⇦Chinese Physics Letters, 2020, Vol. 37, No. 10, Article code 107201Express Letter Giant Spin Transfer Torque in Atomically Thin Magnetic Bilayers Weihao Cao (曹伟豪)1,2, Matisse Wei-Yuan Tu (涂维元)1,3*, Jiang Xiao (肖江)4,5, and Wang Yao (姚望)1,3* Affiliations 1Department of Physics, University of Hong Kong, Hong Kong, China 2Department of Physics, University of California San Diego, La Jolla, CA 92093-0319, USA 3HKU-UCAS Joint Institute of Theoretical and Computational Physics at Hong Kong, Hong Kong, China 4Department of Physics and State Key Laboratory of Surface Physics, Fudan University, Shanghai 200433, China 5Institute for Nanoelectronics Devices and Quantum Computing, Fudan University, Shanghai 200433, China Received 1 September 2020; accepted 12 September 2020; published online 15 September 2020 Supported by the Research Grants Council of Hong Kong (Grant Nos. HKU17303518 and C7036-17W), and the University of Hong Kong (Seed Funding for Strategic Interdisciplinary Research).
*Corresponding authors. Email: wangyao@hku.hk; kerustemiro@gmail.com Citation Text: Cao W H, Tu W Y, Xiao J and Yao W 2020 Chin. Phys. Lett. 37 107201    Abstract In cavity quantum electrodynamics, the multiple reflections of a photon between two mirrors defining a cavity is exploited to enhance the light-coupling of an intra-cavity atom. We show that this paradigm for enhancing the interaction of a flying particle with a localized object can be generalized to spintronics based on van der Waals 2D magnets. Upon tunneling through a magnetic bilayer, we find that the spin transfer torques per electron incidence can become orders of magnitude larger than $\hbar /2$, made possible by electron's multi-reflection path through the ferromagnetic monolayers as an intermediate of their angular momentum transfer. Over a broad energy range around the tunneling resonances, the damping-like spin transfer torque per electron tunneling features a universal value of $(\hbar/2)\tan (\theta /2)$, depending only on the angle $\theta$ between the magnetizations. These findings expand the scope of magnetization manipulations for high-performance and high-density storage based on van der Waals magnets. DOI:10.1088/0256-307X/37/10/107201 PACS:72.25.-b, 79.60.Jv, 73.40.Gk, 73.40.-c © 2020 Chinese Physics Society Article Text Magnetic information storage technology exploits the magnetization direction of ferromagnet to encode information, an example of which is the magnetic random access memory (MRAM) based on magnetic tunnel junctions (MTJs).[1,2] An MTJ consists of two ferromagnetic layers: a fixed reference layer and a free switching layer. The transfer of spin angular momentum between the ferromagnets and the passing electrons realizes both reading and writing functions in an MTJ. Through such a transfer, a fixed ferromagnet acting as a polarizing layer aligns the spin of incident electrons, which, in passing through a free magnetic layer, can have two distinct values of resistance depending on the relative alignment of the magnetizations.[1] This allows electric reading of the magnetic information. In turn, the angular momentum of the electron can be transferred to the magnet as a back-action [Fig. 1(a)], a phenomenon known as spin-transfer torque,[2–5] which is actively explored for all electric manipulation of magnetizations (writing) in storage devices.[2,6] The recent discovery of atomically thin 2D magnets[7–14] has provided an ideal platform for spintronic applications. A great asset of these van der Waals magnets is the ability to engineer magnetic multilayer devices with ultimate miniaturization in thickness, implying the potential to further shrink device sizes for high density storage. The atomically sharp and clean interface[15] between magnetic and non-magnetic sublayers is advantageous for removing device fluctuations and imperfections. Preliminary forms of atomically thin van der Waals spintronic devices are first demonstrated using chromium trihalides.[16–20] The three-atom thick monolayer is a ferromagnet, a thinnest possible unit to encode magnetic information, while adjacent layers separated by van der Waals gaps can be arranged in parallel or anti-parallel configurations through the antiferromagnet interlayer coupling and magnetic field. Sandwiched between graphene contacts, the few-layer chromium trihalides function as multiple-spin-filter MTJs, and record-high tunneling magnetoresistance ratios are measured,[16–20] highly favored for the reading. Manipulation of magnetization orientations in these few-layer magnets by electrostatic gating has also been demonstrated.[21–23] Concerning the writing controls, van der Waals magnet has another significance: the transmission through the magnetic multilayer of atomic thickness can be fully quantum coherent, which is a new realm to explore current induced torques. In conventional MTJ devices,[24–27] the angular momentum transfer per electron incidence from the polarizing layer to the free layer is upper bounded by $\hbar /2$, the maximum amount an electron can carry [Fig. 1(a)]. Here we show that the quantum coherent tunneling through atomically thin magnetic bilayers can lead to an analog of optical microcavity for enhancing spin transfer torque (STT), well exceeding the above limit [Fig. 1(b)]. The scattering wavefunctions at the tunneling resonances have a spin current density profile peaked between the two ferromagnetic monolayers, resembling the intra-cavity field for a resonant photon. As a result, the torque per electron incidence becomes orders of magnitude larger than $\hbar /2$. Electron's role as an intermediate for angular momentum transfer is radically amplified through its multiple reflections between the two monolayers [Fig. 1(b)]. We find a field-like torque that is equivalent to the RKKY interaction mediated by the quasi-bound electron, as well as a damping-like STT that tends to align the downstream layer parallel to the upstream layer, and the upstream layer anti-parallel to the downstream one. When we turn to the damping-like torque per electron transmission, a figure of merit for the current driven torque, a universal value of $\frac{\hbar }{2}\tan ({\theta }/{2})$, is found over a broad range of incident energy, depending only on the angle $\theta$ between the two magnetizations. This finding points to highly efficient manipulation of 2D magnets by tunneling current for spintronics applications.
cpl-37-10-107201-fig1.png
Fig. 1. Cavity enhancement of spin transfer torque. (a) In a conventional magnetic tunnel junction, the angular momentum transfer per electron incidence from the polarizing layer to the free layer is upper bounded by electron's angular momentum $\hbar /2$. (b) Schematic of two high reflective barriers sandwiched in the atomically thin magnetic bilayer. The multiple reflections can radically amplify electron's role as an intermediate of angular momentum transfer, and the torque on each magnetic layer can well exceed $\hbar /2$ per electron incidence.
Method of Analysis. Figure 2(a) schematically shows an exemplary van der Waals (vdW) tunnel junction device consisting of a magnetic bilayer sandwiched between graphene/graphite contacts. The plane of a magnetic monolayer is defined to be the $x$–$y$ plane [Fig. 2(b)] so the vertical tunneling direction is the $z$-direction. Without loss of generality, we set the magnetization of layer 1, $\boldsymbol{M}_{1}$, in the $-z$ direction, and the magnetization of layer 2, $\boldsymbol{M}_{2}$, in $x$–$z$ plane with angle $\theta$ from $\boldsymbol{M}_{1}$, see Fig. 2(b). In a ferromagnetic monolayer, the majority and the minority spin states are separated by a large energy gap of hundreds of meV.[16] The majority (minority) spin in layer 1 is denoted by $| \!\downarrow \rangle$ ($|\!\uparrow \rangle$). Tilted by an angle $\theta$, the majority spin in layer 2 is thus $| \theta \rangle =\cos({\theta }/{2})| \!\downarrow \rangle -\sin({\theta }/{2})|\!\uparrow \rangle$. We first concentrate on the tunneling transport facilitated by the majority spin states in the two wells, while the effects of minority spin states will be included later. Such a tunneling transport can be described by the Hamiltonian: $$\begin{align} \hat{H}=\,&\sum\limits_{\boldsymbol{k}} \left({E_{1}(\boldsymbol{k})\hat{a}}_{1,\boldsymbol{k}}^{† }\hat{a}_{1,\boldsymbol{k}}+E_{2}(\boldsymbol{k})\hat{a}_{2,\boldsymbol{k}}^{† }\hat{a}_{2,\boldsymbol{k}}\right) \\ &+\sum\limits_l E_{l} \hat{a}_{l,\sigma }^{† }\hat{a}_{l,\sigma }+\sum\limits_r E_{r} \hat{a}_{r,\sigma }^{† }\hat{a}_{r,\sigma }\\ &+\Big[ \sum\limits_{l,\boldsymbol{k}} {g_{l,\boldsymbol{k}}\hat{a}_{l,\downarrow}\hat{a}_{1,\boldsymbol{k}}^{† }} +\sum\limits_{r,\boldsymbol{k}} g_{r,\boldsymbol{k}}\hat{a}_{2,\boldsymbol{k}}\hat{a}_{r,\theta }^{† }\\ &+\sum\limits_{\boldsymbol{k}} {t_{12}(\boldsymbol{k})\hat{a}_{1,\boldsymbol{k}}\hat{a}_{2,\boldsymbol{k}}^{† }} +\mathrm{h.c.} \Big], \end{align} $$ where $\hat{a}_{1,\boldsymbol{k}}^†$ and $\hat{a}_{2,\boldsymbol{k}}^†$ create carriers with in-plane momentum $\boldsymbol{k}$ in the first layer with spin $| \!\downarrow \rangle$ and in the second layer with spin $| \theta \rangle$, respectively. Since the in-plane momentum is conserved between the two lattice-matched magnetic layers, their interlayer hopping couples only states of the same $\boldsymbol{k}$ from the two layers. Interlayer hopping also conserves spin, so the coupling is a function of the angle between the magnetizations, i.e., $t_{12}(\boldsymbol{k})=t(\boldsymbol{k})\cos({\theta }/{2})$, as the two quantum well states have spin configuration $| \!\downarrow \rangle$ and $| \theta \rangle$, respectively. For the anti-parallel configuration ($\theta =\pi$), $t_{12}$ vanishes, while at other $\theta$ values, it leads to layer hybridization of the spin majority states. Here $\hat{a}_{l,\sigma }^†$ ($\hat{a}_{r,\sigma }^†$) create the continuum modes in the left (right) contact of spinor $\sigma$. Since the tunneling between the magnetic layer and the adjacent contact is spin conserving, the state in the first (second) magnet couples to the left (right) contact states of the same spin configuration only, denoted as $\hat{a}_{l,\downarrow}^†$ ($\hat{a}_{r,\theta }^†$). Without lattice matching between the contact materials and the magnetic layers, the quantum well state of a given $\boldsymbol{k}$ in the magnetic layer can be coupled to a continuum of modes in the contact. At every $\boldsymbol{k}$, we thus have an effective one-dimensional tunneling channel, characterized by the 1D Hamiltonian $$\begin{align} \hat{H}_{\boldsymbol{k}}=\,&E_{1}(\boldsymbol{k})\hat{a}_{1}^{† }\hat{a}_{1}+E_{2}(\boldsymbol{k})\hat{a}_{2}^{† }\hat{a}_{2}+\sum\limits_l E_{l} \hat{a}_{l,\sigma }^{† }\hat{a}_{l,\sigma }\\ &+\sum\limits_r E_{r} \hat{a}_{r,\sigma }^{† }\hat{a}_{r,\sigma }+\Big[\sum\limits_l {g_{l,\boldsymbol{k}}\hat{a}_{l,\downarrow}\hat{a}_{1}^{† }} +t_{12}\hat{a}_{1}\hat{a}_{2}^{† }\\ &+\sum\limits_r {g_{r,\boldsymbol{k}}\hat{a}_{2}\hat{a}_{r,\theta }^{† }} +\mathrm{h.c.}\Big]. \end{align} $$ With the strong vertical confinement of carriers in each monolayer and their weak vdW coupling in the heterostructures, for the tunneling transport, we model this device as coupled quantum wells as illustrated in Fig. 2(b). Each monolayer corresponds to a quantum well with a spin-dependent potential profile determined by the magnetization direction. The two magnetic quantum wells are separated from each other and from the contacts by barriers that correspond to the vdW gaps. The barrier profile is spin independent, and tunneling between the lattice-matched bilayer conserves in-plane momentum. The latter effectively reduces the tunneling transport to a one-dimensional problem, where the coupled quantum wells feature tunneling resonances at the quasi-bound states. With the lattice mismatch between the graphene electrode and the magnetic layer, the continuity of choices of states in the electrodes is translated into the continuously varied incident energies in the one-dimensional problem.
cpl-37-10-107201-fig2.png
Fig. 2. Cavity amplification of spin current in coupled double magnetic quantum wells. (a) Schematic of a van der Waals heterostructure consisting of two ferromagnetic monolayers, each corresponding to a quantum well, and graphene contacts. (b) Potential profile of the coupled quantum wells in anti-parallel magnetization configurations ($\theta =\pi$). The barriers correspond to the van der Waals (vdW) gaps. The blue solid (red dashed) line is for spin $\uparrow$ ($\downarrow$) electron. Here $\psi_{1}$ and $\psi_{2}$ denote the wavefunction of quantum well states in the first and second layers, respectively; $d_{1}=d_{2}=2.76$ Å, and $w=5.52$ Å. (c) A scattering wavefunction $\varPsi _{\downarrow }\otimes| \!\downarrow \rangle +\varPsi _{\uparrow }\otimes|\!\uparrow \rangle$, for a spin $\downarrow$ incidence at tunneling resonance (denoted by the dot-dashed line in (b), under $\theta =0.99\pi$. The shades mark barrier regions (vdW gaps). (d) Spin current profiles for the wavefunction in (c). $Q_{i}$ denotes the spin current propagating in the $z$ direction carrying spin component $i$, normalized by the incident probability current, and is in unit of $\hbar /2$. (e) Spin currents for a spin $\uparrow$ incidence at the same energy as in (c).
Such a model has been adopted for description of the giant tunneling magneto-resistance through bilayer and few-layer CrI$_{3}$ where neighboring monolayers are of anti-parallel (parallel) magnetization at zero (finite) field,[16] reproducing well the $I$–$V$ curves when the contact Fermi energies are set below the tunneling resonances. The calculations of the CrI$_{3}$/graphene heterostructures therein have used the barrier width $d=2.76$ Å, quantum well width $w= 5.52$ Å, and the spin-splitting $\varDelta = 0.5$ eV in each well. The height of the vdW barriers roughly corresponds to the electron affinity, taken to be 4 eV. We adopt the same set of parameters here in the numerical calculations, while a parameter independent universal value for the STT near the tunneling resonance will be established. We set the potential energy matrix to be $V_{0}+\Delta \hat{\sigma }_{z}$ in the first well, and $V_{0}+\Delta (\hat{\sigma }_{z}\cos\theta +\hat{\sigma }_{x}\sin\theta)$ in the second well. The potential profile generated with these parameters is shown in Fig. 2(b) for an anti-parallel configuration ($\theta =\pi$). The scattering state wavefunction at this spin-dependent potential, $\varPsi (z)=\varPsi _{\downarrow}\otimes | \!\downarrow \rangle +\varPsi _{\uparrow} \otimes |\!\uparrow \rangle$, is solved by the wavefunction matching between the well, barrier and contact regions. By examining $\langle \hat{\sigma}_i \rangle \equiv {\varPsi }^{\ast }\hat{\sigma }_{i}\varPsi$ using the Schrödinger equation, we find $$ \frac{\hbar}{2}\frac{d\langle \hat{\sigma}_i \rangle }{dt}=-\nabla \cdot Q_{i}. $$ This then gives $$ Q_{i}(z)\equiv \frac{\hbar^{2}}{2\,m}\mathrm{{\rm Im}}\Big(\varPsi ^{\ast }\hat{\sigma }_{i} \otimes\frac{\partial }{\partial z}\varPsi\Big), $$ the spin current propagating in the out-of-plane $z$ direction for spin component $i=x,y,z$. Results. In Figs. 2(c)–2(e), we show an example of the wavefunction and spin currents calculated under an angle $\theta =0.99\pi$ between the two magnetizations. Figure 2(c) plots the wavefunction at a tunneling resonance for an electron incident on the $| \!\downarrow \rangle$ state (parallel to the magnetization in layer 1, ${\boldsymbol M}_{1}$). From the calculations of the transmission probabilities, we see that there is nearly perfect transmission at the concerned energy, while the transmitted electron is found to have its spin parallel to the magnetization of layer 2 (${\boldsymbol M}_{2}$). This corroborates the construction of $\hat{H}_{\boldsymbol{k}}$ including only the spin majority states in the two layers. The corresponding spin currents are shown in Fig. 2(d). $Q_{x}$ and $Q_{y}$ both have a standing-wave-like profile between the two outer barriers, analogous to the intra-cavity optical field in a microcavity. The maximum spin current is obtained inside the middle barrier between the two monolayer magnets, reaching a value that is 15 times of the incident spin current. In contrast, the spin currents remain small everywhere for a spin $\uparrow $ incidence at the same energy [Fig. 2(e)]. From the calculations of the reflection coefficients, we find that the spin $ \uparrow $ electron is almost fully reflected. For a spin unpolarized incidence, the first magnet can filter out a spin current, which is then resonantly amplified by the multiple reflections through the two magnets. The spin current profile in Fig. 2(d) implies a sizable torque exerted on each of the magnetic monolayer. The torques on the first and second layers are $\boldsymbol{\tau }^{1}=\boldsymbol{Q}(z_{1})-\boldsymbol{Q}(z_{2})$ and $\boldsymbol{\tau }^{2}=\boldsymbol{Q}(z_{2})-\boldsymbol{Q}(z_{3})$, respectively, where $\boldsymbol{Q=}Q_{x}\hat{x}+Q_{y}\hat{y}+Q_{z}\hat{z}$, and $z_{l}$ is a coordinate inside the $l$th barrier. Our calculations find that the torque is always perpendicular to the magnetization of the corresponding layer, $\boldsymbol{\tau }^{i} \bot \boldsymbol{M}_{i}$. Each torque can be decomposed into two components perpendicular and parallel to the ($x$–$z$) plane spanned by $\boldsymbol{M}_{1}$ and $\boldsymbol{M}_{2}$.[6] The perpendicular components are known as the field-like torques ($\tau_{f}^{1,2}$), which tend to rotate the magnetizations with respect to each other, keeping $\theta$ unchanged. The parallel components are known as the damping-like spin transfer torques ($\tau_{d}^{1,2}$), which tend to change the angle $\theta$. These torque components are schematically illustrated in the inset of Fig. 3. Figures 3(a)–3(c) plot the magnitudes of different torque components as functions of angle $\theta$ and incident energy $E_{\rm in}$, for spin-unpolarized incidence, and Fig. 3(d) shows the corresponding transmission probability $T$. At a general angle $\theta$, all torque components show two resonant peaks, corresponding respectively to the tunneling resonances at the bonding and anti-bonding states in the coupled double wells [see Fig. 3(d)]. Both the torques and transmission probability vanish for the collinear configuration with $\theta =\pi$. In a general non-collinear configuration, we find that at both tunneling resonances $\tau_{d}^{1}$ tends to enlarge the angle $\theta$ between the magnetizations, while $\tau_{d}^{2}$ tends to reduce it [Fig. 3(a)]. These qualitative behaviors are consistent with those observed in the conventional MTJ and spin valve, where the damping-like STT tends to align the downstream layer parallel to the upstream layer, and the upstream layer anti-parallel to the downstream one. For the field-like torque, we find $\boldsymbol{\tau }_{f}^{1}=-\boldsymbol{\tau }_{f}^{2}$, so they tend to rotate the plane spanned by $\boldsymbol{M}_{1}$ and $\boldsymbol{M}_{2}$ about the axis of $\boldsymbol{M}_{1}+\boldsymbol{M}_{2}$. It is worth noting that the field-like torque has opposite sign at the bonding and the anti-bonding tunneling resonances [Fig. 3(b)].
cpl-37-10-107201-fig3.png
Fig. 3. Giant spin transfer torque. The (positive) directions for the torque components are illustrated in the right upper inset. (a) Magnitude of the damping-like spin transfer torque $\boldsymbol{\tau }_{d}^{1}$ on layer 1. Here $\boldsymbol{\tau }_{d}^{2}$ has nearly the same magnitude, and its different direction is noted in the inset. (b) Field-like torque $\boldsymbol{\tau }_{f}^{1}$ on layer 1. The one on layer 2 is exactly its opposite: $\boldsymbol{\tau }_{f}^{2}=-\boldsymbol{\tau }_{f}^{1}$. (c) Angular momentum $S$ carried by each out-going electron, averaged over transmission and reflection. (d) Transmission probability $T$. All quantities are calculated under a spin-unpolarized incidence. The torques are normalized by the incident probability current, and are given in units of $\hbar /2$ (i.e., torque value of 1 corresponds to angular momentum transfer of $\hbar /2$ per incident electron). $E_{\rm in}$ is the incident energy whose reference 0 is set to the middle of the two resonant energies, which corresponds to $[E_{1}(\boldsymbol{k})+E_{2}(\boldsymbol{k})]/2$ for the in-plane momentum $\boldsymbol{k}$ concerned.
The torques plotted in Fig. 3 have been normalized by the incident probability current, so the value can be taken as angular momentum transfer per electron incidence. We find an accelerated increase of the damping-like STT $\tau_{d}^{1,2}$ as a function of $\theta$, reaching a remarkable peak value of 15 quanta of $\frac{\hbar }{2}$ per electron incidence at $\theta \sim 0.98\pi$, followed by a rapid decay towards $\theta =\pi$. With the different directions of $\boldsymbol{\tau }_{d}^{1}$ and $\boldsymbol{\tau }_{d}^{2}$ noted ($\bot \boldsymbol{M}_{1}$ and $\bot \boldsymbol{M}_{2}$ respectively in the $x$–$z$ plane), the quantitative behaviors of $\tau_{d}^{1}$ and $\tau_{d}^{2}$ are nearly the same, and only one is shown in Fig. 3(a). In comparison, the field-like torque $\tau_{f}^{1,2}$ is a smoother function of $\theta$ [Fig. 3(b)]. To understand the above behaviors in more depth, we inspect the spin current density inside the middle barrier, where the wavefunction is in general a superposition of the evanescent tails of the quantum well states from the two layers [cf. Fig. 2(b)], and can be expressed as $$ \langle z|{\varPsi} \rangle =\beta_{1}e^{-\kappa (z-z_{0})}\otimes| \downarrow \rangle +\beta_{2}e^{\kappa (z-z_{0})}\otimes| \theta \rangle, $$ where $z_{0}$ the coordinate of barrier center. Evaluating the spin currents for this wavefunction, we find $$\begin{align} &Q_{x}(z_{0})=i(\beta_{1}^{\ast }\beta_{2}-\beta_{1}\beta_{2}^{\ast })\frac{\hbar^{2}\kappa }{2\,m}\sin({\theta }/{2}),\\ &Q_{y}(z_{0})=(\beta_{1}^{\ast }\beta_{2}+\beta_{1}\beta_{2}^{\ast })\frac{\hbar^{2}\kappa }{2\,m} \sin({\theta }/{2}),\\ &Q_{z}(z_{0})=-i(\beta_{1}^{\ast }\beta_{2}-\beta_{1}\beta_{2}^{\ast })\frac{\hbar^{2}\kappa }{2\,m}\cos({\theta }/{2}).~~ \tag {1} \end{align} $$ For the isolated coupled wells, $\beta_{1}$ and $\beta_{2}$ are real numbers, so $Q_{x}=Q_{z}=0$, and the non-zero $Q_{y}$ gives rise to a field-like torque which takes opposite signs at the bonding and anti-bonding resonances. Through the Landau–Lifshitz–Gilbert equation,[6] one can realize that this torque is equivalent to a Hamiltonian term $J\boldsymbol{M}_{1} \cdot \boldsymbol{M}_{2}$, i.e., an RKKY coupling between the two magnetizations mediated by the coupled quantum well state of the electron. The field-like torque we obtained in the scattering state solution under finite coupling to the contacts [Fig. 3(b)] is a reminiscence of this RKKY coupling. For a scattering state, the associated probability current $j$ corresponds to the transmission probability $T$, and reads $$ j(z_{0})=-i(\beta_{1}^{\ast }\beta_{2}-\beta_{1}\beta_{2}^{\ast })\frac{\hbar \kappa }{m}\cos({\theta }/{2})~~ \tag {2} $$ when evaluated inside the middle barrier. Its finite value requires the spin currents $Q_{z}$ and $Q_{x}$ to be finite as well in the middle barrier. Equations (1) and (2) together lead to $$ \frac{\hbar }{2}\tan ({\theta }/{2}) =-\frac{Q_{x}(z_{0})}{j(z_{0})}=\frac{\tau_{d}^{1}}{T} .~~ \tag {3} $$ Remarkably, the STT on the upstream magnet per electron transmission, $\tau_{d}^{1}/T$, has a universal value independent of the incident energy and the model parameters. We can define two regimes of resonance tunneling through the coupled wells, depending on the relative strength of their coupling $t_{12}$ with respect to the broadening $\varGamma$ of the tunneling resonance. In the strong coupling regime, i.e., $\varGamma < t_{12}=t\cos({\theta }/{2})$, the bonding and anti-bonding resonances are separated as two peaks [Fig. 3(d)], with an on-resonance transmission probability $T_{\mathrm{res}}=0.5$ corresponding to fully transmitted $| \!\downarrow \rangle$ incidence, and fully blocked $|\!\uparrow \rangle$. From Eq. (3), $\tau_{d}^{1}=T\frac{\hbar }{2}\tan ({\theta }/{2})$, i.e., the spin transfer torque per incidence features two resonance peaks accordingly. The on-resonance torque value is then $\frac{\hbar }{4}\tan ({\theta }/{2})$, in excellent agreement with the rising behavior displayed in Fig. 3(a). The small neighborhood of $\theta =\pi$ is the weak coupling regime $t_{12} < \varGamma$, where the two tunneling resonances merge into one [Fig. 3(d)]. In this regime $T_{\mathrm{res}}$ drops from 0.5 to $0$, which is responsible for the rapid decay of $\tau_{d}^{1}$ as $\theta$ approaches $\pi$ [Fig. 3(a)]. Figure 4(a) plots the on-resonance transmission probability $T_{\mathrm{res}}$ and the on-resonance $\tau_{d}^{1}$ value as functions of $\theta$, and their ratio (black curve) is found in perfect agreement with the curve (red dashed) dictated by Eq. (3).
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Fig. 4. Universal value of a damping-like spin transfer torque per electron transmission. (a) On-resonance spin transfer torque $\tau_{d}^{1}$ (green), and transmission probability $T$ (blue). The black curve plots the ratio $\frac{1}{2}\frac{\tau_{d}^{1}}{T}$, and the red dashed one is the function $\frac{1}{2}\tan ({\theta }/{2})$ for comparison. Here $\tau_{d}^{1}$ value is given in units of $\frac{\hbar }{2}$. The shaded area marks the weak coupling regime where $T$ drops from 0.5 (perfect transmission of $\downarrow$ incidence) to 0. (b) Behavior of $\tau_{d}^{1}$ as a function of incident energy $E_{\rm in}$ plotted at the angle $\theta_{\rm m}$ where the maximum torque value is reached, see (a). Curves in different colors correspond to different outer barrier thicknesses $d_{2}$ [see Fig. 2(b)]. The corresponding values of $\theta_{\rm m}$ are shown in the inset. (c) Characteristics of $\frac{\tau_{d}^{1}}{T}$ (dashed) and $T$ (solid) as functions of $E_{\rm in}$, when the inner barrier thickness $d_{1}$ is varied, for $\theta =0.99\pi$.
Figures 4(b)–4(c) further examine the effects of different quantum-well parameters. In Fig. 4(b), under various outer barrier widths $d_{2}$, the damping-like torque $\tau_{d}^{1}$ as a function of incident energy is plotted at the angle $\theta_{\rm m}$ where $\tau_{d}^{1}$ reaches its maximum. Wider $d_{2}$ leads to higher quality factor $Q$ of the tunneling resonance, where an electron has more rounds of reflection between the outer barriers before exiting, hence a larger angular momentum transfer between the two magnetic layers becomes possible. The increase of the $\tau_{d}^{1}$ peak value with $Q$ is indeed observed in Fig. 4(b). We also change the width $d_{1}$ of the middle barrier, equivalent to the change of the coupling constant $t$ between the two quantum-well states, and the increase of $\tau_{d}^{1}$ with $t$ is observed in Fig. 4(c). The intuitive trends reflected in Figs. 4(b) and 4(c) can also be understood from the perspective offered by Eq. (3). With the increase of either $Q$ or $t$, the weak coupling regime shrinks towards $\theta =\pi$ so that $\tau_{d}^{1}$ can pick up higher values on the $\tan ({\theta }/{2})$ curve.
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Fig. 5. Effect of asymmetric configuration on the damping-like spin transfer torque per electron transmission: $\frac{\tau_{d}^{1}}{T}$ (dashed) and $T$ (solid) as functions of $E_{\rm in}$, where curves in different colors correspond to different energy detunings between the two wells $\delta$ in units of meV in (a), different asymmetries between two outer barrier heights in units of meV in (b), and different asymmetries between two outer barrier widths in units of Å in (c). In (b) the two outer barrier heights $V_{\rm L}$ and $V_{\rm R}$ are set to $({V_{\rm L}+V_{\rm R}})/{2}=4$ eV and $\Delta V=V_{\rm R}-V_{\rm L}$. In (c) the two outer barrier widths $d_{\rm L}$ and $d_{\rm R}$ are set to $({d_{\rm L}+d_{\rm R}})/{2}=0.276$ nm and $\Delta d=d_{\rm R}-d_{\rm L}$. For all plots $\theta =0.85\pi$, and the corresponding $\tan ({\theta }/{2})$ is marked by the constant solid horizontal lines.
In Fig. 5(a), we allow a detuning $\delta \equiv E_{1}-E_{2}$ between the two wells, which can quench their hybridization and hence the on-resonance transmission. Nevertheless, over a large range of incident energy, the STT per transmission ${\tau_{d}^{1}}/{T}$ only has modest deviation from the universal value $\frac{\hbar }{2}\tan ({\theta }/{2})$. This allows a flexible bias window to utilize the sizable STT driven by the tunneling current. In Figs. 5(b) and 5(c), we perform the calculation with asymmetry between the tunneling barriers with the top and bottom contacts, in height and width, respectively. While the transmission probability can be affected by such asymmetry, the spin transfer torque per electron transmission is still well described by the universal value. Indeed, the derivation of Eq. (3) has not relied on any assumption that the tunneling barriers with top and bottom contacts are identical, so the results are applicable in the presence of inevitable asymmetry in realistic devices. It is interesting to note that the non-collinear magnetization configurations function as a highly efficient spin pump at the tunneling resonances. Figure 3(c) plots the angular momentum $\boldsymbol{S}$ carried by each out-going electron, averaged over transmission and reflection. With incidence being unpolarized, $\boldsymbol{S+}\boldsymbol{\tau }_{d}^{1}+\boldsymbol{\tau }_{f}^{1}+\boldsymbol{\tau }_{d}^{2}+\boldsymbol{\tau }_{f}^{2}=0$, as a result of the angular momentum conservation. Remarkably, $S$ at the tunneling resonances is a sizable fraction of $\hbar /2$ in almost the entire range of $\theta$, meaning that the bilayer has turned an unpolarized incoming flux into a net outgoing spin current. This spin pump is contributed by the transmission part of the electron where the spin configuration is changed from $| \!\downarrow \rangle$ to $\cos({\theta }/{2})| \!\downarrow \rangle -\sin({\theta }/{2})|\!\uparrow \rangle$, from which we can find $$ \frac{S}{T}=\hbar \cos \Big(\frac{\pi -\theta }{2}\Big).~~ \tag {4} $$ The ratio of $S$ and $T$ from the numerical calculation [Figs. 3(c) and 3(d)] well satisfies the above relation. In the strong coupling regime, $T_{\mathrm{res}}=0.5$, so the on-resonance $S=\frac{\hbar }{2}\cos \frac{\pi -\theta }{2}$ [cf. Fig. 3(c)], and its upper bound $\frac{\hbar }{2}$ is approximately reached at $\theta \sim 0.97\pi$, the boundary of the strong and weak coupling regimes. In the weak coupling limit, $S\cong \hbar T$, where the on-resonance value drops to 0 at $\theta =\pi$. Discussions. Concerning the 2D nature of the tunneling transport, we restore the in-plane momentum $\boldsymbol{k}$ conserved between the two magnetic layers. Referring to $\hat{H}$ introduced in Method of Analysis, the bonding and anti-bonding resonances are given by $$ E_{\pm}(\boldsymbol{k})=\frac{E_{1}(\boldsymbol{k})+E_{2}(\boldsymbol{k})}{2}\pm \sqrt {\Big(\frac{E_{1}(\boldsymbol{k})-E_{2}(\boldsymbol{k})}{2}\Big)^{2}+t_{12}^{2}}, $$ and the contributions to the STT and transmission have the same behavior as presented in Figs. 2–5, except the overall shift of the resonances. $E_{\pm}(\boldsymbol{k})$ give the dispersions of the lowest energy spin polarized band in the bilayer split by the interlayer coupling $t_{12}$. The total STT and the tunneling current are given, respectively, as follows: $$\begin{align} \tau_{\mathrm{sum}}=\,&\int {dEd\boldsymbol{k}} [f_{l}(E)-f_{r}(E)]\tau_{d}^{1}(E,\boldsymbol{k}),\\ j=\,&\int {dEd\boldsymbol{k}} [f_{l}(E)-f_{r}(E)]T(E,\boldsymbol{k}), \end{align} $$ where $f_{l(r)}(E)$ is the Fermi distribution function in the left (right) contact. When the bias window overlaps with the bands $E_{\pm}(\boldsymbol{k})$, the dominant contributions to $\tau_{\mathrm{sum}}$ and $j$ are the resonant tunneling through a range of $\boldsymbol{k}$. Here the ratio of on-resonance $\tau_{d}^{1}(E,\boldsymbol{k})$ and $T(E,\boldsymbol{k})$ is given by Eq. (3), determined by the angle $\theta$ only. In such a case, the damping-like STT per tunneling is still of the universal value: $$ \frac{\tau_{\mathrm{sum}}}{j}=\frac{\hbar }{2}\tan ({\theta }/{2}).~~ \tag {5} $$ When the bias window is below the band edge, quantitative drop of the torque from the universal value can be expected with the increase of detuning, as shown in Fig. 4(c). These results can be straightforwardly generalized to tunneling through the spin minority states in the quantum well. The only difference lies in the opposite signs of the STT, while the torque magnitude is still described by Eq. (3), if the incident energy is near resonance with the spin minority quantum well state. The overall damping-like STT per transmission is then $$ \frac{\tau_{\mathrm{sum}}}{j}=\frac{j_{\downarrow}-j_{\uparrow}}{j_{\downarrow}+j_{\uparrow }} \frac{\hbar }{2}\tan ({\theta }/{2}),~~ \tag {6} $$ where $j_{\downarrow}$ ($j_{\uparrow}$) is the tunneling current for electron incident in the $\downarrow$ ($\uparrow $) state, the majority (minority) spin species of the upstream monolayer. The spin quantization direction can always be chosen according to the magnetization direction, not necessarily the same as the propagation. The spin subspace is decoupled from the coordinate space in the absence of spin-orbit coupling. The results are therefore applicable to arbitrary configurations of the bilayer magnetizations, and to van der Waals magnets of either out-of-plane anisotropy such as CrI$_{3}$,[7] Fe$_{3}$GeTe$_{2}$[9,10] or in-plane anisotropy as the lately discovered CrCl$_{3}$.[28–30] Versatile current induced magnetization dynamics can be expected from the interplay of the giant STT with the distinct magnetization anisotropy and interlayer magnetic coupling in various 2D magnets. We also note that the asymmetry across the contact and a neighboring magnetic layer may cause in-plane Rashba spin-orbit interactions. Within the ferromagnetic layer, the spin-splitting energy is expected to be much larger than the strength of the possible in-plane Rashba spin-orbit interaction, so the effect of the latter can be quenched. In the non-magnetic contact, the Rashba interaction can cause spin depolarization, which can play a deleterious role on the functionalities such as spin filtering and spin pumping that aim at harvesting the transmitted or reflected spin current. Nevertheless, concerning the functionality of the spin transfer torque, it is realized by the multiple scattering of electron between the two magnetic layers upon spin-unpolarized incidence, which is not affected by the Rashba interaction in the contacts. In summary, we have shown that atomically thin magnetic devices can feature universally a giant spin transfer torque, owning to the multiple reflections of electrons in the quantum coherent tunneling through the magnets. This finding introduces the classical paradigm of cavity quantum electrodynamics into magnetization manipulations, where the interaction of a flying object with a localized one is enhanced through multiple reflections inside the cavity. The roles of atoms and flying photon in cavity QED are now played by the magnets and the current carrying electron respectively, where van der Waals interfaces naturally realize the “cavity mirrors”. Electron's functionality as an intermediate for angular momentum exchange between ferromagnetic layers is radically amplified by its multiple impacts on the magnets. The torque generated per electron incidence can become orders larger than $ \hbar /2$ at the tunneling resonance. This regime has not been possible for conventional magnetic tunneling junction where the overall thickness exceeds the coherence length. This mechanism of amplifying spin transfer torque is also distinguished from existing scheme exploiting the resonance tunneling through engineered barrier,[31,32] where magnets are outside the multi-reflection path and hence torque generated per electron incidence is always upper bounded by $ \hbar /2$. In fact, the torque amplification here does not fully rely on the tunneling resonance. Over a rather broad energy range around the tunneling resonance, we find that the figure of merit, i.e., torque generated per electron tunneling, has a sizable and universal value $ (\hbar /2)\tan {(\theta /2)}$, determined by the angle $ \theta $ between the magnetizations only, independent of model parameters. References Giant room-temperature magnetoresistance in single-crystal Fe/MgO/Fe magnetic tunnel junctionsSpin-transfer torque RAM technology: Review and prospectCurrent-driven excitation of magnetic multilayersEmission of spin waves by a magnetic multilayer traversed by a currentSpin transfer torquesCurrent-induced torques in magnetic materialsLayer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limitDiscovery of intrinsic ferromagnetism in two-dimensional van der Waals crystalsGate-tunable room-temperature ferromagnetism in two-dimensional Fe3GeTe2Two-dimensional itinerant ferromagnetism in atomically thin Fe3GeTe2Ising-Type Magnetic Ordering in Atomically Thin FePS 3Raman spectroscopy of atomically thin two-dimensional magnetic iron phosphorus trisulfide (FePS 3 ) crystalsRoom Temperature Intrinsic Ferromagnetism in Epitaxial Manganese Selenide Films in the Monolayer LimitStrong room-temperature ferromagnetism in VSe2 monolayers on van der Waals substratesVan der Waals heterostructuresGiant tunneling magnetoresistance in spin-filter van der Waals heterostructuresProbing magnetism in 2D van der Waals crystalline insulators via electron tunnelingVery large tunneling magnetoresistance in layered magnetic semiconductor CrI3Magnon-assisted tunnelling in van der Waals heterostructures based on CrBr3One Million Percent Tunnel Magnetoresistance in a Magnetic van der Waals HeterostructureElectric-field switching of two-dimensional van der Waals magnetsElectrical control of 2D magnetism in bilayer CrI3Electric-field control of magnetism in a few-layered van der Waals ferromagnetic semiconductorDescription of current-driven torques in magnetic tunnel junctionsSpin-transfer torque in magnetic tunnel junctions: Scattering theoryAnomalous Bias Dependence of Spin Torque in Magnetic Tunnel JunctionsAb Initio Studies of the Spin-Transfer Torque in Magnetic Tunnel JunctionsAtomically Thin CrCl 3 : An In-Plane Layered Antiferromagnetic InsulatorEnhancement of interlayer exchange in an ultrathin two-dimensional magnetEvolution of interlayer and intralayer magnetism in three atomically thin chromium trihalidesBand-pass Fabry-Pèrot magnetic tunnel junctionsEnhancement of Spin-transfer torque switching via resonant tunneling
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