Diego García Ovalle
email: diego-fernando.garcia-ovalle@univ-amu.fr

Armando Pezo

Aurélien Manchon
email: aurelien.manchon@univ-amu.fr

Spin-orbit torque for field-free switching in C 3v crystals

Spin-orbit torques in noncentrosymmetric polycrystalline magnetic heterostructures are usually described in terms of field-like and damping-like torques. However, materials with a lower symmetry point group can exhibit torques whose behavior substantially deviates from the conventional ones. In particular, based on symmetry arguments it was recently proposed that systems belonging to the C3v point group display spin-orbit torques that can promote field-free switching [Liu et al. Nature Nanotechnology 16, 277 (2021)]. In the present work, we analyze the general form of the torques expected in C3v crystals using the Invariant Theory. We uncover several new components that arise from the coexistence of the three-fold rotation and mirror symmetries. Using both tight binding model and first principles simulations, we show that these unconventional torque components arise from the onset of trigonal warping of the Fermi surface and can be as large as the damping-like torque. In other words, the Fermi surface warping is a key indicator to the onset of field-free switching in low symmetry crystals.

I. INTRODUCTION

Electrical manipulation of the magnetization in single magnetic thin films using spin-orbit torques has become routinely available in the past decade [START_REF] Manchon | Current-induced spin-orbit torques in ferromagnetic and antiferromagnetic systems[END_REF]. In perpendicularly magnetized systems, the most suitable configuration for memory applications, achieving reversible current-driven switching necessitates the combination of spin-orbit torque with an external magnetic field [START_REF] Miron | Current-driven spin torque induced by the Rashba effect in a ferromagnetic metal layer[END_REF][START_REF] Liu | Spin-Torque Ferromagnetic Resonance Induced by the Spin Hall Effect[END_REF]. As a matter of fact, whereas the spin-orbit torque tends to bring the magnetization in the plane, applying an additional external field along the current direction provides the necessary force that completes the reversal process in a deterministic manner. The need for this external field is considered as a hurdle for memory applications and several strategies have been proposed to circumvent this difficulty. Field-free current-driven switching has been realized using exchange bias from a neighboring antiferromagnet [START_REF] Oh | Field-free switching of perpendicular magnetization through spin-orbit torque in antiferromagnet/ferromagnet/oxide structures[END_REF][START_REF] Fukami | Magnetization switching by spin-orbit torque in an antiferromagnet/ferromagnet bilayer system[END_REF], exchange coupling [START_REF] Lau | Spin-orbit torque switching without an external field using interlayer exchange coupling[END_REF][START_REF] Wei | Field-Free Spin-Orbit Torque Switching in Perpendicularly Magnetized Synthetic Antiferro-magnets[END_REF] or anomalous Hall torque from a proximate ferromagnet [START_REF] Baek | Spin currents and spin-orbit torques in ferromagnetic trilayers[END_REF][START_REF] Ryu | Efficient spin-orbit torque in magnetic trilayers using all three polarizations of a spin current[END_REF]. The latter takes advantage of an interfacial spin rotation of the incoming spin current [START_REF] Amin | Interface-Generated Spin Currents[END_REF], sometimes called spin swapping [START_REF] Lifshits | Swapping Spin Currents: Interchanging Spin and Flow Directions[END_REF][START_REF] Saidaoui | Spin-Swapping Transport and Torques in Ultrathin Magnetic Bilayers[END_REF] (see also Refs. 13 and 14). In addition, structural engineering has been successfully exploited to design lateral [START_REF] Yu | Switching of perpendicular magnetization by spin-orbit torques in the absence of external magnetic fields[END_REF][START_REF] Bose | Observation of Anomalous Spin Torque Generated by a Ferromagnet[END_REF][START_REF] Cui | Field-Free Spin-Orbit Torque Switching of Perpendicular Magnetization by the Rashba Interface[END_REF][START_REF] Razavi | Deterministic spin-orbit torque switching by a light-metal insertion[END_REF][START_REF] Kang | Electric-field control of field-free spin-orbit torque switching via laterally modulated Rashba effect in Pt/Co/AlOx structures[END_REF] and geometrical [START_REF] Safeer | Spin-orbit torque magnetization switching controlled by geometry[END_REF] symmetry breaking, tilted anisotropy [START_REF] You | Switching of perpendicularly polarized nanomagnets with spin orbit torque without an external magnetic field by engineering a tilted anisotropy[END_REF][START_REF] Liu | Current-induced magnetization switching in all-oxide heterostructures[END_REF][START_REF] Li | Field-Free Deterministic Magnetization Switching with Ultralow Current Density in Epitaxial Au/Fe 4 N Bilayer Films[END_REF] and longitudinal (compositional or structural) gradient [START_REF] Shu | Field-Free Switching of Perpendicular Magnetization Induced by Longitudinal Spin-Orbit-Torque Gradient[END_REF][START_REF] Chen | Free Field Electric Switching of Perpendicularly Magnetized Thin Film by Spin Current Gradient[END_REF].

Whereas most of these works considered multilayers made out of polycrystalline materials, recent experiments demonstrated that low symmetry crystals are endowed with unconventional spin-orbit torques that can play the role of an external field, thereby completing the currentdriven switching process. The impact of the crystalline symmetries on the spin-orbit torque is well-known since its initial observation in the non-centrosymmetric magnetic semiconductors (Ga,Mn)As [START_REF] Chernyshov | Evidence for reversible control of magnetization in a ferromagnetic material by means of spin-orbit magnetic field[END_REF][START_REF] Endo | Current induced effective magnetic field and magnetization reversal in uniaxial anisotropy (Ga,Mn)As[END_REF] and in the Heusler alloy MnNiSb [START_REF] Ciccarelli | Room-temperature spin-orbit torque in NiMnSb[END_REF], where the bulk inversion symmetry breaking promotes a so-called Dresselhaus-like spin-orbit torque. In fact, further lowering of the crystalline symmetries can lead to unusual torques that turn out to be instrumental to achieve field-free switching. For instance, WTe 2 has been shown to display a "perpendicular damping-like torque" [START_REF] Macneill | Control of spin-orbit torques through crystal symmetry in WTe2/ferromagnet bilayers[END_REF][START_REF] Macneill | Thickness dependence of spin-orbit torques generated by WTe2[END_REF] that enables fieldfree switching, an effect confirmed in several experiments [START_REF] Shi | All-electric magnetization switching and Dzyaloshinskii-Moriya interaction in WTe2/ferromagnet heterostructures[END_REF][START_REF] Xie | Field-free magnetization switching induced by the unconventional spin-orbit torque from WTe2[END_REF][START_REF] Kao | Deterministic switching of a perpendicularly polarized magnet using unconventional spin -orbit torques in WTe2[END_REF]. This torque, also present in MoTe 2 [START_REF] Xue | Unconventional spin-orbit torque in transition metal dichalcogenide-ferromagnet bilayers from first-principles calculations[END_REF] and NbSe 2 [START_REF] Guimarães | Spin-Orbit Torques in NbSe2/Permalloy Bilayers[END_REF], is associated with a crystalline mirror symmetry breaking perpendicular to the interface plane. When a current is injected along this mirror, it may generate a nonequilibrium spin density contained in this mirror plane and normal to the interface. Antiferromagnets are also currently attracting attention from this standpoint. Indeed, the combination of crystalline and magnetic symmetries tend to produce spin currents with a polarization different from what is dictated by the conventional spin Hall effect [START_REF] Zelezný | Spin-Polarized Current in Noncollinear Antiferromagnets[END_REF][START_REF] Zhang | Spin Hall effect emerging from a noncollinear magnetic lattice without spin-orbit coupling[END_REF], an effect sometimes called "magnetic" spin Hall effect [START_REF] Kimata | Magnetic and magnetic inverse spin Hall effects in a non-collinear antiferromagnet[END_REF][START_REF] Ghosh | Unconventional Robust Spin-Transfer Torque in Noncollinear Antiferromagnetic Junctions[END_REF]. These spin currents can in turn exert "unconventional" torques on an adjacent ferromagnet, as observed in collinear (Mn 2 Au [START_REF] Chen | Observation of the antiferromagnetic spin Hall effect[END_REF], RuO 2 [START_REF] Bose | Tilted spin current generated by the collinear antiferromagnet ruthenium dioxide[END_REF][START_REF] Bai | Observation of Spin Splitting Torque in a Collinear Antiferromagnet RuO 2[END_REF]), and non-collinear antiferromagnets (Mn 3 GaN [START_REF] Nan | Controlling spin current polarization through non-collinear antiferromagnetism[END_REF], Mn 3 Pt [START_REF] Bai | Control of spin-orbit torques through magnetic symmetry in differently oriented noncollinear antiferromagnetic Mn3Pt[END_REF] and Mn 3 Sn [START_REF] Kondou | Giant field-like torque by the out-of-plane magnetic spin Hall effect in a topological antiferromagnet[END_REF]).

Recently, Liu et al. [START_REF] Liu | Symmetry-dependent field-free switching of perpendicular magnetization[END_REF] studied the current-driven magnetization reversal in a crystalline CuPt/CoPt bilayer in the L1 1 phase grown along the (111) direction. They reported that field-free switching could be achieved when the current was applied along low-symmetry crystallographic directions. Intriguingly, the polarity of the magnetization reversal loop displayed a periodic pattern depending on the crystallographic direction along which the current was applied. This unusual behavior was interpreted as arising from an unconventional torque, tagged "3m" torque, which appears in crystals with C 3v point group [START_REF] Železný | Spin-orbit torques in locally and globally non-centrosymmetric crystals: Antiferromagnets and ferromagnets[END_REF]. Nonetheless, no microscopic explanation was proposed to explain the emergence of the "3m" torque in this bilayer. Such an explanation is highly desired, especially with the acceleration of the research in twodimensional van der Waals magnets [START_REF] Gong | Two-dimensional magnetic crystals and emergent heterostructure devices[END_REF]. As a matter of fact, most of the van der Waals magnets possess a hexagonal or trigonal point group and are therefore entitled to display such a torque. For instance, the "3m" torque was identified in Fe 3 GeTe 2 monolayer [START_REF] Johansen | Current Control of Magnetism in Two-Dimensional Fe3GeTe2[END_REF][START_REF] Zhang | Gigantic Current Control of Coercive Field and Magnetic Memory Based on Nanometer-Thin Ferromagnetic van der Waals Fe3GeTe2[END_REF] and is associated with an unconventional form of Dzyaloshinskii-Moriya interaction [START_REF] Laref | Elusive Dzyaloshinskii-Moriya interaction in monolayer Fe3GeTe2[END_REF]. Nonetheless, mere symmetry consideration is not sufficient and a microscopic description is needed. Indeed, recent first principles calculation in the Janus monolayer VSeTe demonstrated that although this material possesses the C 3v symmetry, no "unconventional" torque can be obtained and only the usual field-like and damping-like torques are present [START_REF] Smaili | Janus monolayers of magnetic transition metal dichalcogenides as an all-in-one platform for spin-orbit torque[END_REF]. Therefore, understanding the physical origin of the "3m" torque in C 3v crystals and suggesting guidelines to enhance it is of crucial interest.

E 2C3 3σv Linear Quadratic Cubic A1 1 1 1 z x 2 + y 2 , z 2 z 3 , z(x 2 + y 2 ), x(x 2 -3y 2 ) m 2 x + m 2 y , m 2 z my(3m 2 x -m 2 y ) A2 1 1 -1 y(3x 2 -y 2 ) mz - mx(m 2 x -3m 2 y ) E 2 -1 0 (x, y) (x 2 -y 2 , xy), (xz, yz) (z(x 2 -y 2 ), xyz), (xz 2 , yz 2 ), (x(x 2 + y 2 ), y(x 2 + y 2 )) (mx, my) (m 2 x -m 2 y , mxmy) (mz(m 2 x -m 2 y ), mxmymz), (mxm 2 z , mym 2 z ) (mxmz, mymz) (mx(m 2 x + m 2 y ), my(m 2 x + m 2 y ))
In this work, we intend to clarify the nature of the spin-orbit torque in crystals with C 3v point group, i.e., its vectorial form and its microscopic origin. We first determine the general form of the spin-orbit torque up to the third order in magnetization using the Invariant Theory applied on the C 3v character table. We then consider a minimal model for a magnetic gas with C 3v symmetries. In this model, the spin texture is governed by the cooperation between linear (Rashba) and cubic spinmomentum locking terms. The Fermi surface is characterized by trigonal warping that appears close to the top of the band structure. We show that the unconventional "3m" torque is associated with the cubic spin-momentum locking when the Fermi surface displays strong trigonal warping. We therefore suggest that trigonal warping can be used as a good indicator for the search of "3m" torques in C 3v crystals and two-dimensional van der Waals magnets.

II. SYMMETRY ANALYSIS

We first determine the general form of the torque using the representation theory [START_REF] Dresselhaus | Group Theory: Application to the Physics of Condensed Matter[END_REF][START_REF] Lax | Symmetry Principles in Solid State and Molecular Physics[END_REF]. The C 3v point group is characterized by the identity E, the three-fold rotation along z, C 3 , and the mirror symmetry normal to, say, y, σ v . It has three irreducible representations A 1 , A 2 and E, i.e., matrices representing the action of the symmetry operations E, C 3 and σ v . Although a given symme-try operation can be represented by an infinite number of matrices, the trace of these representative matrices is unique for a given operation. Therefore, each irreducible representation can be identified by a unique set of traces called "characters". Table I gives the character table of the C 3v point group. The (equilibrium and nonequilibrium) properties of a given crystal are written as the combination of polar and axial vectors. For instance, in the case of the spin-orbit torque these vectors are the electric field (E x , E y , E z ) (polar vector) and the magnetization (m x , m y , m z ) (axial vector). Concretely, they transform in the following way,

(E x , E y ) σv -→ (E x , -E y ), (1)
(E x , E y ) C3 --→ (-1 2 E x - √ 3 2 E y , √ 3 2 E x -1 2 E y ), (2)
and

(m x , m y , m z ) σv -→ (-m x , m y , -m z ), (3)
(m x , m y , m z ) C3 --→ (-1 2 m x - √ 3 2 m y , √ 3 2 m x -1 2 m y , m z ). (4)
When applying the symmetry operations on these vectors' components, they transform according to the irreducible representations A 1 , A 2 and E so that one can define basis functions for each representation. In Table I, we give the basis functions of the irreducible representation of the C 3v point group up to the third order in magnetization.

Let us determine the general form of the torque based on Table I. The spin-orbit torque τ is associated with an effective field, τ = -γm × h. This effective field h is an axial vector, like m, so its component h z and (h x , h y ) belong to A 2 and E, respectively, and can be expanded as combinations of the invariant basis functions given in Table I. The only possible combinations of magnetization components with E that are invariant under the symmetries of the group are m z , m 3 z and m x (m 2 x -3m 2 y ), that all belong to A 2 . Conversely, z × E is an axial vector, and hence its allowed combinations are 1, m z and m y (3m 2

x -m 2 y ), that all belong to A 1 . Accounting for all combinations involving polar vector components at the first order and axial vector components up to the third order in magnetization, we obtain

h ∥ = h ∥ FL (1 + η FL m 2 z + δ FL m y (3m 2 x -m 2 y ))z × E + h ∥ DL ((1 + η DL m 2 z )m z + δ DL m x (m 2 x -3m 2 y ))E (5) +[h ∥ 3m m x (1 + η 3m m 2 z ) + h z 3m m z m y -2h ∥ PH m x m y + h ∥ χ m z (m 2 x -m 2 y )](E x x -E y y) +[-h ∥ 3m m y (1 + η 3m m 2 z ) + h z 3m m z m x + h ∥ PH (m 2 x -m 2 y ) + 2h ∥ χ m x m y m z ](E y x + E x y), h ⊥ = h z DL (1 + η z m 2 z )E • m + h z FL m z m • (z × E) (6) +h z PH ((m 2 x -m 2 y )E y + 2m x m y E x ) + h z χ m z ((m 2 x -m 2 y )E x -2m x m y E y ) z.
The formulas given above are general and they do not rely on any specific microscopic mechanism. We recognize the field-like torque (h In addition, we also identify two additional torques that we refer to as in-plane (h ∥ PH ) and out-of-plane (h z PH ) planar Hall torque, and chiral torques (h ∥,z χ ). The planar Hall torque possesses symmetries comparable to the planar Hall effect: it is active when the magnetization lies in the (x, y) plane and its magnitude depends on the angle between the electric field and the magnetization. The chiral torque necessitates to cant the magnetization away from the plane and it changes sign when reversing the magnetization (m z → -m z ).

To clarify the impact of the torque on the magnetization dynamics, we analyze its expression in two illustrative situations. When the magnetization lies out-of-plane (m = z), which is typical of perpendicularly magnetized systems at rest [see Fig. 1(a)], the two torque components up to first order in magnetization read

τ ∥ = -γh ∥ FL z × (z × E), (7)
τ ⊥ = -γh ∥ DL z × E (8)
We see that only the conventional field-like and damping-like torques are active in this configuration. One can see that the field-like torque is always along the electric field, ∼ m × (z × E), whereas the damping-like torque is perpendicular to it ∼ m × [(z × E) × m]. These two torques are the ones that destabilize the magnetization from its rest position and tend to bring it in the plane, normal to the applied electric field [see Fig. 1(b)].

Once the magnetization is in-plane, at ϕ = φ E + π 2 , where φ E is the in-plane angle of the electric field with respect to x and (θ, ϕ) are the polar and azimuthal angles of the magnetization unit vector, the torques τ = -γm× h = τ θ e θ + τ ϕ e ϕ in regular spherical coordinates are

τ θ /E = γ[h DL δ DL -h 3m ] sin 3φ E , (9)
τ ϕ /E = γh z PH cos 3φ E . (10)
In this configuration, the conventional field-like and damping-like torques are quenched, and the only active torques are the "3m" torque (h ∥ 3m ), identified in Ref. [START_REF] Liu | Symmetry-dependent field-free switching of perpendicular magnetization[END_REF], the trigonal anisotropy correction to the dampinglike torque (h ∥ DL δ DL ), and the perpendicular planar Hall torque (h z PH ). Here, only τ θ induces the deterministic switching, which means that the "3m" torque and the trigonal anisotropy correction to the damping-like torque are the active contributions in this process. Remarkably, in this frame the two other torques identified in Eqs. ( 5)-( 6), i.e. the planar Hall torque (h ∥,z PH ) and the chiral torque (h ∥,z χ ), are only active when θ ̸ = 0, π/2 and should therefore impact the magnetization dynamics itself. Their influence could modify the current-driven auto-oscillation [START_REF] Liu | Magnetic Oscillations Driven by the Spin Hall Effect in 3-Terminal Magnetic Tunnel Junction Devices[END_REF][START_REF] Demidov | Magnetic nano-oscillator driven by pure spin current[END_REF], a phenomenon that we leave to future studies.

III. PHYSICAL ORIGIN OF THE UNCONVENTIONAL TORQUES

A. Minimal tight-binding model for C3v magnets

The symmetry analysis provided above does not give information about the relative magnitude of the different torques. To better understand which microscopic mechanisms control these different components, we now turn our attention towards a minimal model for the spin-orbit torque. We consider a ferromagnetic system defined in a hexagonal lattice as depicted on Fig. 2(a) with C 3v symmetry modeled by the Hamiltonian

H 0 = ε k + ∆σ • m + H R + H R3 , (11)
with

H R = -i t R a u,s=± sσ • (z × u)e isk•u = t R a η k • (σ × z), (12)
H R3 = -it R3 u,s=± sσ • ze isk•u = t R3 λ k σ z . ( 13
)
The sum is taken over the nearest neighbors, i.e., u = a, b, c, sketched on Fig. 2(a), and a is the lattice parameter. Explicitly,

ε k = -2t(cos k • a + cos k • b + cos k • c), η k = 2(a sin k • a + b sin k • b + c sin k • c) and λ k = 2(sin k • a + sin k • b + sin k • c).
Here, t is the nearestneighbor hopping parameter, ∆ is the exchange between the conduction electrons and the magnetization m, t R is the linear Rashba spin-orbit coupling coming from inversion symmetry breaking normal to the (a, b) plane and t R3 is its cubic correction that is associated with the mirror symmetry normal to y axis. This model enhances the conventional free-electron Rashba gas by adding two ingredients: (i) the hexagonal symmetry and (ii) the cubic spin-orbit coupling. of parameters with (red lines) and without (black lines) cubic spin-orbit coupling. One directly sees that the cubic spin-orbit coupling only modifies the band structure close to the top of the band. The Fermi surface at three characteristic fillings are sketched on Figs. 2(c-e), with (red lines) and without (black lines) cubic spin-orbit coupling. At low band filling [Fig. 2(c)], where the dispersion is mostly quadratic, the Fermi surface is spherical and the cubic spin-orbit coupling has almost no impact. We therefore expect that the torque reduces to its most conventional form, field-like and damping-like. Upon increasing the band filling [Fig. 2(d,e)], the Fermi surface starts displaying hexagonal warping and the cubic spin-orbit coupling modifies the energy contours. In this context, at high band filling the warping is strong with Fermi pockets appearing away from Γ-point. Turning on the cubic spin-orbit coupling modifies the Fermi surface, resulting in a strong trigonal warping. It is clear that the unconventional torques identified in the previous section are expected to emerge in this regime.

The impact of Fermi surface warping on the spinorbit torque has been addressed theoretically in the context of the topological insulator surfaces. Kurebayashi and Nagaosa [START_REF] Kurebayashi | Theory of currentdriven dynamics of spin textures on the surface of a topological insulator[END_REF], Imai et al. [START_REF] Imai | Spintronic properties of topological surface Dirac electrons with hexagonal warping[END_REF] investigated the influence of warping on the spin-transfer torque and spinorbit torque, respectively, in magnetic domain walls and skyrmions to the first order of the magnetization gradient. The spin-orbit torque discussed presently is not addressed in these works. Zhou et al. [START_REF] Zhou | Nonlinear antidamping spinorbit torque originating from intraband transport on the warped surface of a topological insulator[END_REF] investigated the appearance of a damping-like torque that is nonlinear in electric field and directly induced by the warping. Li et al. [START_REF] Li | In-plane magnetization effect on current-induced spinorbit torque in a ferromagnet/topological insulator bilayer with hexagonal warping[END_REF] investigated the impact of the hexagonal warping on the spin-orbit torque, linear in electric field, and observed that the torque does not vanish when the magnetization lies in the plane. This is consistent with the analysis performed in the previous section, although a direct connection with the general form provided in Eqs. ( 5)-( 6) remains difficult.

Let us now compute the effective spin-orbit field driven by the current, h = (∆/V M s ) ⟨σ⟩, where M s is the saturation magnetization of the ferromagnet, V is the volume of the unit cell and ⟨...⟩ denotes nonequilibrium quantum statistical averaging. ⟨σ⟩ is computed within the linear response formalism considering the symmetrized decomposition of Kubo-Bastin formula proposed in Ref. [START_REF] Bonbien | Symmetrized decomposition of the Kubo-Bastin formula[END_REF], which takes the form

⟨σ i ⟩ Int = - eℏ 4π f (ϵ)dϵ Re Tr v(G R-A )σ i (∂ ϵ G R+A ) , ( 14
)
⟨σ i ⟩ Ext = - eℏ 8π ∂ ϵ f (ϵ)dϵ Re Tr v(G R-A )σ i (G R-A ) . (15)
Here v = ∂ k H is the velocity operator in the direction of the applied electric field, f (ϵ) is the equilibrium Fermi distribution function, G R(A) is the retarded (advanced) Green function and G R±A = G R ± G A . Notice that Eq. ( 14) gives the intrinsic contribution whereas Eq. ( 15)

gives the extrinsic one. Based on time-reversal symmetry, the spin-orbit field components that are odd in magnetization are of intrinsic origin, such as the dampinglike torque h ∥,z DL , the "3m" torque h ∥ 3m and the chiral torque h ∥,z χ , whereas the terms that are even in magnetization are extrinsic, such as the field-like torque h ∥,z FL and the planar Hall torque h ∥,z PH . To assess the relative magnitude of the different torque components and identify their physical origin, we compute the angular dependence of the fields when the magnetization rotates in the (x, y) plane, while the electric field is applied along x. Our results are reported in Fig. 3 for both intrinsic (a,b) and extrinsic (c,d) contributions in the cases of low (left panels) and high(right panels) band fillings. We analyze these results based on the angular dependence of the spin-orbit fields given by Eqs. ( 5)-( 6), when there is linear and cubic spin-orbit coupling. In this scenario, we deduce that

h =   h ∥ 3m cos ϕ -h ∥ PH sin 2ϕ + h ∥ DL δ DL cos 3ϕ -h ∥ 3m sin ϕ + h ∥ PH cos 2ϕ + h ∥ FL (1 + δ FL sin 3ϕ) h z DL cos ϕ + h z PH sin 2ϕ   (16)
From Fig. 3(a) it is clear that for h x and h y a 3-fold dependence related to δ DL and δ FL dominates at low band filling, while for h z the term h z DL is predominant in this regime. Besides, h ∥ 3m can be identified through the h y contribution at high band filling [Fig. 3(b)]. Regarding the extrinsic contributions, we notice from Fig. 3(c) that h x and h y are not trivial due to h ∥ PH and h ∥ FL at low band filling, whereas h z PH becomes relevant at high band filling in Fig. 3(d).

Analyzing the in-plane angular dependence of the field components, Fig. 3, with Eq. ( 16), one can extract the different torque contributions, reported on Fig. 4 explained above, this material has been recently experimentally demonstrated to host a sizable "3m" torque [START_REF] Liu | Symmetry-dependent field-free switching of perpendicular magnetization[END_REF]. We considered a CuPt/Co slab containing 12 layers, such that the L1 1 phase is made up of stacking elemental fcc layers along the [111] direction. We determine the band structure and spin textures by employing fully relativistic density functional theory. We describe the spinorbit coupling within a fully relativistic pseudo-potential formulation and used the generalized gradient approximation (GGA) for the exchange-correlation functional, the calculations are converged for a 400 Ry plane-wave cut-off for the real-space grid with a 13×13×1 k-points sampling of the Brillouin zone. We used the conjugate gradient algorithm to minimize the atomic forces below 0.01 eV/ Å. The momentum-resolved spin texture at the Fermi level is reported in Fig. 5 and displays a very clear hexagonal symmetry, suggesting an effectively large cubic spin-orbit coupling interaction. The angular dependence of the intrinsic and extrinsic spin-orbit fields is reported on Fig. 6(a) and (b), respectively, when the magnetization is rotated in the (x,y) plane. The calculations are performed with broadening Γ = 0.025 eV in the zero temperature limit. The angular dependence is well reproduced by Eq. ( 16). The intrinsic spin-orbit torque is composed of the damping-like torque (h z ) and the "3m" torque (h x,y ), with h ∥ 3m /h z DL ≈ 0.67, indicating that the "3m" torque is about the same order of magnitude as the damping-like torque. The extrinsic torque is one order of magnitude larger and is composed of the field-like torque and the planar Hall torque. The possible differences between our numerical predictions and our symmetry analysis in Eq. ( 16) can be explained by the neglect of higher-order terms in the character table expansion and the large values of cubic spin-orbit coupling. Nevertheless, we can extract h

∥ PH /h ∥ FL ≈ 1 and h z PH /h ∥ PH ≈ 0.4
, meaning that the planar Hall torque is anisotropic and as large as the fieldlike torque, and shall therefore impact the magnetization switching and dynamics. We leave this question to further studies. We emphasize that the relative magnitude of the intrinsic to extrinsic torques is not meaningful since the extrinsic torque is inversely proportional to the disorder broadening Γ, which is taken as (small) free parameter in our model.

C. Discussion and conclusion

The presence of these unconventional torques is particularly interesting for applications as they not only enable field-free switching but also impact the current-driven auto-oscillations. Our minimal model suggests that C 3v crystals could host such torques. Nonetheless, we emphasize that this is not a sufficient condition. As a matter of fact, in a previous study, we computed the spin-orbit torque in vanadium-based Janus transition metal dichalcogenides VSeTe and found no such torque, in spite of the similar crystal symmetries [START_REF] Smaili | Janus monolayers of magnetic transition metal dichalcogenides as an all-in-one platform for spin-orbit torque[END_REF]. We attributed this absence to the fact that in this material, the electronic transport here is mostly driven by states at Γ-point and therefore the crystal symmetries are not imprinted on the Bloch states. In contrast, in the L1 1 CuPt the Fermi surface shows a very strong warping, indicating that the Bloch states have a strong symmetry character and enabling the onset of the "3m" torque as well as other unconventional torques. Since the indicator to the presence of this torque is the trigonal warping of the Fermi surface, many other materials could display such effects: For example, Bi-based topological insulators (Bi,Sb) 2 /(Se,Te) 3 [START_REF] Chen | Experimental realization of a three-dimensional topological insulator, Bi2Te3[END_REF][START_REF] Hsieh | A tunable topological insulator in the spin helical Dirac transport regime[END_REF][START_REF] Alpichshev | STM imaging of electronic waves on the surface of Bi2Te3: Topologically protected surface states and hexagonal warping effects[END_REF], and Bi 4 Te 3 [START_REF] Chagas | Multiple strong topological gaps and hexagonal warping in Bi4Te3[END_REF], but also possibly in the recently grown LaAlO 3 /EuTiO 3 /SrTiO 3 all-oxide heterostructure [START_REF] Chen | Ferromagnetic Quasi-Two-Dimensional Electron Gas with Trigonal Crystal Field Splitting[END_REF].

We conclude this work by emphasizing that other unconventional torques are yet to be found in low-symmetry crystals that could lead to original current-driven dynam-ics, as already reported in WTe 2 /Py [START_REF] Macneill | Control of spin-orbit torques through crystal symmetry in WTe2/ferromagnet bilayers[END_REF][START_REF] Shi | All-electric magnetization switching and Dzyaloshinskii-Moriya interaction in WTe2/ferromagnet heterostructures[END_REF][START_REF] Xie | Field-free magnetization switching induced by the unconventional spin-orbit torque from WTe2[END_REF] and Fe 3 GeTe 2 [START_REF] Johansen | Current Control of Magnetism in Two-Dimensional Fe3GeTe2[END_REF][START_REF] Zhang | Gigantic Current Control of Coercive Field and Magnetic Memory Based on Nanometer-Thin Ferromagnetic van der Waals Fe3GeTe2[END_REF]. In this context, one needs to keep in mind that the general form of the spin-orbit field used in this work is obtained via a low-order expansion of the character table that is formally valid only when the spinorbit coupling is smaller than the exchange. In materials where the spin-orbit coupling and the exchange are of the same order of magnitude, much more complex torques are expected.

  ∥,z FL ), the damping-like torque (h ∥,z DL ) and the "3m" torque reported in Refs. 46 and 49 (h ∥,z 3m ). At higher orders, these terms are modulated by a planar anisotropy term, ∼ η α , and a trigonal anisotropy term, ∼ δ α . In addition, the magnitude of the fieldlike and damping-like torques are different in-plane (h ∥ FL , h ∥ DL ) and out-of-plane (h z FL , h z DL ). By removing these anisotropies, i.e., by setting η FL,DL = 0, δ FL,DL = 0 and h ∥ FL,DL = h z FL,DL , one retrieves the effective fields associated with the conventional field-like and damping-like torques, i.e., ∼ z × E and ∼ (z × E) × m.

FIG. 1 .

 1 FIG. 1. (Color online) Schematics of the torque components when the magnetization is (a) perpendicular to the plane and (b) in the plane, at 90°of the applied electric field.

FIG. 2 .

 2 FIG. 2. (Color online)(a) The unit cell of the minimal model for the C3v crystal. The grey atoms represent the hexagonal lattice sites and the red atoms break the plane inversion symmetry while conserving the three-fold rotation along z and the mirror symmetry normal to y. (b) Band structure of the tight-binding model described in the text with tR = 0.1t and ∆ = 0.5t, for the cases tR3 = 0 (black lines), and tR3 = 0.1t (red lines). The magnetization direction is set to θ = π 2 and ϕ = 0. The horizontal dashed lines correspond to µ = -5t, µ = t and µ = 2t, respectively. (c-e) Fermi surfaces with (tR3 = 0.1t -red lines) and without (tR3 = 0 -black lines) cubic spin-orbit coupling for (c) µ = -5t, (d) µ = t and (e) µ = 2t.

  Figure 2(b) represents the band structure for a standard set

  as a function of t R3 and µ. From Figs. 4(a, b), h ∥ 3m requires cubic Rashba coupling and increases with the band filling, confirming its sensitivity to the trigonal warping of the Fermi surface. This behaviour is different from h ∥ α δ α (α=DL, FL), which is displayed in Fig. 4(c) and reaches a maximum close to µ = 0 for t R3 ̸ = 0. The planar contributions h ∥ PH and h z PH are depicted in Figs. 4(d) and they exhibit different behaviors with t R3 and µ: whereas h PH does not require t R3 and it follows a similar tendency to h z DL and h ∥ FL [Fig. 4(e, f)], h z PH increases with the band filling and requires t R3 ̸ = 0. The salient features of the different torque components in C 3v systems are summarized in Table II. B. First principles case study: CuPt(111)/Co We conclude this work by computing the spin-orbit torque in L1 1 CuPt(111)/Co from first principles. As

FIG. 3 .

 3 FIG. 3. (Color online) Angular dependence of the effective field components hx (black lines), hy (blue lines) and hz (red lines) when the magnetization rotates in the (x, y) plane. We indicate a scaling factor of a given component whenever is necessary. The system's parameters are t = 1, tR = 0.1t, tR3 = 0.05t, ∆ = 0.5t and the homogeneous disorder Γ = 0.1t. The intrinsic (a, b) and extrinsic (c, d) contributions are plotted for µ = -5t (left panels) and µ = 2t (right panels).Our calculations reproduce our symmetry predictions in Eq. (16).

FIG. 4 .

 4 FIG. 4. (Color online) Effective field's components extracted from fitting the numerical results with Eq. (16). The panels show (a) h ∥ 3m , (c) h ∥ DL δDL (solid lines) and h ∥ FL δFL (dotted lines), (d) h ∥ PH (solid lines) and h z PH (dotted lines), (e) h z DL , and (f) h ∥ FL as a function of the chemical potential µ, for different values of the cubic spin-orbit coupling tR3 (black, blue and red lines stand for tR3 = 0, 0.05t, 0.075t). For completeness, panel (b) shows h ∥ 3m as a function of tR3 for different values of µ (black, blue and red lines stand for µ = -5t, t, 2t).

FIG. 5 .

 5 FIG. 5. (Color online) Spin texture in momentum space close to Fermi level a selected band of CuPt(111)/Co slab computed from first principles. A strong hexagonal symmetry is obtained suggesting the presence of a large cubic spin-orbit coupling interaction.

FIG. 6 .

 6 FIG. 6. (Color online) Angular dependence of the intrinsic (top) and extrinsic (bottom) spin-orbit field components when the magnetization is rotated in the (x, y) plane. The black, blue and red curves represent the x, y and z components the effective fields, respectively .

TABLE I .

 I Character table of the C3v point group. (x, y, z) are the components of a polar vector whereas (mx, my, mz) are the components of an axial vector.

TABLE II .

 II Summary of the minimal model analysis.

	Component Physical origin	Source
	h	∥ FL , h z FL , h h z PH , h z 3m ∥ PH	Extrinsic Extrinsic	Linear Rashba Linear + cubic Rashba
		h	∥ DL , h z DL	Intrinsic	Linear Rashba
			δFL, δDL	Intrinsic	Linear + cubic Rashba
	h	∥ 3m , h	∥ χ, h z χ	Intrinsic	Linear + cubic Rashba

ACKNOWLEDGMENTS

The authors thank L. Liu, J. Chen, K. Belashchenko, J. Medina, J.H. Garcia and S. Roche for fruitful discussions. This work was supported by the ANR ORION project, grant ANR-20-CE30-0022-01 of the French Agence Nationale de la Recherche. In addition, this research was partially financed by the ANR MNEMOSYN project, grant ANR-21-GRF1-0005. D. G.O. and A. M. acknowledge support from the Excellence Initiative of Aix-Marseille Université -A*Midex, a French "Investissements d'Avenir" program.


Gokaran Shukla
email: gokaran.shukla@kaust.edu.sa

Hasan M Abdullah

Avijeet Ray

Shubham Tyagi

Aurélien Manchon

Stefano Sanvito
email: sanvitos@tcd.ie

Udo Schwingenschlögl
email: udo.schwingenschlogl@kaust.edu.sa

ZnSe and ZnTe as tunnel barriers for Fe-based spin valves

come

I. INTRODUCTION

ZnSe and ZnTe are extensively used semiconductors in thin-film solar cells and optoelectronics devices operating in the blue and green regions of the visible spectrum [START_REF] Eason | High-brightness blue and green light-emitting diodes[END_REF][START_REF] Okerke | ZnSe buffer layer deposition for solar cell application[END_REF][START_REF] Chen | Solution-processed CdTe thin-film solar cells using ZnSe nanocrystal as a buffer layer[END_REF]. Both compounds exhibit high electron mobility (ZnSe, 200-280 cm 2 V -1 s -1 ; ZnTe, 340 cm 2 V -1 s -1 ) and low reflectance [START_REF] Georgescu | Analysis of thickness influence on refractive index and absorption coefficient on zinc selenides thin films[END_REF] for large photo-generated currents for both p and n-type dopants [START_REF] Rose | Electron mobility in II-VI semiconductors[END_REF][START_REF] Krasnov | p-Type conductivity in ZnSe[END_REF][START_REF] Samantilleke | Growth of n-type and p-type ZnSe thin films using an electrochemical technique for applications in large area optoelectronics devices[END_REF][START_REF] Chang | Aluminum-doped n-type ZnTe layers grown by molecular-beam epitaxy[END_REF][START_REF] Olusola | Growth of n-and p-type ZnTe semiconductors by intrinsic doping[END_REF][START_REF] Robinson | Wide band gap chalcogenide semiconductors[END_REF]. In addition, these semiconductors possess a small carrier effective mass at the Γ-point, a property used for the enhancement of infrared-laser optical gains by Cr doping [START_REF] Berry | Fabrication and power scaling of a 1.7 W Cr:ZnSe waveguide laser[END_REF][START_REF] Fang | Low loss and dispersion engineered ZnSe waveguides at telecom wavelengths[END_REF] and high energy X-ray and gamma-ray detectors [START_REF] Eason | High-brightness blue and green light-emitting diodes[END_REF][START_REF] Robinson | Wide band gap chalcogenide semiconductors[END_REF][START_REF] Cho | Comparative study of a CsI and a ZnSe(Te/O) scintillation detector's properties for a gamma-ray measurement[END_REF][START_REF] Linardatos | On the optical response of tellurium activated zinc selenide ZnSe:Te single crystal[END_REF]].

Here we explore the possibility to integrate the materials into spintronics devices.

In spintronics one controls the device operation and performance using both the electron charge and spin [START_REF]Handbook of Spin Transport and Magnetism[END_REF][START_REF] Peng | Magnetic tunnel junctions for spintronics: Principle and applications[END_REF]. The most common device is perhaps the magnetic tunnel junction (MTJ), which consists of two ferromagnetic leads separated by a thin tunnel barrier. An MTJ exhibits low (high) resistance when the magnetic vectors of the two leads are oriented parallel (antiparallel). There is the possibility to swap the sign of the signal by appropriate choice of the leads and tunnel barrier [START_REF] Teresa | Role of metal-oxide interface in determining the spin polarization of magnetic tunnel junctions[END_REF]. In binary terms, a low (high) resistance maps to a 0 [START_REF] Eason | High-brightness blue and green light-emitting diodes[END_REF]. The magnetoresistance effect in MTJs is the backbone of current spintronics devices, such as magnetic random access memories (MRAMs) [START_REF] Bhatti | Spintronics based random access memory: A review[END_REF], microwave generators [START_REF] Kiselev | Microwave oscillations of a nanomagnet driven by a spin polarized current[END_REF], stochastic oscillators [START_REF] Kaiser | Hardware-aware in-situ Boltzmann machine learning using stochastic magnetic tunnel junctions[END_REF], neuromorphic computers [START_REF] Vincent | Spin-transfer torque magnetic memory as a stochastic memristive synapse for neuromorphic systems[END_REF][START_REF] Srinivasan | Magnetic tunnel junction based long-term short-term stochastic synapse for a spiking neural network with on-chip STDP learning[END_REF][START_REF] Toreejon | Neuromorphic computing with nanoscale spintronic oscillators[END_REF][START_REF] Cai | Robust ultra-low power non-volatile logic-inmemory circuits in FD-SOI technology[END_REF], MRAM-based ultrafast embedded memories [START_REF] Garello | Spin-orbit torque MRAM for ultrafast embedded memories: From fundamentals to large scale technology integration[END_REF], spin-based NANDs [START_REF] Wang | High-density NAND-like spin transfer torque memory with spin orbit torque erase operation[END_REF], nonvolatile flip-flops [START_REF] Montesi | Building blocks to use in innovative non-volatile FPGA architecture based on MTJs[END_REF][START_REF] Iyengar | MTJ-based state retentive flip-flop with enhanced-scan capability to sustain sudden power failure[END_REF][START_REF] Cai | Low power magnetic flip-flop optimization with FDSOI technology boost[END_REF], non-volatile full adders [START_REF] Meng | A spintronics full adder for magnetic CPU[END_REF], non-uniform clock generators [START_REF] Lee | A spintronic voltage-controlled stochastic oscillator for event-driven random sampling[END_REF], radio frequency spectrum optimizers [START_REF] Marques | A review of sparse recovery algorithms[END_REF], spin transfer torque devices [START_REF] Berger | Emission of spin waves by a magnetic multilayer traversed by a current[END_REF][START_REF] Ralph | Spin transfer torques[END_REF], magnetic sensors [START_REF] Fujiwara | Magnetocardigraphy and magnetoencephalography measurements at room temperature using tunnel magneto-resistance sensors[END_REF], comparators [START_REF] Jiang | Spin analogto-digital converter using magnetic tunnel junction and spin Hall effect[END_REF], thermal sensors [START_REF] Sengupta | Magnetic tunnel junction as an on-chip temperature sensor[END_REF], and analog-to-digital converters [START_REF] Chakraborty | Design of a low-voltage analog-to-digital converter using voltagecontrolled stochastic switching of low barrier nanomagnets[END_REF][START_REF] Salehi | Energy-aware adaptive rate and resolution sampling of spectrally sparse signals leveraging VCMA-MTJ devices[END_REF][START_REF] Salehi | SLIM-ADC: Spin-based logic-in-memory analog to digital converter leveraging SHE-enabled domain wall motion devices[END_REF]. Nonvolatility [START_REF] Bhatti | Spintronics based random access memory: A review[END_REF], low power consumption [START_REF] Maciel | Magnetic tunnel junction applications[END_REF], large area scalability [START_REF] Maciel | Magnetic tunnel junction applications[END_REF], and practically infinite endurance [START_REF] Chappert | The emergence of spin electronics in data storage[END_REF] make MTJs promising candidates for memory elements in nano-electronics.

The main quality factor of a MTJ is the tunneling magneto resistance ratio, T M R = G P -G AP G P +G AP , where G P (G AP ) is the total conductance in the parallel (P) (anti-parallel; AP) configuration. The total conductance can be written as sum of the spin-majority and spin-minority contributions, following a two-spin-fluid approximation that neglects spin-flip scattering, G P(AP) = G

P(AP) ↑ + G P(AP) ↓ , with G P(AP) σ
being the conductance for spin σ. In the nineties, MTJs were mostly based on amorphous Al 2 O 3 tunnel barriers [START_REF] Miyazaki | Giant magnetic tunneling effect in Fe/Al2O3/Fe junction[END_REF][START_REF] Moodera | Large magnetoresistance at room temperature in ferromagnetic thin film tunnel junctions[END_REF]. The TMR ratio in these devices is 2P1P2 1-P1P2 in Julliere's model, which is valid for thick and high enough tunnel barriers. P 1 and P 2 denote the spinpolarized densities of states in the two magnetic leads at the Fermi level, E F [START_REF] Julliere | Tunneling between ferromagnetic films[END_REF]. In general, P = n ↑ -n ↓ n ↑ +n ↓ , where n σ is the density of states at E F for spin σ [START_REF] Mazin | How to define and calculate the degree of spin polarization in ferromagnets[END_REF]. Since P in most transition-metal ferromagnets hardly exceeds 50%, the TMR ratio of an amorphous tunnel barrier is limited to ∼70%. This, however, does not apply to crystalline tunnel barriers in which the TMR ratio is determined by the symmetry-filtering mechanism [START_REF] Butler | Spin-dependent tunneling conductance of Fe/MgO/Fe sandwiches[END_REF][START_REF] Mathon | Theory of tunneling magnetoresistance of an epitaxial Fe/MgO/Fe(001) junction[END_REF].

In a crystalline tunnel barrier the transverse wave vector, k ∥ , remains a good quantum number. This means that only the longitudinal k z component of the wave vector (which is complex in an insulator) changes across the barrier, while the transverse component is conserved.

Symmetry-filtering then allows Bloch states with a certain angular momentum to tunnel with higher probability than others [START_REF] Butler | Spin-dependent tunneling conductance of Fe/MgO/Fe sandwiches[END_REF][START_REF] Butler | Tunneling magnetoresistance from a symmetry filtering effect[END_REF][START_REF] Shukla | Fe and Co-based magnetic tunnel junctions with AlN and ZnO spacers[END_REF]. The angular-momentumdependent tunneling probability is mainly governed by the overlap integral between the wavefunction of the incident electron and the wavefunctions at the valence band maximum and conduction band minimum of the tunneling barrier [START_REF] Butler | Tunneling magnetoresistance from a symmetry filtering effect[END_REF]. In the case of exact symmetry matching the overlap integral is maximal, resulting in high tunneling probability [START_REF] Butler | Tunneling magnetoresistance from a symmetry filtering effect[END_REF]. Notably, the current commercial MTJs are mostly based on the Fe/MgO/Fe junction, [START_REF] Parkin | Giant tunnelling magnetoresistance at room temperature with MgO (100) tunnel barriers[END_REF][START_REF] Yuasa | Giant room-temperature magnetoresistance in single-crystal Fe/MgO/Fe magnetic tunnel junctions[END_REF] in which the exact symmetry matching condition is fulfilled. In this case the ∆ 1↑ states tunnel with higher probability than the ∆ 5↑ and ∆ 2↓ states, while the ∆ 1↓ states are not available in Fe at E F [START_REF] Butler | Spin-dependent tunneling conductance of Fe/MgO/Fe sandwiches[END_REF]. This means that the tunnel current is completely spin-polarized for a thick enough MgO tunnel barrier.

In addition to the Fe/MgO/Fe junction, the symmetry-filtering argument has been tested in various wide bandgap oxide and multiferroic MTJs [START_REF] Maclaren | Spindependent tunneling in epitaxial systems: Band dependence of conductance[END_REF][START_REF] Maclaren | Layer KKR approach to Bloch-wave transmission and reflection: Application to spin dependent tunneling[END_REF][START_REF] Mavropoulos | Complex band structure and tunneling through ferromagnet/insulator/ferromagnet junctions[END_REF][START_REF] Sacconi | Full-band tunneling in high-κ oxide MOS structures[END_REF][START_REF] Velev | Magnetic tunnel junctions with ferroelectric barriers: Prediction of four resistance states from first principles[END_REF][START_REF] Caffrey | Prediction of large bias-dependent magnetoresistance in all-oxide magnetic tunnel junctions with a ferroelectric barrier[END_REF][START_REF] Liu | Giant tunneling magnetoresistance in epitaxial Co2MnSi/MgO/Co2MnSi magnetic tunnel junctions by half-metallicity of Co2MnSi and coherent tunneling[END_REF][START_REF] Caffrey | Coexistance of giant tunneling electroresistance and magnetoresistance in an-all oxides magnetic tunnel junction[END_REF][START_REF] Caffrey | Spin-filtering efficiency of ferrimagnetic spinels CoFe2O4 and NiFe2O4[END_REF][START_REF] Shukla | HfO2 and SiO2 as barriers in magnetic tunneling junctions[END_REF]. There are many reasons behind the success of the Fe/MgO/Fe junction in spintronics, particularly the ease of lattice-matched crystalline growth, the large inplane/perpendicular magneto-crystalline anisotropy, the wide bandgap of MgO, and the availability of a controlled lithographic process for large area. However, the Fe/MgO/Fe junction also encounters limitations at the fabrication level. In particular, growing a compound with four-fold rotation symmetry (C 4 ) on a compound with six-fold rotation symmetry (C 6 ) requires intermediate layers for interface strain minimization, demanding high-end lithographic processes. Therefore, it is imperative to explore other MTJs, which may require less expensive lithographic steps. Here we investigate ZnSe and ZnTe as tunnel barriers in MTJs. These are unlikely to replace the Fe/MgO/Fe junction for data-storage applications, but we argue that they may be integrated into system on chip and internet of things applications [START_REF] Zhao | Spin-MTJ based Non-volatile flip-flop[END_REF][START_REF] Zarei | Power and area-efficient design of VCMA-MRAM based full-adder using approximate computing for IoT applications[END_REF] for their low power dissipation due to small interfacial resistance (∼ 1 Ωµm 2 ) [START_REF] Jiang | Magnetic tunnel junctions with ZnSe barriers[END_REF].

ZnSe and ZnTe exist in zinc-blende (cubic) and hexagonal close-packed (hcp) crystal structures, the zinc-blende structure being more stable at room temperature and crystallographically better compatible with the elementary ferromagnets bcc-Fe, hcp-Co, and fcc-Ni. A large inplane lattice mismatch with the ferromagnet could result in a high interfacial defect concentration, which changes the Coulomb potential profile across the MTJ and assists spin flipping through a Rashba field or through spin-orbit coupling. As for the ferromagnet, we consider Fe, since it has a high Curie temperature and provides a 100% spin-polarized tunnel current for ∆ 1 -selecting tunnel barriers. Our study thus extends previous theoretical work on the Fe/ZnSe/Fe junction [START_REF] Maclaren | Layer KKR approach to Bloch-wave transmission and reflection: Application to spin dependent tunneling[END_REF][START_REF] Herper | Electric transport in Fe/ZnSe/Fe heterostructures[END_REF] and introduces the Fe/ZnTe/Fe junction.

The paper is organized as follows. We first present our computational method and details of the systems investigated. Then, we discuss our results using the real and complex band structures of ZnSe, ZnTe, and Fe. Next, we analyze the TMR ratio. In the end, we conclude and summarize our main findings.

II. COMPUTATIONAL DETAILS

The ground-state electronic properties of ZnSe, ZnTe, and Fe are calculated by density functional theory (DFT) with the exchange-correlation energy taken in the local density approximation (LDA) as parametrised by Ceperly and Alder [START_REF] Ceperly | Ground state of the electron gas by a stochastic method[END_REF]. We use the siesta DFT code, which is based on norm-conserving pseudo-potentials and a numerical atom-centered localized basis set [START_REF] Soler | The SIESTA method for ab initio order N materials simulation[END_REF]. The s, p, and d orbitals of Fe are expanded in a double-ζ basis set, while the s and p orbitals of Zn, Se, and Te are expanded in a double-ζ plus polarization basis set. We take 400 Ry as a real-space energy cutoff and employ a 8 × 8 × 8 Monkhorst-Pack k-point mesh. To correct the underestimation of the bandgap in the LDA we apply the atomic self-interaction correction (ASIC) [START_REF] Pemmaraju | Atomic-orbital-based approximate selfinteraction correction scheme for molecules and solids[END_REF][START_REF] Filippetti | Variational pseudo-selfinteraction-corrected density functional approach to the ab initio description of correlated solids and molecules[END_REF] to both ZnSe and ZnTe when computing the band structure and transport properties.

We construct C 4 -symmetric Fe/ZnSe/Fe and Fe/ZnTe/Fe junctions with lattice parameters of 5.7 Å, 6.1 Å, and 2.9 Å for ZnSe [START_REF] Jiang | Magnetic tunnel junctions with ZnSe barriers[END_REF], ZnTe [START_REF] Szwacki | Structural properties of MnTe, ZnTe, and ZnMnTe[END_REF], and Fe [START_REF] Jiang | Magnetic tunnel junctions with ZnSe barriers[END_REF], respectively (see Fig. 1). A 2 × 2 (001) Fe cell is adopted. The lattice mismatch at the Fe/ZnSe interface is 1.6% with Fe being compressed and ZnSe left free of strain. The lattice mismatch at the Fe/ZnTe interface is 5% with Fe being stretched and ZnTe left free of strain. We absorb the strain at the interface entirely into Fe, since its electronic properties (work function, band structure, etc.) do not change significantly. The tunnel barriers are then 31.07 Å (ZnSe) and 39.93 Å (ZnTe) thick. Note that there exist several MTJs made from materials with large lattice mismatch, including Co/ZnO/Co [START_REF] Li | Spin memristive magnetic tunnel junctions with CoO-ZnO nano composite barrier[END_REF], CoFe/AlN/CoFe [START_REF] Schwickert | Magnetic tunnel junctions with AlN and AlNxOy barriers[END_REF], and Co/ZnO/Ni [START_REF] Althammer | Spin transport and spin dephasing in zinc oxide[END_REF], indicating that also Fe/ZnSe/Fe and Fe/ZnTe/Fe junctions can be fabricated. Our MTJs are asymmetric with different interfaces to the two electrodes. Such broken inversion symmetry is likely to occur in an actual MTJ and lifts the spin degeneracy for the AP configuration. Using the conjugate gradient method, we relax the atomic coordinates in the scattering region until the forces on each atom become less than 0.01 eV/ Å. Since ZnSe and ZnTe also crystallize in hcp crystal structures, we perform additional complex band structure calculations to verify that our results remain robust in this case. However, we do not perform quantum transport calculations for the hcp crystal structures due to the large lattice mismatch with Fe.

For the quantum transport calculations we use the nonequilibrium Green's functions method with Kohn-Sham Hamiltonian (NEGF+DFT) scheme as implemented in the smeagol code [START_REF] Rocha | Towards molecular spintronics[END_REF][START_REF] Rocha | Spin and molecular electronics in atomically generated orbital landscapes[END_REF][START_REF] Rungger | Algorithm for the construction of self-energies for electronic transport calculations based on singularity elimination and singular value decomposition[END_REF], which integrates siesta as DFT engine. The spin-dependent current in the Landauer-Büttiker formalism is

I σ (V ) = e h dE T σ (E; V )[f L -f R ] , (1)
where e is the electron charge, h is the Planck constant, and T σ (E; V ) is the energy-and voltage-dependent total transmission coefficient. Moreover,

f L (f R ) is the Fermi- Dirac distribution function of the left (right) lead, which is evaluated at E -µ L (E -µ R ), where µ L/R = E F ± eV 2
is the chemical potential of the left/right lead at bias voltage V . Since the MTJs are translationally invariant perpendicular to the transport direction, we integrate over the 2D Brillouin zone (BZ) in the transverse plane to obtain

T σ (E; V ) = 1 Ω BZ BZ dk ∥ T σ k ∥ (E; V ) , (2)
where Ω BZ is the volume of the 2D BZ and T σ k ∥ (E; V ) is the k ∥ -dependent transmission coefficient. In terms of the coupling matrix of the leads, Γ σ α (E; V ) (α = L and R), and the effective Green's function of the scattering region, G σ C (E; V ), we can write

T σ k ∥ (E; V ) = Tr[Γ σ L (E; V )G σ † C (E; V )Γ σ R (E; V )G σ C (E; V )], (3)
where Tr is the trace operation and all quantities are assumed to be k

∥ -dependent. G σ C (E; V ) is related to the Kohn-Sham Hamiltonian, H σ C , as G σ C (E; V ) = lim η→0 [E + iη -H σ C -Σ σ L (E; V ) -Σ σ R (E; V )] -1 .
(4) The self-energy of the leads, Σ σ α (E; V ), is related to the coupling matrix as

Γ σ α (E; V ) = i(Σ σ+ α (E; V ) -Σ σ- α (E; V )), (5)
where a superscript + (-) indicates the retarded (advanced) part of the self-energy. The quantum transport calculations are performed in the zero bias limit (V = 0) after converging the charge density on a 8 × 8 × 1 kpoint mesh with a density matrix tolerance of 10 -4 . The transmission coefficient is then integrated on a 50 × 50×1 k-point mesh. Integration on a 100×100×1 k-point mesh yields no significant change in T σ (E; V ) or in the TMR ratio.

III. RESULTS AND DISCUSSION

A. ZnSe and ZnTe as tunnel barriers

Fe is a high-temperature ferromagnet, while ZnSe and ZnTe are II-VI chalcogenide semiconductors [START_REF] Robinson | Wide band gap chalcogenide semiconductors[END_REF]. The optical bandgaps of zinc-blende ZnSe and ZnTe are 2.7 eV and 2.3 eV, respectively [START_REF] Robinson | Wide band gap chalcogenide semiconductors[END_REF], values that match well our ASIC [START_REF] Pemmaraju | Atomic-orbital-based approximate selfinteraction correction scheme for molecules and solids[END_REF][START_REF] Filippetti | Variational pseudo-selfinteraction-corrected density functional approach to the ab initio description of correlated solids and molecules[END_REF] direct bandgaps of 2.2 eV and 2.0 eV. We examine the decay amplitude of the complex wave vector of the tunneling electrons, k z = iκ. In general, k z does not only depend on the barrier height, energy, and transverse wave vector but also on the curvature of the wavefunction in the plane perpendicular to the transport direction [START_REF] Butler | Tunneling magnetoresistance from a symmetry filtering effect[END_REF]. We can write [START_REF] Butler | Tunneling magnetoresistance from a symmetry filtering effect[END_REF]

κ(E, k ∥ ) = 2m(U -E)/h 2 + k 2 || -⟨ϕ| ∂ 2 dx 2 + ∂ 2 dy 2 |ϕ⟩/⟨ϕ|ϕ⟩
, where U is the barrier height and the last term under the square root represents the effect of the transverse oscillation of the wavefunction (angular momentum quantum number). The higher the contribution of the Laplacian, the larger is κ(E, k || ) and the smaller is the tunneling probability through the barrier. We can write

T σ k ∥ (E; V ) = T σ 0,k ∥ (E; V )e -2κ(E,k ∥ )d
, where T σ 0,k ∥ (E; V ) is the interfacial transmission probability, which strongly depends on the atomic composition and crystallinity of the interface, and d is the effective thickness of the tunnel barrier [START_REF] Parkin | Giant tunnelling magnetoresistance at room temperature with MgO (100) tunnel barriers[END_REF][START_REF] Yuasa | Giant room-temperature magnetoresistance in single-crystal Fe/MgO/Fe magnetic tunnel junctions[END_REF].

In the two semiconductors considered, the valence band maximum and conduction band minimum at the Γpoint are predominantly made of Zn s and chalcogenide p orbitals, respectively, so that one expects a complex band with ∆ 1 symmetry to connect the real bands across the bandgap. This is verified by calculating the map of κ over the (k x , k y )-plane at E F , as presented in Fig. 2 for zinc-blende ZnSe and ZnTe in panels (a) and (b) and for hcp ZnSe and ZnTe in panels (c) and (d), respectively. In all cases we find the minimal κ at the Γ-point, as expected from the bandgap position in k-space. Furthermore, the variation of κ in the 2D BZ is similar in all the cases investigated. In more detail, we find κ to vary between 0.8 Å-1 and 2.3 Å-1 for zinc-blende ZnSe, between 0.8 Å-1 and 2.0 Å-1 for zinc-blende ZnTe, between 0.8 Å-1 and 2.1 Å-1 for hcp ZnSe, and between 0.8 Å-1 and 2.1 Å-1 for hcp ZnTe. Analysis based on the symmetry-filtering argument is only semi-quantitative, since T σ (E; V ) is not only calculated at the Γ-point but in the entire 2D BZ. However, the fact that κ is always minimal at the Γ-point indicates that in all cases most of the current is due to electrons incident to the barrier with k ∥ = 0.

B. Symmetry of the complex bands

Having established that the transport is dominated by the center of the BZ, we now investigate the angular momentum symmetry of the complex bands at the Γ-point and E F . This is done by projecting the Bloch wavefunctions onto the plane perpendicular to the transport direction, and then by visual inspection of the spatial distribution. The ∆ 1 symmetry corresponds to states carrying zero angular momentum about the transport direction, therefore presenting s, p z , and d 3z 2 -r 2 distributions. Similarly, wavefunctions with p x , p y , d xz , and d yz distributions are mapped to the ∆ 5 symmetry, while those with d xy and d x 2 -y 2 distributions are mapped to the ∆ 2 and ∆ 2 ′ symmetries, respectively [START_REF] Bosoni | Complex band structure with nonn-orthogonal basis set: Analytical properties and implementation in the SIESTA code[END_REF].

The calculated real and complex bands are presented in Fig. 2. We also report the symmetries of the bands most relevant for the tunneling. The complex band with the smallest κ across the bandgap has always ∆ 1 symmetry. For both zinc-blende ZnSe and ZnTe the bands of ∆ 2 and ∆ 5 symmetries are also present at the top of the valence band, i.e., there are branches of complex bands with low κ close to the bottom of the bandgap. These are relevant for the transport only for energies at the lower end of the bandgap. A different situation is encountered for the hcp crystal structures, where the real bands with ∆ 1 and ∆ 5 symmetries are almost degenerate at the top of the valence band. Note that in no case we find socalled spurious flat bands, which may arise when using non-orthogonal atom-centered localised orbitals [START_REF] Bosoni | Complex band structure with nonn-orthogonal basis set: Analytical properties and implementation in the SIESTA code[END_REF].

C. Real bands of Fe

The real bands of Fe are plotted along the transport direction in Fig. 3. Bands of ∆ 5 symmetry are present at E F for both spin directions, although they just touch E F in the majority spin channel. The exchange splitting of these bands is about 2 eV. The band of ∆ 2 symmetry crosses E F only in the minority spin channel, while it remains completely occupied in the majority spin channel. Most important for the transport are the bands of ∆ 1 symmetry. Here, only a highly dispersive majority spin band crosses E F , while the minority spin bands remain empty. This band alignment suggests that the transport in the Fe/ZnSe/Fe and Fe/ZnTe/Fe junctions is characterized by symmetry-filtering of the band of ∆ 1 symmetry, similar to the case of the Fe/MgO/Fe junction.

Additional information can be obtained from the distribution of the open scattering channels at E F in the 2D BZ. As expected, the maps shown in Fig. 3 for the two spin channels exhibit a four-fold rotational symmetry around the Γ-point. In the case of the majority spin channel the open scattering channels are distributed across the entire 2D BZ, with some concentration around the Γ-point. In contrast, in the case of the minority spin channel most of the open scattering channels are found around the Γ-point and there are large regions of the 2D BZ where no open scattering channels are available.

D. Zero bias transmission and TMR ratio

We now analyse the zero bias transmission coefficients at E F in the 2D BZ, shown as heat maps in Fig. 4 for both the Fe/ZnSe/Fe and Fe/ZnTe/Fe junctions. The heat maps display a two-fold rotation symmetry, reflecting the two-fold rotation symmetry of the zinc-blend structure about the transport direction. The broken inversion symmetry of the junctions lifts the spin degeneracy in the AP configurations, i.e., the transmission profile is different for the two spin channels. We set the spin quantization axis with respect to the right lead.

E F of the Fe/ZnSe/Fe junction is close to midgap, i.e., the low bias transport is determined by electron tunneling and the symmetry-filtering argument comes into play. As a consequence, for both spin channels and magnetic configurations the maximal transmission is found around the Γ-point. Differences, however, emerge between the spin-dependent transmission functions. We consider first the P configuration and the majority spin channel. In this case the complex band of ∆ 1 symmetry determines the minimal κ (see Fig. 2 (a)), i.e., the maximal transmission appears exactly at the Γ-point with a rapid decay away from the center of the 2D BZ. The band of ∆ 1 symmetry, however, is not available in the Fe band structure for the minority spin channel. Therefore, the transmission proceeds mainly through the bands of ∆ 2 and ∆ 5 symmetries, and the amplitude is maximal in an annular region around the Γ-point, but not exactly at the Γ-point, meaning that the tunneling electrons in the minority spin channel have a non-vanishing transverse wave vector. In addition, a minor contribution to the transmission (about 1% of the maximum) originates from two arc regions placed along the diagonal of the 2D BZ. Thus, the maximal transmission appears in an annular region around the Γ-point, with a minor contribution from the arc regions placed along the diagonal of the 2D BZ.

A different situation is encountered for the Fe/ZnTe/Fe junction. Here, E F lies just at the edge of the conduction band of ZnTe, i.e., the transport is not dominated by electron tunneling but by conduction through the tunnel barrier. As a consequence, the transmission coefficients approach unity and symmetry-filtering is no longer at play (being still at play away from E F , where electron tunneling takes place). The zero bias transmission maps in Fig. 4(b) indeed reveal significant amplitudes in different regions of the 2D BZ, particularly in a star-shaped region around the Γ-point and close to the corners.

Next, we analyze T σ (E; V = 0) and the resulting TMR ratio as functions of the energy, which are plotted for both spin channels and magnetic configurations in Fig. 5. E F of the Fe/ZnSe/Fe junction lies close to the middle of the bandgap of ZnSe and the transport thus is tunnelinglike. For the P configuration we notice that the trans- mission coefficient is larger for the majority spin channel than for the minority spin channel between -0.3 eV and 0.3 eV (E F = 0 eV). This is the region where the majority spin band with ∆ 1 symmetry of Fe overlaps with the complex band of minimal κ, which also has ∆ 1 symmetry. In the rest of the bandgap the tunneling current is carried by bands of ∆ 2 and ∆ 5 symmetries, resulting in a lower transmission. For E < -0.3 eV the minority spin current has a predominant ∆ 2 character and the majority spin current has a predominant ∆ 5 character. There is a resonant transmission peak near -0.3 eV, associated with a majority spin surface state. For the AP configuration, as expected, the transmission coefficient is almost spindegenerate (not exactly, because the inversion symmetry is broken). We observe a strong energy dependence of the TMR ratio. Interestingly, we find that the TMR ratio at E F is about 200% and relatively constant within 0.4 eV around E F . This suggests that a large TMR ratio is realized in the typical bias range applied to MTJs. Note that our calculated TMR ratio is lower than what usually is found for the Fe/MgO/Fe junction with a barrier of similar thickness. This is due to the fact that the bandgap of ZnSe is significantly smaller than that of MgO, meaning that the wavefunction decay in ZnSe is slower than in MgO and the symmetry-filtering is less effective. Since symmetry-filtering is at play, one expects the TMR ratio to grow with the thickness of the tunnel barrier.

According to Fig. 5(d), unfortunately, E F of the Fe/ZnTe/Fe junction is placed at the bottom of the conduction band of ZnTe, i.e, symmetry-filtering is not at play at E F . Therefore, the junction is similar to a standard giant magnetoresistance (GMR) spin valve and the zero bias GMR ratio is mainly determined by the spin polarisation of the leads. This results in a zero bias GMR of around 60%, which is a modest value. We expect this result to be rather robust, since the bottom of the conduction band is characterised by electrons with s symmetry, i.e., spin-orbit coupling is not at play. There is spin-filtering in the bandgap region, in particular for energies between -1 eV and -1.7 eV, where the transmission coefficient in the P configuration is dominated by the minority spin channel. This is associated with the band of ∆ 2 symmetry, which has its maximal dispersion in this energy window for the minority spin channel. The resulting TMR ratio for -1.7 eV < E < -1.0 eV touches 200%, but this is not accessible experimentally under normal bias.

IV. CONCLUSION

We have explored the potential of chalcogenide semiconductors as tunnel barriers in future MTJs. Specifically, we have studied ZnSe and ZnTe, which are extensively used in the solar cell and display industries, opening the possibility for integration in optical technologies.

Our theoretical analysis, based on density functional theory and the non-equilibrium Green's functions method, relates the tunneling properties of the junctions to the electronic structures of the constituent compounds. In particular, we find for the Fe/ZnSe/Fe junction that the transport is tunneling-like, while in the case of the Fe/ZnTe/Fe junction E F is pinned to the conduction band of ZnTe and only a GMR effect with a modest GMR ratio of around 60% is achieved. In contrast, symmetryfiltering is at play in the case of the Fe/ZnSe/Fe junction, resulting in a TMR ratio of the order of 200% at E F , due to the predominant transmission of electrons with ∆ 1 symmetry, similar to the celebrated Fe/MgO/Fe platform. Unfortunately, since the bandgap of ZnSe is smaller than that of MgO, the TMR ratio never reaches very large values, but it is still useful for applications, particularly when considering that the current density achievable by this junction is large.

FIG. 1 :

 1 FIG. 1: (Colour online) Simulation cells of the scattering regions of the (a) Fe/ZnSe/Fe and (b) Fe/ZnTe/Fe junctions. The junctions are asymmetric with broken inversion symmetry, which results in different chemical environments at the two ends of the tunnel barrier. Color code: Fe red, Zn magenta, Se yellow, and Te grey.

FIG. 2 :

 2 FIG. 2: (Colour online) Electronic structure analysis of (a) zinc-blende ZnSe, (b) zinc-blende ZnTe, (c) hcp ZnSe, and (d) hcp ZnTe. The left panel always shows the map of the decay coefficient, κ(E, k ∥ ), as a function of the transverse wavevector, k ∥ , at E F . The black box marks the 2D BZ. The value of κ is encoded as a heat map from the minimal to the maximal value. The right panel always shows the real (red) and complex (black) band structures along the [001] direction. The results are obtained by LDA+ASIC calculations.

5 FIG. 3 :

 53 FIG. 3: (Colour online) Electronic structure analysis of bcc-Fe. The left panel shows the real band structure along the [001] direction for both the majority (red lines) and minority (black lines) spin channels. The symmetry of the bands is established as described in the text. The right panels show maps of the number of open majority (top) and minority (bottom) spin scattering channels at E F over the 2D BZ. Note that the real band structure refers to the primitive unit cell of bcc-Fe, while the total number of open spin scattering channels refers to the 16-atom cubic cell used in the transport calculations.

FIG. 4 :

 4 FIG. 4: (Colour online) Zero bias transmission coefficient at E F for the (a) Fe/ZnSe/Fe and (b) Fe/ZnTe/Fe junctions plotted in the 2D BZ, i.e., as a function of k ∥ = (k x , k y ). Results are shown for the two spin channels and the P and AP configurations.

FIG. 5 :

 5 FIG. 5: (Colour online) Total transmission coefficient of the Fe/ZnSe/Fe junction in the (a) P and (b) AP configurations and of the Fe/ZnTe/Fe junction in the (d) P and (e) AP configurations. For the AP configuration the spin quantization axis is set with respect to the right lead. Panels (c) and (f) show the energy dependence of the zero bias TMR ratio for the Fe/ZnSe/Fe and Fe/ZnTe/Fe junctions, respectively.

Acknowledgments. The research reported in this publication was supported by funding from King Abdullah University of Science and Technology. SS thanks the Irish Research Council for financial support through the Advanced Laureate Award (IRCLA/2019/127).


Basile Ploteau
email: basile.ploteau@supagro.fr

Guilhem Brunel

Léo Pichon

Arnaud Ducanchez

Thomas Crestey

Simon Moinard

Bruno Tisseyre

Experiences with teaching precision agriculture through reverse pedagogy

Keywords: active-learning, competences, education, flipped classrooms, skills

Precision agriculture (PA) technologies have been widely developed in agriculture over recent decades. Therefore, understanding the challenges of PA is a major issue in the training of agricultural students. Reversal pedagogy (RP) is an efficient pedagogical approach used to make a group of students understand key concepts when teaching courses that deal with complex and multi-disciplinary issues. An RP approach was implemented in a two-week introductory teaching unit on PA for a Master's degree course in France. The method allowed the students to be able, in a short period of time to, i) understand the main concepts of PA, ii) produce structured and scientifically relevant teaching resources and iii) train other students. The results demonstrated that RP is an interesting approach for teaching PA at universities.

Introduction

Precision Agriculture (PA) uses advanced and emerging technologies, such as physical sensors, remote sensing, and geolocation, to improve the efficiency and productivity of agricultural practices. Therefore, understanding the issues of PA and the physical principles of key PA technologies is a major issue in the education of agricultural students. However, recent studies in several countries have shown that the majority of graduating agricultural students do not feel adequately trained in PA [START_REF] Bournaris | Current Skills of Students and Their Expected Future Training Needs on Precision Agriculture: Evidence from Euro-Mediterranean Higher Education Institutes[END_REF]. The teaching of PA is complex because it involves many disciplines (physics, computer science, economics, sociology, etc.) framed in a multidisciplinary approach. Consequently, many traditional pedagogical approaches are not necessarily well-suited to teaching PA with its complex systems and technologies. [START_REF] Mccubbins | Examining Student Perceptions of Their Experience in a TBL Formatted Capstone Course[END_REF] described how a Team Based Learning (TBL) approach was used to teach farm management to a group of undergraduate students. In the TBL approach, the traditional model of the classroom is reversed, students construct their own knowledge from resources provided to them and teachers can use class time to follow students through hands-on activities and projects that help the students to apply their knowledge and skills. It has been proven that students' involvement is higher and their results are better with a TBL approach [START_REF] Sisk | Team-Based Learning: Systematic Research Review[END_REF]. [START_REF] Mccubbins | Examining Student Perceptions of Their Experience in a TBL Formatted Capstone Course[END_REF] also demonstrated that this approach helps students to develop a deeper understanding of the complex concepts and could be used to teach a range of topics in agricultural education. The TBL approach has proven to be effective in a variety of fields, including engineering (Najdanovic-Visak, 2017), and can be particularly well suited to teaching PA. Reversal Pedagogy (RP) (Caillez and Hénin, 2017) is a special case of TBL in which students construct knowledge themselves and then teach it to other students. The purpose of this approach is to get a group of students understanding key challenges and concepts in a relatively short period of time when teaching courses deal with complex issues and multiple topics. The RP approach includes i) a step of familiarisation with the subject by reading documents, ii) a step of organisation and prioritisation of the content and, iii) a step of production of teaching resources. This approach has never been reported for teaching tertiary-level PA courses. The RP approach was implemented in a two-week introductory teaching unit of a master degree course entitled 'information and communication technologies in agriculture (AgroTIC)'. The objective of this paper is to evaluate the relevance of the RP approach to educate students on key PA issues and technologies.

Material and methods

General RP approach

The general pedagogical approach of this study was derived from the Reverse Pedagogy (RP) approach as described by [START_REF] Cailliez | La classe renversée : l'innovation pédagogique par le changement de posture (reverse classroom : pedagogical innovation through attitude change)[END_REF]. Students were brought to an active attitude inspired by flipped classrooms [START_REF] Divjak | Flipped classrooms in higher education during the COVID-19 pandemic: findings and future research recommendations[END_REF] and active-learning strategies [START_REF] Gleason | An active-learning strategies primer for achieving ability-based educational outcomes[END_REF]. Their objective was to design a training session for other students. The approach consisted of three main steps: i) Students acquired a first level of knowledge on the basis of activities and documents (documents, videos, experts, etc.) provided by teachers, ii) they designed a teaching unit including theoretical and practical activities and finally iii) they led a training session for other students.

Implementation of the RP approach

The RP approach has been implemented since 2016 and was evaluated in 2022 in a teaching unit for master degree students in France specialising in digital tools for agriculture. The unit is called AgroTIC and jointly taught between L'Institut Agro Montpellier and Bordeaux Science Agro. The teaching unit is an introduction to PA lasting 5 days over a two-week period. It was the first teaching unit of the semester in 2022 and 14 students attended. In parallel, student visits to companies that provide precision agriculture services were held. The objective for the students was to develop general skills about the main topics of PA and to understand some key concepts (Table 1). Three technical topics (geolocation, optical and physical properties of the soil) in relation with farming practices and one transversal topic on services and uses of digital tools in a farming context were targeted. The three steps of the RP approach were implemented in five consecutive sessions (Fig. 1). The first step (acquiring a minimum level of knowledge) was carried out over two sessions (90-100 min sessions). Session 1 (Fig. 1a) was an introduction with a presentation of the deliverables and a self-assessment of the students' skills. Students also expressed their views and representations on PA during a moving debate (Fig. 1a). Session 2 was dedicated to the exploration of the three technical topics by three groups of students. The concepts of the transversal topic about services and uses were addressed during the exploration of the three technical topics. The objective was to identify the key concepts and main resources on these topics with the support of the teachers (Fig. 1b). The second step of the RP approach (designing the teaching unit) was carried out in two further sessions. The objective of the third session (0.5 day) was to stimulate the students' creativity (Fig. 1c). Firstly, the students worked in groups on an initial design of the teaching unit. After a first full draft, students were stopped and invited to look at and to improve the work in progress from the other groups. The aim of this cross-fertilisation was to encourage the emergence of new ideas. Consequently, the resulting second drafts for each group contained ideas from the three different groups. At the end of the third session, the students all came together to define a common design for the training sessions. This led into a fourth session (run over three half-days) where each of the three groups designed in detail a training session related to one of the three topics (Fig. 1d). At the end of the fourth session, students were considered as "expert students''' on the specific topic they had worked on. They were considered as non-expert on the other two topics.

Table 1. List of the topics and the associated key concept to be learned by students The third and final step in the RP approach was implemented in a fifth session. (Fig. 1e). It took place on a farm where fields and farming equipment were at the students' disposal. The groups of students were mixed up so that each new group contained expert students and non-expert students for each technical topic. During the training session, each student was alternatively in the position of a "teacher" for the topic he/she was expert on and a "student" for the topics he/she were non-expert on.

Evaluation of the approach

The relevance of the proposed approach was assessed by measuring the students' progression on the key concepts. This progression was estimated by self-assessment of the students before and after the training. For each self-assessment, they estimated a level of expertise for each of the four topics (Table 1) and each key concept. They chose their level of expertise as either: 1) unaware of the existence of the tool/key concept, 2) aware of the existence of the tool/key concept, 3) understands how the tool/key concept works, 4) is able to explain how the tool/key concept works or, 5) is able to give recommendations in a professional context taking into account the key concept.

Results and discussion

During the introduction, students attended workshops where they were able to express and discuss their views on PA (Fig 2a). Each student was briefed with basic knowledge of PA. Following this, the students created a complete 'hands-on' training workshop for one of the topics (geolocation, optical properties, soil physical properties) to understand the main concepts and to manipulate the associated tools and technologies (Fig. 2b). For example, the students were able to handle optical proxy detection sensors or GNSS receivers with different correction systems. The key concepts were presented by the expert student of each topic using a large format printed poster. As an illustration, Figure 2c presents the poster made for the "optical properties" topic. Similar posters were produced for the other two topics. During the workshop, the expert students made a presentation and led a practical training session. Then they moderated a discussion with the other students and answered their questions. A teacher was present in each workshop and did complete or correct the expert student's answers when it was necessary. The RP approach allowed students to learn concepts that were meaningful for them and appeared to encourage a better long-term understanding. Students were also developing professional skills needed by agronomists, such as an ability to find information on their own, to present figures and posters precisely, to concisely deliver key messages and to lead a training program. It is likely that by using classical teaching, it would have been more difficult to teach all the main concept in this short period of time. It is also likely that it would have been complicated to train students on transversal topic like time management and summarising the explanation of complex concepts.

The students' self-assessment of their global capability for each of the three topics allowed them to estimate and monitor their progress over the five days of training. For each of the three topics, students felt that they had made progress, which was reflected in a shift to a higher level of understanding. As an illustration of this, Figure 3 presents the selfassessment results for the geolocation topic showing the cohorts shift in improved understanding of the topic. As can be seen in Figure 3, before the course, a majority of students were aware of the existence of geolocation tools but did not necessarily understand how they worked. After the course, most of them were able to explain how geolocation technologies work with 79% of the students self-assessing as being able to explain geolocation technology (category 4 or higher) and approximately a third of students were confident in their ability to recommend it (with an adapted correction service if necessary) in a professional context (category 5). It is worth noting that all the expert-students who designed the geolocation training workshop were in category 4 or 5. Similar results were obtained for the two other topics. After the training 71 % and 79 % of students self-assessed in category 4 or 5 respectively for "soil physical properties" and "optical properties" topics. Out of the 14 students, only two for the "soil physical properties" topic and one for the "optical properties" topic considered they had understood how the tools work but were not confident in explaining it to other students. For these three students, the five days of training might have been insufficient to master the key concepts studied. All the concepts introduced during this teaching unit were also developed during the following semester (in a more traditional setting) and allowed all students to ultimately master the topics. These results on student progression can be refined by studying the knowledge acquisition of the corresponding key concept for each topic. For most of the key concepts, the large majority of the students improved during the course and reached a minimum level of understanding (category 3 or higher) (Table 2). For a few key concepts, there was no progression made among the group of students. These concepts fell into two broad categories. Firstly, there were some technical key concepts that were just missed during the training sessions. For example, the "factors influencing the power autonomy of a sensor" were overlooked during the designing session (Fig. 1d), even if the teachers were being attentive, so no student expertise was developed and it was not taught within the delivered workshop. The second broad category involved key concepts related to the services and uses. These are complex and multidisciplinary concepts such as "A way to identify and classify the uses of digital tools on a farm", which are difficult to fully develop in a restricted timeframe. Interestingly, some students had their own ideas and self-trust about it before the course, but appeared to arrive at a decision that it is more complicated than they thought by the end of the course. This, in itself, demonstrates a much deeper appreciation and understanding of the delivery of PA services. Some technologies, such as mapping or wireless sensor networks, were omitted voluntarily during the teaching unit as they were developed in specific teaching units in the following semester. A noted advantage to the RP approach was an apparent higher level of student motivation as no absentees or late arrivals were observed (which is consistent with previous years as well). The students viewed this course as an opportunity to get to know each other and to know how to work efficiently in groups, which does help them in subsequent courses/semesters. Table 2. Change in the percentage of students self-assessing at level 3, 4 or 5 of knowledge acquisition before and after the teaching unit for each key concept. In this context, the reverse engineering pedagogy did have some specific requirements to be relevant for students. i) The teaching period had to be grouped in a two-week period as it was considered overly complicated to keep the motivation of students if the training was spread out over several months. ii) It required a multidisciplinary teaching staff. In the case presented above, nine teaching staff were involved, one professor who coordinated the course, three lecturers were involved in training the students on technical topics and a further five experts/teaching staff were required to provide technical advice to the students to enable them to deliver the workshops and field demonstrations. iii) Easy access to sensors and PA tools for students was necessary in order to let them design adapted practical exercises. These requirements do impose restrictions on the ability to apply the RP approach universally across the entire AgroTIC Masters course.

Conclusions

This introductory PA teaching unit has three purposes: i) to use the RP approach to train students about PA within a professional context, ii) to introduce multi-disciplinary PA concepts and, iii) to enable students to better know each other in order to create a group dynamic for the coming year. For these three objectives, reverse engineering pedagogy proved to be an effective and interesting method. In particular, the RP approach allowed the students to obtain a better overview and understanding of PA technologies by promoting a progression in the mastering of several key concepts.

  of trilateration positioning -The main sources of error with geolocation -The difference between relative and absolute positioning error -Accuracy of correction systems and their different uses in agriculture -The functioning of the correction systems and their limits in agriculture Optical property measurements in precision agriculture -The working principle of an optical sensor -The different levels of information on a crop using ground-based measurements or remote sensing -The definition and the meaning of an indicator such as NDVI -The field impact of using a passive or an active optical sensor -The uses allowed by the spatial, temporal and spectral resolution Measurement of soil physical properties in precision agriculture -The stages of the data acquisition chain -The influence of the metrological characteristics of a sensor (precision, accuracy) on the agronomic interpretation -The difference between direct and indirect measurement of a variable -Differences between physical variables (electrical capacitance, conductivity, resistivity) on soil knowledge -Factors influencing the power autonomy of a sensor Services and uses of digital tools for crop, water and soil monitoring -Factors in the adoption of digital solutions on a farm -The obstacles to the adoption of digital solutions on a farm -A way to identify and classify the uses of digital tools on a farm -The range of prices of the main types of digital services -The services that support the deployment of a digital tool

Figure 1 .

 1 Figure 1. Implementation of Reverse Pedagogy for teaching Precision Agriculture to agronomy students: (a) Introduction and evaluation of the group's initial perception of PA, (b) Exploration of the topics in thematic groups, (c) Conceptualisation of the design of the training session, (d) Design of a training session for each topic (geolocation, optical measurements and soil physical measurements) and (e) Delivery of the training session.

Figure 2 .

 2 Figure 2. Illustrations of activities and productions carried out by students during precision agriculture teaching unit: a) students stand according to their agreement (front row) or disagreement (back row) with a statement made during the moving debate, b) students learned how to use a proximal sensor to understand optical properties and key associated concepts during presentation day and c) an example of a poster made by students for teaching to other students the key concepts of "optical properties" in PA.

Figure 3 .

 3 Figure 3. Students' self-assessment of their global capability concerning geolocation in Precision Agriculture before and after the teaching unit.