Spin-modulated torque waves in ferrimagnetic tunnel junctions

P. Merodio, A. Kalitsov, H. Béa, V. Baltz, and M. Chshiev

Phys. Rev. B 90, 224422 – Published 24 December 2014

Abstract

Spin-transfer torque (STT) in tunnel junctions with ferromagnetic leads is one of the essential underlying phenomena of modern spintronics. Here, we present a theoretical study of STT in ferrimagnet- (FI-) based tunnel junctions where two FI metal electrodes are separated by a thin nonmagnetic insulating barrier. We show that electronic structure parameters, such as bandwidths and exchange splittings of the FI leads strongly influence STT. In particular, the STT spatial distribution within the leads shows a striking spin-modulated wavelike behavior resulting from the interplay between the exchange splittings of the two FI sublattices. Additionally, we identify the fundamental parameter for quantifying STT characteristic lengths in FI metals, which will also be accessible to experiments, for instance, by ferromagnetic resonance and spin pumping measurements.

Received 22 April 2014
Revised 4 November 2014

DOI:https://doi.org/10.1103/PhysRevB.90.224422

Authors & Affiliations

P. Merodio^1,*, A. Kalitsov^1,2, H. Béa¹, V. Baltz¹, and M. Chshiev^1,†

¹Univ. Grenoble Alpes, INAC-SPINTEC, F-38000 Grenoble, France; CNRS, SPINTEC, F-38000 Grenoble, France; and CEA, INAC-SPINTEC, F-38000 Grenoble, France
²MINT Center, University of Alabama, P.O. Box 870209, Tuscaloosa, Alabama 35487-0209, USA

^*pablomerodio@gmail.com
^†mair.chshiev@cea.fr

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Vol. 90, Iss. 22 — 1 December 2014

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Images

Figure 1
Schemes adapted from Ref. [3] for the specific case of FI leads. (Top) Schematic of the FI-MTJ consisting of left and right semi-infinite FI leads separated by a nonmagnetic barrier of $N_{b}$ atomic layers. The magnetization $M^{'}$ of the right lead points along the $z$ direction. The nonequilibrium on-site torques are represented by the in-plane $(T_{| |})$ and out-of-plane $(T_{⊥})$ components, and the spin current densities are also indicated $(I)$ . (Middle) Schematic of sublattices $A$ and $B$ in the FI leads with different spin splittings $Δ_{A}$ and $Δ_{B}$ corresponding to up and down localized spins, respectively. The magnetization $M$ of the left lead is parallel to the $FI / B$ interface and is rotated by an angle θ around the $y$ axis. The greek primed and unprimed letters denote atomic sites in the right and left leads, respectively, and the latin letters denote the sites in the barrier. (Bottom) Schematic of the bands and the potential profile where the 1D densities of states for up and down spins are split by $δ = Δ_{A} - Δ_{B}$ and the lower and upper bands for a given spin are separated by a gap of $Δ_{A} + Δ_{B}$ . $ɛ_{B}$ is the spin-independent on-site energy in the barrier, and $V$ is the potential applied through the junction. The lower dashed line indicates the Fermi level at equilibrium, set at 0 eV in all regions, when no voltage is applied.
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Figure 2
Spatial distribution of the in-plane $(T_{| |})$ and out-of-plane $(T_{⊥})$ nonequilibrium torque components and local spin-density components $(S_{⊥}$ and $S_{| |})$ for a 1D system; $θ = π / 2$ , $Δ_{A} = 0.5, Δ_{B} = 0.25 eV$ , and $V = 0.1 V$ . The equilibrium torques and local spin densities were subtracted: $T_{| |} (0 V)$ and $S_{⊥} (0 V)$ equal zero in contrast to $T_{⊥} (0 V)$ and $S_{| |} (0 V)$ that result from interlayer equilibrium RKKY interactions. The red and blue curves (symbols) refer to the right- (left-) hand ordinates.
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Figure 3
For six different 1D FI leads pairs, $1 / δ$ dependence of the in-plane torque component period $(P)$ for $θ = π / 2$ and $V = 0.1 V$ . The double symbols at 4 and $10 e V^{- 1}$ represent FI electrodes with unequal spin splittings $Δ_{A}, Δ_{B}$ but the same $δ = Δ_{A} - Δ_{B}$ . The spin splittings take the following values ${Δ_{A}, Δ_{B}} = {0.5, 0.1; 0.5, 0.25; 0.6, 0.35; 0.5, 0.4; 0.4, 0.3; 0.5, 0.45}$ . Upper (lower) insets: spatial distribution of the parallel torque wave for $1 / δ = 4 e V^{- 1} (1 / δ = 10 e V^{- 1})$ . Τhe dashed line is a linear fit to the data.
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Figure 4
(a) Dependence with the hopping parameter $(t)$ of the in-plane torque component waves period $P$ for $θ = π / 2$ and $V = 0.1 V$ . (b) Corresponding dependence with $t^{5}$ of the torque waves amplitude. The three FI leads of (a) and (b) have $Δ_{A}$ and $Δ_{B}$ given in eV in the caption of (a). The lines are linear fits to the data.
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Figure 5
For four 3D FI leads, $1 / δ$ dependence of the torque characteristic length (ξ) for $θ = π / 2$ and $V = 0.1 V$ . The spin splittings take the following values ${Δ_{A}, Δ_{B}} = {0.5, 0.25; 0.6, 0.43; 0.5, 0.4; 0.4, 0.34}$ . Upper (lower) inset: spatial distribution (in $μ eV$ per unit area) of the in-plane torque wave for $1 / δ = 4 e V^{- 1} (1 / δ = 16.7 e V^{- 1})$ . The dashed line is a linear fit to the data.
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