well discussed in many places, Stiles and Zangwill (2002); Slonczewski (2002) we repeat the argument here in the context of the two-ferromagnetic description of antiferromagnet. Ignoring the intersublattice overlap , an electron in F can only tunnel into a superposition of up and down spin states in one sublattice, which have different de Broglie wavelengths. Accordingly they dephase as they propagate and induce precession of the transverse spin component. The precession frequency differs for different orbital indices . Upon averaging over , the transverse component of the injected spin current is rapidly lost and absorbed into the magnetizations as required by the overall spin conservation, resulting in the torque.

Note that the torque appears as the expectation value of the staggered spin operator . It is due to the opposite handedness of the dephasing-induced precession in the two sublattices as depicted in Fig. 5. Our generalized expression (30) shows that the spin transfer torque in antiferromagnets is as effective as in ferromagnets even when and multiple tunneling processes are taken into account.

### iii.2 Non-staggered field-like torque

As stated above, the transverse component of spin is rapidly lost upon entering the antiferromagnet according to the two-ferromagnet description. Next we discuss the fate of transverse spin conservation in the presence of by looking at , which is essentially the expectation value of the component of the spin and given by

(33) | |||||

Note that the second term in the second line is one order higher in compared to the first term. It has been kept, nevertheless, since the leading order term is proportional to , which implies that this contribution would have been absent in the two-ferromagnet model and is unique to antiferromagnets.

In AF with non-vanishing , all the four bands have a nonzero amplitude at both and sublattice sites as shown in Fig. 3(b). Thus a ferromagnetic electron may tunnel into a superposition of up and down states of exactly the same energy and wavelength, say and . Alternatively, if one treats as a perturbation, an electron tunnels into a superposition of, e.g. and , then via , the state hops onto that has exactly the same wavelength as due to the sublattice symmetry. Either way, after the tunneling, the two electron states, representing a single electron, propagate with exactly the same phase evolution, dephasing is thus avoided, and the transverse spin is conserved (Fig. 6). We reiterate that this is a consequence of the complete sublattice symmetry assumed in our model. Consequently, there will be a nonvanishing expectation value of the component of spin proportional to the fraction of electrons undergoing the intersublattice hopping , which is represented by the first term in (33). This also explains the factor in Eq. (30), which coincides with the fraction of electrons propagating with different wavelengths and affected by the dephasing.

We note that a related mechanism was discussed in the context of antisymmetric F/N/F spin valves.Brataas et al. (2003)

The physics behind the second term should then be related to intrasublattice processes as it also comes with the factor of . It represents the residual component of spin that has managed to survive the dephasing. One way to interpret this term is to consider the tilt of electron quantization axes in AF due to the influence of F. It should not be confused with the tilt of as they are assumed fixed in the electron time scale. The part of self-energy proportional to can be considered as an additional Zeeman term in the direction of the ferromagnetic moment . In the leading order approximation in , taking it into account yields the direction of the effective magnetic field (preferred quantization axis) for the antiferromagentic electrons (Fig. 6). While the spin transverse to the quantization axis is lost by dephasing, the longitudinal component is conserved by definition. Thus the fraction of the injected spin component will be conserved and contribute to the field-like torque.

It is helpful to write down in the weak ferromagnetic tunneling limit , yielding

(34) | |||||

As one can see, the intersublattice contribution (the first term) has the relaxation time in the numerator. When this first term dominates , its relative magnitude compared to the spin transfer torque is given by . Therefore could in principle greatly exceed depending on how and compare with . This is one of the reasons for specifying the origin of the relaxation in the present work. The intrasublattice contribution (the second term) is always suppressed by a factor of with respect to . This contribution exists for ferromagnets as well, though mostly neglected because it is always subdominant.

### iii.3 Néel field-like torque

As we have explained, the component of the spin expectation value is related to the conservation of the injected spin current. In the picture given above that is based on first order tunneling processes, the generated expectation value is the same for the two sublattices, which therefore leads to a non-vanishing . If the higher order effects of the tunneling from F are considered, however, they break the symmetry between the sublattices and result in different expectation values of and . This breaking of the sublattice symmetry manifests itself through two different parameters. The first is the energy split , whose first order effect gives an equal magnitude of and as we have just seen, but at the second order leads to an asymmetric change of quantization axis for and sublattices, and in turn to different values of . The other parameter is , which represents the different exit rates to F for the antiferromagnetic electrons in and sublattices. This is also expected to give an asymmetric correction to the conserved , given that have a finite expectation value, since if electrons in one sublattice escape faster than those in the other, it should reduce the spin expectation value for the former compared to the latter.

In our notation, the asymmetric part of the conserved component corresponds to and is given in terms of the expectation value of , which reads

(35) | |||||

The similarity of this formula to the intrasublattice contribution of is apparent; the differences are the factor , the different power of , and the sign in front of . The latter two features are related to the fact that is an order higher also in than . Hence it is reasonable to interpret this correction as arising from the escape rate difference. Since the same estimate as for the intrasublattice term of applies, is expected to be subdominant in the entire parameter space and we do not go deeper in its physical interpretation.

### iii.4 Unconventional anti-damping-like torque

Similarly to the field-like torques, the leading order effect of the ferromagnetic tunneling gives a sublattice symmetric contribution to and , albeit the expectation values are staggered . Then the symmetry breaking effects discussed in the previous subsection should generate a non-staggered component , which corresponds to . To the leading order in , one derives

(36) | |||||

whose close connection to (30) is clear. This contribution can be interpreted as caused by the different electron escape rates between and sublattices limiting the expectation value of the component of the spin. The factor is also reasonable as there is no symmetry breaking when the ferromagnetic moment is perpendicular to the Néel vector.

### iii.5 Comparison of the torques

The expressions for the four torques given in Eqs. (30), (33), (35), (36) can be evaluated explicitly once the parameters of the model Hamiltonian such as the spectra of the leads and the antiferromagnetic island and the tunneling amplitudes are specified. Here, we aim to make general statements about the relative importance of the different kinds of spin torque.

A brief look at Eqs. (30), (33), (35), (36) reveals that the summation in involves terms that are even in for , Eq. (36) and odd in for the remaining three torques. Since can take both positive and negative values, cancellations may in principle occur when evaluating , and . This fact makes general statements about their magnitude difficult. For the further discussion, we therefore assume that does not change sign within the relevant interval of energies.

First, we will be concerned with the comparison of the two damping-like torques. For this case, one arrives at the inequality , which is a direct consequence of the following hierarchy of relaxation rates, . Let us discuss the ratio in two limiting cases. In the limit of weak ferromagnetic tunneling, one has . This implies that when the first order tunneling approximation is justified, is likely negligible compared to the conventional spin transfer torque . In the opposite regime, i.e. , the ratio can be estimated as . The estimate (16) for the relaxation rates suggests that in this case , where is the exchange splitting in the ferromagnetic lead.

Next, we will be concerned with the comparison of the torques entering Eq. (1) for , and . The relevant control parameter is . This estimate suggests that the Néel field-like torque in the absence of spin-orbit coupling is insignificant in almost all circumstances.

We will now turn to the two torques entering Eq. (2) for , and . Here, the control parameter is (assuming that the first term dominates in expression for , Eq. (34)). This ratio is proportional to and therefore enhanced for weak coupling to F. Further, the ratio is strongly affected by the value of , which is difficult to estimate from microscopic considerations. Since is likely negligible when , the detection of a significant might be used as an experimental probe into the extent of microscopic intersublattice wave function overlap, although it could be practically difficult to eliminate the possible field-like contributions from spin-orbit interactions.

For the comparison of and , one can probably assume as .

In summary, the discussion presented above suggests the hierarchy . As for the non-staggered field-like torque, a sizeable requires large intersublattice overlap amplitudes . Indeed, its relative magnitude compared to , for which one estimates and compared to , for which