Enhanced Spin Conductance of a Thin-Film Insulating Antiferromagnet

# Enhanced Spin Conductance of a Thin-Film Insulating Antiferromagnet

## Abstract

We investigate spin transport by thermally excited spin waves in an antiferromagnetic insulator. Starting from a stochastic Landau-Lifshitz-Gilbert phenomenology, we obtain the out-of-equilibrium spin-wave properties. In linear response to spin biasing and a temperature gradient, we compute the spin transport through a normal metalantiferromagnetnormal metal heterostructure. We show that the spin conductance diverges as one approaches the spin-flop transition; this enhancement of the conductance should be readily observable by sweeping the magnetic field across the spin-flop transition. The results from such experiments may, on the one hand, enhance our understanding of spin transport near a phase transition, and on the other be useful for applications that require a large degree of tunability of spin currents. In contrast, the spin Seebeck coefficient does not diverge at the spin-flop transition. Furthermore, the spin Seebeck coefficient is finite even at zero magnetic field, provided that the normal metal contacts break the symmetry between the antiferromagnetic sublattices.

###### pacs:
72.25.Mk, 72.20.Pa, 05.40.-a, 75.76+j

Introduction. Antiferromagnets have recently garnered increasing interest in the spintronics community, both for their novel intrinsic properties and their technological potential. Their appealing features are their lack of stray magnetic fields, fast dynamics (relative to ferromagnets), and robustness against external fields Baltz et al. (2016). This last property, however, is a double-edged sword, as the lack of response to an external field makes control of antiferromagnets challenging. Recent theoretical and experimental work has instead sought to generate and/or detect antiferromagnetic dynamics optically Gómez-Abal et al. (2004); ?; ? and electrically Wadley et al. (2016); ?; ?; ?; ?; ?.

In ferromagnets, equilibrium thermal fluctuations generate spin waves that can drive coherent magnetic dynamics Yan et al. (2011); ? or transport spin Goennenwein et al. (2015); ?; ?. It is of interest to see if, and how, thermal magnons can provide long-range spin-transport in ferromagnetic insulators. However, in antiferromagnets, at zero magnetic fields, spin-wave excitations are doubly degenerate. The two branches carry opposite spin polarity. Thus, to realize spin transport by thermally generated spin waves, the symmetry between the antiferromagnetic sublattices must be lifted. One means of achieving this is to employ a ferromagnetic layer, controlled by a magnetic field Lin et al. (2016). Alternatively, the magnetic field itself suffices to break the sublattice symmetry, eliminating the need for a ferromagnetic component. The spin Seebeck effect Ohnuma et al. (2013); Rezende et al. (2016), in which angular momentum is driven by a temperature gradient, was measured Wu et al. (2016); ? in bipartite electrically insulating antiferromagnets at finite magnetic fields.

We consider two different scenarios for electrical and thermal experimental generation of spin transport in antiferromagnets without the need for a ferromagnetic component. The realization of these scenarios can open the doors towards long-range transport of magnons in antiferromagnets that can be strongly enhanced by appropriate tuning of the magnetic field. Such achievements could further cultivate antiferromagnets in a more vital role within spintronics.

The first option is to inject thermal magnons by a spin accumulation in an adjacent metal. While spin accumulation-induced thermal magnon injection in ferromagnetnormal metal heterostructures has been the subject of recent research Cornelissen et al. (2016); Zhang and Zhang (2012), theoretical (and experimental) studies of the ferromagnetic analogue are currently lacking and are restricted to coherent magnetic dynamics of the antiferromagnetic order and the resulting spin superfluidity Takei et al. (2014); Qaiumzadeh et al. (2016). Here, we show that the spin conductivity of thermal magnons is strongly enhanced upon approaching the spin-flop transition, and in the ideal case diverges at the transition. This leads to a large amount of tunability of the magnon transport by an external field which may be desirable for applications.

A second possibility for engineering magnon spin transport in antiferromagnets is to break the interface sublattice symmetry. Magnetically uncompensated antiferromagnetmetal interfaces have been studied theoretically Cheng et al. (2014). Nevertheless, the possibility of realizing a spin Seebeck effect by breaking the sublattice symmetry at the interface has not been proposed until now.

In this Letter, we use a stochastic Landau-Lifshitz-Gilbert theory Hoffman et al. (2013); ? to study these phenomena. We treat spin transport through a normal metalinsulating antiferromagnetnormal metal heterostructure in the linear response to a spin accumulation and temperature gradient. The magnon conductance is found to be strongly enhanced (and in the ideal case diverges) near the onset of the spin-flop transition, while the spin Seebeck coefficient does not. Including both sublattice symmetry respecting and -breaking terms at the interface, we show that these two contributions may be distinguished by their sign under magnetic field reversal, and that the latter contribution leads to a nonzero Seebeck coefficient at zero magnetic field.

Stochastic Dynamics. We consider a bipartite antiferromagnet (AF). The system is translationally invariant in the plane. There is an interface along the plane on the left with a normal metal (LNM) and an interface along the plane with an identical normal metal (RNM) on the right (see Fig. 1a). Let us suppose that a spin accumulation is fixed by, e.g., spin Hall physics in the left lead, or that a linear phonon temperature profile is established across the structure 1.

We begin by parameterizing the AF spin degrees of freedom in the long wavelength limit by the Nel order unit vector and dimensionless magnetization . At zero temperature, the AF relaxes towards a ground state which is determined by the free energy Hals et al. (2011):

 (1)

Here, is the sum of the saturation spin densities of the and sublattices (in units of ), is the volume of the AF, is the susceptibility, is the Nel order exchange stiffness and is the uniaxial, easy-axis anisotropy. The external magnetic field is taken to be applied along the direction in order to preserve rotational symmetry around the axis in spin space. The bulk symmetry of the bipartite lattice under the interchange of the sublattices, which sends and , is manifest in the form of .

At sufficiently small magnetic fields, , the ground states are degenerate, given by and , and the AF is in the antiferromagnetic phase. In the antiferromagnetic phase, the ground state magnetic texture is insensitive to the spin accumulation in the linear response, and the AF does not support a spin current at zero temperature. At fields , the ground state is “spin-flopped”, with and in the plane. Spin biasing of the spin-flopped state generates a spin super current Takei et al. (2014); Qaiumzadeh et al. (2016) at zero temperature. In order to focus on transport by thermally activated spin waves, we restrict the following discussion to the antiferromagnetic phase. Furthermore, in this phase, the spin waves are circular and therefore simpler to analyze.

At finite temperatures, fluctuations drive the AF texture away from the zero temperature configuration, necessitating equations of motion that incorporate bulk and boundary fluctuations and dissipation. The small amplitude excitations of the Nel order above the ground state are described by the linearized equation of motion in the bulk ():

 (∂2x+q2)n=−fB/A. (2)

(see Supplemental Material). Here, is the Fourier transform (in the coordinates and ) of , while with . The stochastic force , modeling fluctuations of the AF lattice that drive , is connected to the bulk Gilbert damping by the fluctuation dissipation theorem (here in the large exchange regime, ):

 ⟨f∗B(x,q,ω)fB(x′,q,ω′)⟩= ×δ(x−x′)δ(q−q′)δ(ω−ω′)2α(2π)3ℏωtanh[ℏω/2T], (3)

where is the local temperature in units of energy.

Complementing Eq. (2) are boundary conditions on :

 A∂xn+iα′ωd(ℏω−μ)n=−fL(x=−d/2) −A∂xn+iα′ωdℏωn=−fR(x=d/2), (4)

where and correspond to fluctuations by lead electrons at the interfaces. The quantity , describing dissipation of magnetic dynamics at the interfaces (which we have taken to be identical for simplicity), has contributions from both sublattice-symmetry-respecting () and -breaking () microscopics there. For example, in a simple model in which fluctuation and dissipation torques for the two sublattices are treated independently (see Supplemental Material), one finds and ; here (with as the spin mixing conductance) is the effective damping due to spin pumping for sublattice Cheng et al. (2014); Takei et al. (2014)(see Fig. 1b). Such a model corresponds to the continuum limit of a synthetic antiferromagnet (composed of ferromagnetic macrospins separated by normal metals) in which sublattice symmetry breaking may be more carefully controlled (see Fig. 1c).

The effective surface forces and damping coefficient are connected via the fluctuation-dissipations theorems for the interfaces:

 ⟨f∗l(q,ω)fl′(q,ω′)⟩= ×δll′δ(q−q′)δ(ω−ω′)2sα′ω(2π)3(ℏω−μl)tanh[(ℏω−μl)/2Tl], (5)

where we have retained terms up to first order in . Here and are lead electronic temperatures, and, in our setup, and .

To summarize, the bulk equation of motion, Eq. (2), together with the boundary conditions, Eqs. (4), specifies the AF dynamics. The description takes into account the presence of the metallic leads for a particular realization of the noise. For our purposes, we require only the ensemble information about noise contained in the fluctuation-dissipation theorems Eqs. (3) and (5).

Spin Transport. We now obtain the spin current that flows across the right interface in linear response to the spin accumulation at the left interface. Rewriting the equation of motion for the magnetization (Eq. (16) in the Supplementary Material) as a continuity equation for the spin density , one obtains an expression for the spin current: . Solving Eqs. (2)-(5) in the absence of a temperature gradient, retaining terms only up to linear order in , the -spin current flowing through the right interface becomes:

 j≡⟨^z⋅j⟩=AsIm⟨n∗(r)∂xn(r)⟩x=d2=Gμ, (6)

where we have introduced the spin conductance .

In the low-damping/thin-film limit, , where is the imaginary correction to (i.e. with real) due to Gilbert damping, the spin current is carried by well defined spin-wave modes (corresponding to solutions to Eq. (2) in the absence of noise) with frequencies . Here is the transverse wavevector, is an integer denoting spin-wave confinement in the direction, and the labels corresponds to the two spin-wave branches for which rotates in opposite directions, as has the opposite sign of (though a different magnitude when ) In the low damping thin/film limit, the spin conductance is therefore a sum over contributions from each of these modes and can further be broken into “symmetric” and “antisymmetric” (under interchange of the sublattices) pieces:

 G=∑l=0,1,2…∫d2q(2π)2(G(S)lq+G(A)lq). (7)

Defining:

 F(i,j)±(ξ/T)≡(T/ξ)i(ℏω(±)/T)1+jsinh2(ℏω(±)/2T), (8)

with , the symmetric contribution, which is proportional to , is:

 G(S)lq=(ςlα′)22ςlα′+α(1χT)G(S)(ξlq/T), (9)

where and , with for and for (reflecting the exchange boundary conditions Kapelrud and Brataas (2013); Hoffman et al. (2013)). The antisymmetric piece, which is proportional to , reads:

 G(A)lq=ςl~α′2ςlα′(ςlα′+α)(2ςlα′+α)2G(A)(ξlq/T) (10)

where and we have assumed that . From Eqs. (9) and (10) we find an algebraic decay of the spin current with film thickness. In the extreme thin film limit, , the damping at the interface dominates over bulk, and both contributions decay as ; in the opposite regime, , one has that both again decay in the same way, as . The symmetric and antisymmetric contributions may instead be distinguished by reversing the direction of the applied field: changes sign under (and therefore vanishes at zero field), while remains the same. Note however that the antisymmetric contribution, Eq. (10), is suppressed by a factor of relative to the symmetric contribution, Eq. (9).

As a consequence of the divergence of the Bose-Einstein distribution at zero spin-wave gap (and as a precursor to superfluid transport Takei et al. (2014)), the spin conductance diverges as one approaches the spin-flop transition. The boundary for the antiferromagnetic phase is defined by the vanishing of the spin-wave gap for one of the modes (), which determines the critical field, . Then, from Eqs. (9) and (10), both the symmetric and antisymmetric contributions diverge as as (see Fig. 2).

We may compare these results with spin transport driven by a temperature gradient. Supposing a linear temperature gradient , with a continuous profile across the structure so that and , Eqs. (2)-(5) yield a spin current for :

 j=SΔT, (11)

where is the temperature change across the AF. In the low-damping/thin-film limit, the Seebeck coefficient similarly separates into symmetric and antisymmetric sums over discrete spin-wave modes:

 S=∑l=0,1,2…∫d2q(2π)2(S(S)lq+S(A)lq). (12)

where

 S(S)lq=(ςlα′/8)(1/χT)S(S)(ξlq/T), (13)

is the symmetric contribution, with , and

 S(A)lq=ςl(~α′/4)S(a)(ξlq/T), (14)

the antisymmetric contribution, with . In contrast to the spin conductance, there is no divergence in the spin Seebeck coefficient as . Furthermore, the antisymmetric contribution is under (and is generally nonzero at zero field), while the symmetric contribution is (vanishing at zero field, as is required by sublattice symmetry); in contrast to spin biasing, a temperature gradient requires either a field or sublattice symmetry breaking at the interfaces in order to generate a spin current, else the two branches carry equal and opposite spin currents.

In the high temperature limit (), one has that both contributions to the spin conductance increase with temperature, as more thermally occupied modes are available for transport: and . In contrast, the antisymmetric Seebeck coefficient saturates at a constant value, while the symmetric contribution vanishes as . The Seebeck coefficient in Eq. (12), however, ultimately increases with temperature at high temperatures (see Fig. 3), in disagreement with the prediction of Rezende et al. (2016) and measurements of Wu et al. (2016); ?, both of which show a Seebeck signal vanishing at high temperatures. This is a consequence of the long wavelength nature of our continuum Landau-Lifshitz-Gilbert treatment, which can be remedied by, e.g., artificially introducing momentum cutoffs or employing a lattice model. Note that, like the conductance, the antisymmetric contribution to the spin Seebeck coefficient, Eq. (14), is suppressed by a factor of relative to the symmetric, Eq. (13); this suppression, however, does not affect the divergence of the symmetric and antisymmetric spin conductance contributions, which occurs at low frequencies.

Both symmetric and antisymmetric contributions to decay as ; writing , one finds that is constant, reflecting that the Seebeck effect here is driven by bulk fluctuations.

Discussion. The thin-film approximation, Eqs. (7) to (14), in which the structural transport coefficients consist of contributions from well-defined spin-wave modes, are valid for thicknesses . The parameter , which describes the decay of magnons across the thickness of the film, can be estimated from a Heisenberg model on a lattice as (supposing ), where is the Nel temperature and the lattice spacing. For , a low damping factor and a lattice spacing nm, corresponds to a thickness of 50 nm, which grows larger at lower temperatures.

The stochastic Landau-Lifshitz-Gilbert phenomenology we employed, Eqs. (2)-(5), may of course be extended to thicker films, resulting in, for example, an exponential decay over the lengthscale (with as the magnon wavevector) rather than an algebraic decay of the spin conductance with distance. Thicker films, however, introduce additional complications, e.g., elastic disorder scattering and phonon-magnon coupling (e.g. phonon drag). Spin wave interactions (scattering and mean field effects), which are absent in the single particle treatment above, may drastically change transport at higher spin wave densities, e.g. at higher temperatures/thicker films or near the spin-flop transition, where the Bose-Einstein divergence may necessitate a many-body treatment thereby altering the conductance . The scattering times and length scales over which such effects become important remains an open question. In addition, in our model the transition from antiferromagnetic to spin-flop phase is second order; the presence of Dzyaloshinskii-Moriya interaction, spin wave interactions, or in-plane anisotropy can change critical exponents such as in from its value obtained above or even alter order of the phase transition thereby diminishing the divergence of Hohenberg and Halperin (1977).

This work had received funding from the Stichting voor Fundamenteel Onderzoek der Materie (FOM) and the European Research Council via Advanced Grant number 669442 “Insulatronics”.

## Appendix A Derivation of Spin Wave Equations of Motion

In this section, we obtain Eqs. (2)-(5) from the free energy, Eq. (1). We begin with the nonlinear coupled equations for the Nel order and magnetization:

 ℏ˙n=Fm×n+τn, (15)
 ℏ˙m=Fm×m+Fn×n+τm. (16)

Here, and , with representing a functional derivative Hals et al. (2011). The terms and capture thermal fluctuations and dissipation in the bulk and at the interfaces. It is possible to find phenomenological expressions for and by listing out all terms with the appropriate symmetries. We take an alternative approach and construct and from the corresponding torques that would arise on two separate ferromagnetic sublattices Gomonaĭ and Loktev (2008) (see Fig. 1b). Momentarily neglecting the spin accumulation inside the left normal metal, the fluctuating and dissipative torques are:

 ℏ˙mζ∣∣fd=τζ=[fζ−αζℏ˙mζ]×mζ, (17)

where is a unit vector in the direction of the magnetization of the sublattice. The parameter depends on the coordinate :

 αζ(x)=α+α′ζdδ(x+d/2)+α′ζdδ(x−d/2), (18)

where is the bulk damping, while , with as the spin mixing conductance, is the effective damping due to spin pumping Cheng et al. (2014); Takei et al. (2014), which may differ for the two sublattices (see Fig. 1c). (For simplicity, we assume that is the same at the left and right interfaces.) Meanwhile, the fluctuating forces have contributions from both the interfaces and the bulk:

 fζ(r)=fLζ(ρ)δ(x+d/2)+fRζ(ρ)δ(x−d/2)+fBζ(r), (19)

where . The bulk and interface Langevin sources are subject to the fluctuation-dissipation relations:

 ⟨f(i)Bζ(r,t)f(i′)Bζ′(r′,t′)⟩=αδζζ′δii′δ(r−r′)R(x,t−t′) (20) ⟨f(i)lζ(ρ,t)f(i′)l′ζ′(ρ′,t′)⟩=αζδll′δζζ′δii′δ(ρ−ρ′)Rl(t−t′), (21)

with . The bulk and interface noise functions and respectively depend on the bulk and left (right) interface temperatures and . In the white noise limit, these are proportional to ; we will consider colored noise, and because we will require only the Fourier transforms of the these quantities (see Eqs. (3) and (5)), we do not specify the time-dependence of the colored noise functions here.

Then,

 τm=12(τa+τb)=(fm−αSℏ˙m−αAℏ˙n)×m+(fn−αAℏ˙m−αSℏ˙n)×n, (22)

while

 τn=^P2(τa−τb)=(fm−αSℏ˙m)×n−αA(ℏ˙n×n+^Pℏ˙m×m). (23)

Here, projects out components colinear with , ensuring that , while , , , and .

Finally, to include a spin accumulation along the left interface, we replace in the terms with in Eq. (17); correspondingly, and in the terms with in Eqs. (22) and (23). Inserting the expressions for and of Eqs. (22) and (23) into Eqs. (15) and (16), we obtain the full nonlinear equations for the noisy antiferromagnetic dynamics. Note that the terms proportional to in Eqs. (22) and (23) break the symmetry of Eqs. (15) and (16) under , , reflecting the broken sublattice symmetry at the interfaces.

Next, we expand Eqs. (15) and (16) around the ground state, which we have chosen, without loss of generality, as . Writing in order to preserve rotational symmetry of the spin around the axis, we define for , , and the Fourier transform in a circular basis:

 a(x,q,ω)=∫d2ρ(2π)2dt2πeiωt−iq⋅ρ[ax(r,t)+iay(r,t)], (24)

so the linearized equations of motion become:

 ℏωn=−χ−1m−Hn+fm+iℏω(αAn+αSm) (25)
 ℏωm=(A∂2x−Aq2−K)n−Hm+fn+iℏω(αSn+αAm). (26)

Finally, we solve Eq. (25) for , and insert the result into Eq. (26) to obtain a differential equation for . Neglecting terms , , etc., and supposing the strong exchange limit, 2, one obtains (again omitting dependence on , which can be easily restored as above, for brevity):

 0=(A∂2x−K)n−χ(ℏω+H)i2~α′d∑l=L,R(ℏω−μl)δ(x−xl)ℏω+χ(ℏω+H)2n +iαℏωn+fn−χ(ℏω+H)fm+iα′ℏd∑l=L,R(ℏω−μl)δ(x−xl), (27)

with and . In the bulk (), this yields Eq. (2), with and . Neglecting terms and using Eq. (20), one obtains, after Fourier transforming, Eq. (3). Integrating Eq. (27) over the interfaces and restoring dependence on the spin accumulations at the boundaries, one obtains the boundary conditions, Eqs. (4), where for and , with and . From Eq. (21), one obtains the surface fluctuation-dissipation theorems Eq. (5) in the large exchange limit, .

### Footnotes

1. A complete treatment of transport generally requires treating the coupled magnetic, phononic and electronic degrees of freedom of the heterostructure on equal footing. However, if the metallic leads are good spin sinks (so that any spin accumulation driven by magnetic dynamics is quickly relaxed, and is determined exclusively by spin Hall driving) and assuming a large phononic heat conductance throughout the structure, it is reasonable to neglect the feedback on the lead-electrons and phonons from the AF spin wave degrees of freedom in our simple model.
2. Strictly speaking, this is not possible at the interfaces where is nonzero. However, in order parameterize the AF by smoothly varying fields and , it is necessary that is large, so we may consider as a smoothly varying function which is sharply peaked at with ; then Eq. (27) follows.

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