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Proximity Spin–Orbit Torque on a Two-Dimensional Magnet within van der Waals Heterostructure: Current-Driven Antiferromagnet-to-Ferromagnet Reversible Nonequilibrium Phase Transition in Bilayer CrI3

  • Kapildeb Dolui
    Kapildeb Dolui
    Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716, United States
  • Marko D. Petrović
    Marko D. Petrović
    Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716, United States
  • Klaus Zollner
    Klaus Zollner
    Institute for Theoretical Physics, University of Regensburg, Regensburg 93040, Germany
  • Petr Plecháč
    Petr Plecháč
    Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716, United States
  • Jaroslav Fabian
    Jaroslav Fabian
    Institute for Theoretical Physics, University of Regensburg, Regensburg 93040, Germany
  • , and 
  • Branislav K. Nikolić*
    Branislav K. Nikolić
    Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716, United States
    *Email: [email protected]
Cite this: Nano Lett. 2020, 20, 4, 2288–2295
Publication Date (Web):March 4, 2020
https://doi.org/10.1021/acs.nanolett.9b04556
Copyright © 2020 American Chemical Society
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Abstract

The recently discovered two-dimensional magnetic insulator CrI3 is an intriguing case for basic research and spintronic applications since it is a ferromagnet in the bulk but an antiferromagnet in bilayer form, with its magnetic ordering amenable to external manipulations. Using the first-principles quantum transport approach, we predict that injecting unpolarized charge current parallel to the interface of the bilayer-CrI3/monolayer-TaSe2 van der Waals (vdW) heterostructure will induce spin–orbit torque and thereby drive the dynamics of magnetization on the first monolayer of CrI3 in direct contact with TaSe2. By combining the calculated complex angular dependence of spin–orbit torque with the Landau-Lifshitz-Gilbert equation for classical dynamics of magnetization, we demonstrate that current pulses can switch the direction of magnetization on the first monolayer to become parallel to that of the second monolayer, thereby converting CrI3 from antiferromagnet to ferromagnet while not requiring any external magnetic field. We explain the mechanism of this reversible current-driven nonequilibrium phase transition by showing that first monolayer of CrI3 carries current due to evanescent wave functions injected by metallic transition metal dichalcogenide TaSe2, while concurrently acquiring strong spin–orbit coupling via such a proximity effect, whereas the second monolayer of CrI3 remains insulating. The transition can be detected by passing vertical read current through the vdW heterostructure, encapsulated by a bilayer of hexagonal boron nitride and sandwiched between graphite electrodes, where we find a tunneling magnetoresistance of ≃240%.

Introduction

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The recent discovery of two-dimensional (2D) magnets derived from layered van der Waals (vdW) materials(1,2) has opened new avenues for basic research on low-dimensional magnetism(3,4) and potential applications in spintronics.(5−9) Their magnetic phases can substantially differ from those in conventional bulk magnetic materials due to large structural anisotropy, which makes different signs and magnitudes of intralayer Jintra and interlayer Jinter exchange coupling between localized magnetic moments possible. For example, Jintra and Jinter are ferromagnetic and antiferromagetic, respectively, between magnetic moments on Cr atoms within the bilayer of CrI3, which eventually becomes an antiferromagnetic insulator with Néel temperature TN ≃ 61 K.(2) Hence, antiferromagnet spins have an opposite orientation in the two monolayers, whereas monolayer, trilayer, and bulk CrI3 are ferromagnetic. Thus, the bilayer of CrI3 can also be viewed as two monolayer ferromagnets that are antiferromagnetically coupled to each other. The monolayer CrI3 circumvents the Mermin-Wagner theorem,(10) where thermal fluctuations destroy long-range magnetic order in 2D, by exhibiting strong uniaxial perpendicular magnetic anisotropy (PMA) that removes rotational invariance and effectively makes it a realization of the Ising model.(3) The PMA is also required for high-density device applications.
The layer and stacking order(11,12) dependence of electronic and spin structure as a new knob, together with possibilities for external manipulation via gating, straining, and coupling to other 2D materials within vdW heterostructures, allows for dramatic changes of magnetic ordering of 2D magnets, which is not possible with conventional bulk magnetic materials. For example, very recent experiments(13−15) have demonstrated antiferromagnet (AFM) to ferromagnet (FM) phase transition in the bilayer of CrI3 by applying an external electric field via gate voltage or by electrostatic doping. While these effects offer potential building blocks(16) for an ultralow-dissipation nonvolatile memory, at present, they employ cumbersome external magnetic fields that cannot be generated on the nanoscale as required for integration with other elements of a circuit. Furthermore, to read the change of magnetic state of such a device within a circuit, it is still required to pass a current through it,(17) as demonstrated recently using unconventional magnetic tunnel junctions where bilayer CrI3 functions as the spin-filter sandwiched between metallic electrodes (such as graphene) with current flowing perpendicular to the interface.(18−21)
An alternative for magnetization switching is to inject a current through 2D magnet and drive its magnetization dynamics via spin torque, as exemplified by very recent experiments(8,9) on spin–orbit torque (SOT)(22,23) driven magnetization dynamics in Fe3GeTe2/Pt heterostructures. However, the layers employed in these experiments were much thicker than the ultimate limit envisaged using vdW heterostructures composed of just a few atomically thin layers. They are flat and ensure highly transparent interfaces, so that drastically smaller energy consumption per switching cycle can be achieved. These experiments have also relied on Fe3GeTe2 being a metallic vdW ferromagnet,(3) so that the CrI3 bilayer with an energy gap is at first sight not suitable for SOT-operated devices.
Here, we employ first-principle quantum transport framework, which combines(24−26) nonequilibrium Green functions (NEGFs)(27) for two-terminal devices with noncollinear density functional theory (ncDFT) calculations,(28,29) to predict that the AFM–FM nonequilibrium phase transition can be induced by SOT in the bilayer-CrI3/monolayer-TaSe2 vdW lateral heterostructure depicted in Figure 1 where unpolarized charge current is injected parallel to the interface. The monolayer of metallic TaSe2 is chosen in 1H phase for which lattice mismatch between TaSe2 and CrI3 is as small as 0.1%, while inversion symmetry of TaSe2 is broken to create large spin–orbit coupling (SOC).

Figure 1

Figure 1. Schematic view of the CrI3/TaSe2 vdW heterostructure consisting of an insulating antiferromagnetic bilayer of CrI3 and a nonmagnetic metallic monolayer TMD TaSe2. The unpolarized charge current is injected parallel to the interface by a small applied bias voltage Vb between the left and right macroscopic reservoirs. The current flows through the monolayer of TaSe2 as well as through the first monolayer (see the Supporting Information) of CrI3 due to evanescent wave functions injected into it by TaSe2. The unit vectors of magnetizations on the two CrI3 monolayers are denoted by m1 and m2. The heterostructure is assumed to be infinite in the xy-plane.

Both conventional spin-transfer torque (in the absence of SOC and in geometries with two FM layers with noncollinear magnetizations(24,30)) and SOT (in geometries with one FM layer but requiring interfacial or bulk SOC effects(24−26,31,32)) can be described microscopically and independently of a particular physical mechanism(22,26) as a consequence of the interaction between current-driven (CD) nonequilibrium spin density(33−35) of conduction electrons SCD(r) and a nonzero exchange-correlation (XC) magnetic field BXC(r)(28,29) present in equilibrium. Their cross product, SCD(r) × BXC(r), is local torque at some point in space r, so that the total torque is obtained by integration(24,25,30)(1)While BXC(r) is nonzero in both the monolayer and bilayer of CrI3 due to long-range magnetic ordering, SCD(r) appears only on the monolayer of CrI3 that is in direct contact with the monolayer of TaSe2, as demonstrated by Figure 2. This is due to the proximity effect where evanescent wave functions from TaSe2 penetrate [Figures 2 and 3; also see the Supporting Information] up to the first monolayer of CrI3 to make it a current carrier (see the Supporting Information for spatial profiles of local charge current density), while also bringing(36) SOC from TaSe2 to ensure that SCD(r) is not collinear to BXC(r). The giant SOC hosted by TaSe2 itself due to inversion symmetry breaking in ultrathin layers of transition metal dichalcogenides (TMDs)(37,38) is confirmed by large SCD(r) within the spatial region of the TaSe2 monolayer in Figure 2.

Figure 2

Figure 2. Current-driven nonequilibrium spin density SCD = (SCDx,SCDy,SCDz) in the linear-response regime within a bilayer-CrI3/monolayer-TaSe2 vdW heterostructure for (a) m1; (b) m1ŷ; (c) m1. Vertical dashed lines indicate the position of each atomic plane. The area of the common rectangular supercell of the vdW heterostructure in Figure 1 is denoted by , where a = 6.85 Å is the lattice constant of bulk CrI3. Shaded green areas represent rescaled SCDy in the spatial region of the first monolayer of CrI3, which is in direct contact with the monolayer of TaSe2.

Figure 3

Figure 3. First-principles-computed bands of the bilayer-CrI3/monolayer-TaSe2 vdW heterostructure with SOC turned on. We use Hubbard U = 2.0 eV and U = 0 eV for (a) and (c), respectively, in ncDFT + U calculations where the color corresponds to the equilibrium spin expectation value, (σ̂z is the Pauli matrix) in the eigenstates |Ψk⟩ of ncDFT Hamiltonian (for the other two components, Seqx and Seqy, and their texture in k-space, see the Supporting Information). Panels (b) and (d) show the same band structures as (a) and (c), respectively, but with different colored symbols corresponding to projections onto different monolayers.

The SOT vector in eq 1, with its complex angular dependence [Figure 4] on the direction of magnetization (along the unit vector m1 in Figure 1) of the first monolayer of CrI3, is combined in a multiscale fashion(30,39) with the classical dynamics of magnetization governed by the Landau-Lifshitz-Gilbert (LLG) equation to demonstrate reversible switching of m1 [Figure 5b and the accompanying movie in the Supporting Information] from the −z to +z direction by current pulses and, thereby, transition from the AFM to FM phase of the CrI3 bilayer. The dynamics of m1 can be detected by passing the vertical read current along the z-axis,(17) where we compute the tunneling magnetoresistance (TMR) of ≃240% [Figure 6] due to the AFM–FM transition of the CrI3 bilayer. For such a scheme, we assume that the bilayer-CrI3/monolayer-TaSe2 vdW heterostructure is sandwiched between two metallic semi-infinite graphite electrodes along the z-axis with hexagonal BN (hBN) bilayers inserted between the leads and vdW heterostructure [inset of Figure 6].

Figure 4

Figure 4. Azimuthal (θ) and polar (φ) angle dependence of the magnitude of SOT component |To| (odd in m1) for different orientations of the magnetization m1 = (sin θ cosφ, sin θ sinφ, cos θ) on the first monolayer of CrI3. The magnetization is rotated within the (a) xz-plane; (b) xy-plane; (c) yz-plane. Solid blue lines are fit to NEGF + ncDFT-computed SOT values (red dots) using eq 6 with fitting parameters from Table 1.

Figure 5

Figure 5. Classical dynamics of magnetizations (a, b) m1(t) on the first monolayer of CrI3 and (c, d) m2(t) on the second monolayer of CrI3. The two magnetizations are exchange coupled [eq 7], while according to Figure 2 only m1(t) experiences SOT from Figure 4. The dynamics is obtained by solving two coupled LLG equations [eq 8] for m1(t) and m2(t). The bias voltage Vb = 0.2 V is dc in (a) and (c), while in (b) and (d) we use a sequence of short rectangular voltage pulses of the same amplitude with pulse duration δtON = 1.0 ps followed by a pause of δtOFF = 100 ps during which no voltage is applied. A movie animating m1(t) in panel (b) as well as m2(t) in panel (d) is provided in the Supporting Information.

Figure 6

Figure 6. TMR vs angle θ between magnetizations m1 and m2 on two monolayers of CrI3 in Figure 1 for vertical read current(17) flowing perpendicularly (i.e., along the z-axis in Figure 1) through the bilayer-CrI3/monolayer-TaSe2 vdW heterostructure. The heterostructure is sandwiched between two semi-infinite graphite leads with (red squares) and without (blue circles) the bilayer of hBN inserted between the leads and the vdW heterostructure, as illustrated in the inset.

Methodology

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We employ the interface builder in the QuantumATK(40) package to construct a unit cell of the vdW heterostructure in Figure 1 while starting from experimental lattice constants of CrI3 and TaSe2 layers. In order to determine the interlayer distance between CrI3 and TaSe2, we perform DFT calculations with Perdew–Burke–Ernzerhof (PBE) parametrization of the generalized gradient approximation (GGA) for the XC functional, including Grimme D2(41) vdW corrections.
In addition, we employ ncDFT + U calculations using the Quantum ESPRESSO(42) package to examine how nonzero(43) Hubbard U affects the bands of the vdW heterostructures since nonzero U has been utilized before(44,45) as a cure for the band gap problem in CrI3.(46) For this purpose, we use PBE parametrization of GGA for the XC functional; fully relativistic pseudopotentials with the projector augmented wave method(47) for describing electron–core interactions; energy cutoff of 550 Ry for the plane wave basis set; k-point sampling of 30 × 30 × 1 for self-consistent calculations. A comparison of the cases with U = 0 and U = 2.0 eV in Figure 3 shows that U ≠ 0 barely changes the band structure within the energy window of ±1 eV, with only the conduction band of CrI3 experiencing a shift in energy of about ≃0.15 eV [Figure 3b,d] at the Γ point and energies around 0.5 eV. This is because Hubbard U acts(44,45) on d-orbitals of Cr whose bands are higher in energy. States near the Fermi level EEF = 0, which are responsible for charge and spin transport properties in the linear-response transport regime, are formed by a TaSe2 monolayer [Figure 3b,d]. Due to the short-range of the proximity effect, only the bands of the first monolayer of CrI3 hybridize with the bands of TaSe2, such as spin-down bands near the K valley at energies of 0.5–0.8 eV [Figure 3a,b].
The NEGF + ncDFT formalism,(24,25,30) which combines self-consistent Hamiltonian from ncDFT calculations (with U = 0 based on Figure 3) with a nonequilibrium density matrix and current calculations from NEGF calculations, makes it possible to compute spin torque in an arbitrary device geometry at small or finite bias voltage. The single-particle Kohn–Sham (KS) Hamiltonian in ncDFT is given by(2)where σ = (σ̂x, σ̂y, σ̂z) is the vector of the Pauli matrices; Vext(r), VH(r), and VXC(r) = δEXC[n(r),μ(r)]/δn(r) are the external, Hartree, and XC potentials, respectively, and the XC magnetic field, BXC(r) = δEXC[n(r),μ(r)]/δμ(r), is the functional derivative with respect to the vector magnetization density μ(r). The extension of DFT to the case of spin-polarized systems is formally derived in terms of μ(r) and total electron density n(r), where in collinear DFT μ(r) points in the same direction at all points in space, while in ncDFT μ(r) can point in an arbitrary direction.(28,29)
The heterostructure in Figure 1 is split into the central region and left (L) and right (R) semi-infinite leads, all of which are composed of the same CrI3/TaSe2 trilayer. The self-energies of the leads ΣL,R(E) and the Hamiltonian ĤKS of the central region are obtained from ncDFT calculations within the QuantumATK package(40) using PBE parametrization of GGA for the XC functional; norm-conserving fully relativistic pseudopotentials of the type SG15-SO(40,48) for describing electron–core interactions; the SG15 (medium) numerical linear combination of atomic orbitals (LCAO) basis set.(48) Periodic boundary conditions are employed in the plane perpendicular to the transport direction with grids of 1 × 101 k-point (lateral device setup in Figure 1 for SOT calculations) and 25 × 25 k-point (vertical device setup in the inset of Figure 6 for TMR calculations). The energy mesh cutoff for the real-space grid is chosen as 100 hartree.
The lesser Green function (GF), G<(E) = iG(E)[fL(E)ΓL(E) + fR(E)ΓR(E)]G(E), of NEGF formalism makes it possible to construct the nonequilibrium density matrix(27)(3)in the steady state and elastic transport regime. Here, G = [EΛHKSΣL(E,VL) – ΣR(E,VR)]−1 is the retarded GF; fL,R(E) = f(EeVL,R) is the shifted Fermi function of the macroscopic reservoirs into which semi-infinite leads terminate; Vb = VLVR is the applied bias voltage between them; ΓL,R(E) = i[ΣL,R(E) – ΣL,R(E)] is the level broadening matrix. For the lateral heterostructure [Figure 1], all matrices, HKS, G, G<, ΓL,R and ρneq, depend on ky, while for vertical the heterostructure [inset of Figure 6], they depend on (kx,ky). Due to the nonorthogonality of the LCAO basis set |ϕn⟩, we also use the overlap matrix Λ composed of elements ⟨ϕij⟩.
The CD part of the nonequilibrium density matrix, ρCD(ky) = ρneq(ky) – ρeq(ky), is obtained by subtracting the equilibrium density matrix ρeq(ky) for VL = VR. This yields(4)and SOT(5)which we compute by performing trace in the LCAO basis [instead of in real space as in eq 1], and additional integration over the one-dimensional Brillouin zone (BZ) of length ΩBZ is performed.

Nonequilibrium Spin Density

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The xy-plane averaged SCD is plotted in Figure 2 for three representative orientations of the magnetization m1∥{, ŷ, } on the first monolayer of CrI3. The nonequilibrium spin density is zero on the second monolayer of CrI3, which confirms that evanescent wave functions originating from the metallic TaSe2 monolayer and the spin–orbit proximity effect carried by them decay exponentially fast, so they are able to reach only the first monolayer of CrI3. Independently of the orientation of m1, the magnitude of SCD within the first monolayer of CrI3 is mainly dominated by its y-component, which is an order of magnitude smaller than SCD within TaSe2. Concurrently, the magnetic proximity effect from CrI3 induces a small magnetization into the monolayer of TaSe2 with magnetic moments on Ta and Se atoms being 0.008 and 0.001 μB, respectively, where μB is the Bohr magneton. In comparison, we compute magnetic moments on Cr and I atoms as μCr = 3.43 μB and μI = 0.14 μB, respectively. The component SCDy within the TaSe2 monolayer is insensitive to m1, while SCDx remains negligible.
Unlike the surface of the topological insulator(35) or heavy metals where SCD is confined to the plane in accord with the phenomenology of the standard inverse spin-galvanic (or Edelstein) effect,(33,34) TaSe2 exhibits large out-of-plane componentSCDz ,which is sensitive to the orientation of m1 and it is highly sought for SOT-operated device applications.(49) This can be traced to the large Seqz component of the spin texture in k-space for an isolated monolayer TaSe2, which remains large when TaSe2 is covered by the bilayer CrI3 while also acquiring nontrivial in-plane components Seqx and Seqy (see the Supporting Information for more details).

Angular Dependence of SOT

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The SOT vector can be decomposed,(31,50)T = Te + To, into odd (o) and even (e) components with respect to the magnetization m1. They can be computed directly from eqs 4 and 5 by using the respective components of the nonequilibrium density matrix, ρCD = ρCDe + ρCDo, as introduced in refs (24and51). Due to the absence of the bulk in the case of the monolayer TaSe2, the vdW heterostructure in Figure 1 does not generate vertical spin Hall current along the z-axis as one of the mechanisms for Te. Other interfacially based mechanisms(26) for Te ≠ 0 require the backscattering of electrons,(24,52−54) which is absent in the ballistic transport regime we assume, so we find Te → 0.
The nonzero To component, computed from NEGF + ncDFT formalism as dots in Figure 4, can be fitted by a function defined as an infinite series(50)(6)assuming that current flows along the x-axis as in Figure 1. Note that other expansions, such as in terms of orthonormal vector spherical harmonics, can also be employed to define the fitting function.(26) Here, τnαo and τnβo are the fitting parameters, and p is the unit vector along the reference direction set by current-induced nonequilibrium spin density, such that To = 0 when m1p. In simple systems, like the Rashba spin-split 2D electron gas(33) or metallic surface of the topological insulator(35) in contact with the FM layer, pŷ (assuming injected current along ) is determined by symmetry arguments.(25) However, for more complicated systems, it has to be calculated, and we find p ≡ (θ = 88°, φ = 98°) instead of often naı̈vely assumed p ≡ (θ = 90°, φ = 90°)∥ŷ. The lowest order term τo(p × m1) in eq 6 is conventional field-like torque,(22) while higher terms can have properties of both field-like and damping-like torque(26) [the lowest order term τem1 × (p × m1) in the expansion of Te is conventional damping-like torque(50)]. The value of τo, together with other non-negligible parameters in eq 6, is given in Table 1.
Table 1. Non-negligible Coefficients (in units of 10–4 eVb/□) in the Expansion of To in Equation 6 Are Obtained by Fitting (Solid Lines) NEGF + ncDFT-Computed Angular Dependence of SOT (Dots) in Figure 4 for Bilayer-CrI3/Monolayer-TaSe2 vdW Heterostructure
p(θ, φ)τoτoτoτoτoτo
(88°, 98°)77.2217.32–30.3213.54–9.19–6.88

SOT-Driven Classical Dynamics of Magnetization

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The effective anisotropic classical Heisenberg model(3) for magnetic moments m1 and m2 on Cr atoms within two monolayers of CrI3 in Figure 1 is given by(7)where the values for J12 = −0.05 meV, as the interlayer AFM exchange coupling, and A = 1.5 meV, as the PMA constant in the presence of TaSe2 monolayer, are extracted from ncDFT calculations. They are close to the corresponding values obtained for the isolated CrI3 bilayer in previous ncDFT calculations.(11,55)
We simulate the classical dynamics m1(t) by solving the LLG equation(8)where γ is the gyromagnetic ratio; is the effective magnetic field due to interactions in the Hamiltonian in eq 7; λ1 is the Gilbert damping parameter; SOT at an arbitrary direction of m1 is given by eq 6 with parameters in Table 1. The LLG equation for m2 is the same as eq 8, but with and To ≡ 0 because no current flows through the second monolayer of CrI3.
The computed trajectories m1(t) are plotted in Figure 5a for dc bias voltage as well as for rectangular voltage pulses in Figure 5b. The trajectories m2(t) are much simpler, m2z(t) ≈ 1 while m2x(t) and m2y(t) perform small oscillations around zero, as shown in Figure 5c,d for dc bias voltage or rectangular voltage pulses, respectively. The time evolutions m1(t) and m2(t) are also animated in the movie provided in the Supporting Information. For unpolarized charge current injected by dc bias, magnetization m1 switches from being antiparallel to m2 to a noncollinear direction within the yz-plane. The nonequilibrium and noncollinear configuration of m1 and m2 will go to a stable FM phase (both magnetizations pointing along the positive z-axis) when Vb is turned off and the system goes back to equilibrium. On the other hand, using voltage pulses leads to AFM–FM transition with reversal from m1∥–ẑ to m1∥+ẑ while magnetization of the second layer remains m2∥+ẑ. Such current-induced FM phase is stable in-between two pulses, on the proviso that A > |J12| in eq 7, and can be reversed back to the AFM phase by the next pulse [Figure 5b and movie in the Supporting Information]. We assume different Gilbert damping parameters λ1 = 0.01 > λ2 = 0.0001 on two monolayers of CrI3 due to the presence of the TaSe2 monolayer, but the actual value λ1 on the first monolayer of CrI3 is likely smaller. Thus, we anticipate that the time needed to stabilize the FM phase would be on the order of ∼1 ns instead of ∼100 ps in Figure 5 (where λ1 was tuned for such numerical convenience).
Since To forces precession of magnetization around the axis defined by p, magnetization reversal in the voltage pulse setup is achieved by fine-tuning the pulse duration δtON to half of the period of that precession. The Gilbert damping term, λ1m1 × dm1/dt, does not play a role in this type of switching, although λ1 together with the PMA constant is critical to stabilize the FM phase after the pulse is switched off, as shown in the movie in the Supporting Information. For instance, when Gilbert damping is set to zero, the magnetizations in both monolayers never fully align with the z-axis and instead continue to precess around it which renders the FM phase unstable. Note that more detailed LLG simulations would be required to simulate more than two magnetic moments and their inhomogeneous switching in a particular device geometry, as often observed experimentally in ferromagnet/heavy-metal heterostructures,(56) but our two-terminal device is homogeneous and translationally invariant within the xy-plane in Figure 1. Also, in the presence of disorder and thereby induced voltage drop across the central region,(25,26,53) we expect that Te would become nonzero(54) and contribute to switching.

TMR as a Probe of AFM–FM Transition

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Finally, in analogy with the experiments(17) where SOT-driven magnetization switching has been probed by passing additional vertical read current through SOT devices operated by lateral current, we investigate the angular dependence of TMR for vertical current assumed to be injected between semi-infinite graphite leads along the z-axis sandwiching bilayer-CrI3/monolayer-TaSe2 [see inset in Figure 6 for illustration]. We define angular dependence of TMR as TMR(θ) = [R(θ) – R(0)]/R(0), where R(0) is the resistance of the FM phase with m1ẑ∥m2 and R(θ) is the resistance for angle θ between them. We compute as the inverse of two-terminal conductance expressed in terms of the zero-bias transmission function via the standard Landauer-Büttiker formula.(27) The total transmission function per unit interfacial area is obtained by the integration of the ky-resolved transmission function over the BZ, , akin to eq 5. Thus, R(θ = 180°) corresponds to the AFM phase. Note that TMR(θ = 180°) recovers the conventional definition of TMR using only parallel and antiparallel configurations of the magnetizations. In Figure 6, we obtain TMR(θ = 180°) ≃ 240% when using additional hBN bilayers inserted between graphite leads and the vdW heterostructure. When hBN is removed, TMR drops to TMR(θ = 180°) ≃ 40%, while exhibiting a peculiar change of sign for the angles between θ = 0° and θ = 180° in accord with the experimental observation reported in ref (18) of few-layer-graphene/bilayer-CrI3/few-layer-graphene junctions.

Supporting Information

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The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.nanolett.9b04556.

  • Movie, accompanying Figure 5a,b, animates the time evolution of magnetizations, m1(t) and m2(t) in Figure 1, driven by a sequence of rectangular voltage pulses (MP4)

  • Additional details for Figure 2, such as the spatial profile of local current density on different monolayers of the bilayer-CrI3/monolayer-TaSe2 vdW heterostructure, as well as for Figure 3, such as spin textures in the bilayer-CrI3/monolayer-TaSe2 vdW heterostructure vs spin textures in an isolated monolayer-TaSe2 (PDF)

The authors declare no competing financial interest.

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Acknowledgments

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K.D. and B.K.N. were supported by DOE Grant No. DE-SC0016380. M.D.P. and P.P. were supported by ARO MURI Award No. W911NF-14-0247. K.Z. and J.F. were supported the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) SFB 1277 (Project-ID 314695032), DFG SPP 1666, and the European Unions Horizon 2020 research and innovation program under Grant No. 785219. The supercomputing time was provided by XSEDE, which is supported by NSF Grant No. ACI-1053575.

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  • Abstract

    Figure 1

    Figure 1. Schematic view of the CrI3/TaSe2 vdW heterostructure consisting of an insulating antiferromagnetic bilayer of CrI3 and a nonmagnetic metallic monolayer TMD TaSe2. The unpolarized charge current is injected parallel to the interface by a small applied bias voltage Vb between the left and right macroscopic reservoirs. The current flows through the monolayer of TaSe2 as well as through the first monolayer (see the Supporting Information) of CrI3 due to evanescent wave functions injected into it by TaSe2. The unit vectors of magnetizations on the two CrI3 monolayers are denoted by m1 and m2. The heterostructure is assumed to be infinite in the xy-plane.

    Figure 2

    Figure 2. Current-driven nonequilibrium spin density SCD = (SCDx,SCDy,SCDz) in the linear-response regime within a bilayer-CrI3/monolayer-TaSe2 vdW heterostructure for (a) m1; (b) m1ŷ; (c) m1. Vertical dashed lines indicate the position of each atomic plane. The area of the common rectangular supercell of the vdW heterostructure in Figure 1 is denoted by , where a = 6.85 Å is the lattice constant of bulk CrI3. Shaded green areas represent rescaled SCDy in the spatial region of the first monolayer of CrI3, which is in direct contact with the monolayer of TaSe2.

    Figure 3

    Figure 3. First-principles-computed bands of the bilayer-CrI3/monolayer-TaSe2 vdW heterostructure with SOC turned on. We use Hubbard U = 2.0 eV and U = 0 eV for (a) and (c), respectively, in ncDFT + U calculations where the color corresponds to the equilibrium spin expectation value, (σ̂z is the Pauli matrix) in the eigenstates |Ψk⟩ of ncDFT Hamiltonian (for the other two components, Seqx and Seqy, and their texture in k-space, see the Supporting Information). Panels (b) and (d) show the same band structures as (a) and (c), respectively, but with different colored symbols corresponding to projections onto different monolayers.

    Figure 4

    Figure 4. Azimuthal (θ) and polar (φ) angle dependence of the magnitude of SOT component |To| (odd in m1) for different orientations of the magnetization m1 = (sin θ cosφ, sin θ sinφ, cos θ) on the first monolayer of CrI3. The magnetization is rotated within the (a) xz-plane; (b) xy-plane; (c) yz-plane. Solid blue lines are fit to NEGF + ncDFT-computed SOT values (red dots) using eq 6 with fitting parameters from Table 1.

    Figure 5

    Figure 5. Classical dynamics of magnetizations (a, b) m1(t) on the first monolayer of CrI3 and (c, d) m2(t) on the second monolayer of CrI3. The two magnetizations are exchange coupled [eq 7], while according to Figure 2 only m1(t) experiences SOT from Figure 4. The dynamics is obtained by solving two coupled LLG equations [eq 8] for m1(t) and m2(t). The bias voltage Vb = 0.2 V is dc in (a) and (c), while in (b) and (d) we use a sequence of short rectangular voltage pulses of the same amplitude with pulse duration δtON = 1.0 ps followed by a pause of δtOFF = 100 ps during which no voltage is applied. A movie animating m1(t) in panel (b) as well as m2(t) in panel (d) is provided in the Supporting Information.

    Figure 6

    Figure 6. TMR vs angle θ between magnetizations m1 and m2 on two monolayers of CrI3 in Figure 1 for vertical read current(17) flowing perpendicularly (i.e., along the z-axis in Figure 1) through the bilayer-CrI3/monolayer-TaSe2 vdW heterostructure. The heterostructure is sandwiched between two semi-infinite graphite leads with (red squares) and without (blue circles) the bilayer of hBN inserted between the leads and the vdW heterostructure, as illustrated in the inset.

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    The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.nanolett.9b04556.

    • Movie, accompanying Figure 5a,b, animates the time evolution of magnetizations, m1(t) and m2(t) in Figure 1, driven by a sequence of rectangular voltage pulses (MP4)

    • Additional details for Figure 2, such as the spatial profile of local current density on different monolayers of the bilayer-CrI3/monolayer-TaSe2 vdW heterostructure, as well as for Figure 3, such as spin textures in the bilayer-CrI3/monolayer-TaSe2 vdW heterostructure vs spin textures in an isolated monolayer-TaSe2 (PDF)


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