Spin injection and pumping generated by a direct current flowing through a magnetic tunnel junction

TL;DR

This paper theoretically investigates how a direct current in a magnetic tunnel junction can generate spin-polarized currents and induce magnetization precession, leading to potential applications in spin pumping and spintronic devices.

Contribution

It presents a detailed theoretical model of spin dynamics and charge transport in a specific MTJ structure, highlighting electrically induced large-angle magnetization precession at low current densities.

Findings

Large-angle magnetization precession can be electrically induced.

Generated spin pumping produces a detectable transverse voltage.

The proposed spin injector shows high efficiency in charge-to-spin conversion.

Abstract

A charge flow through a magnetic tunnel junction (MTJ) leads to the generation of a spin-polarized current which exerts a spin-transfer torque (STT) on the magnetization. When the density of applied direct current exceeds some critical value, the STT excites high-frequency magnetization precession in the "free" electrode of MTJ. Such precession gives rise to microwave output voltage and, furthermore, can be employed for spin pumping into adjacent normal metal or semiconductor. Here we describe theoretically the spin dynamics and charge transport in the CoFeB/MgO/CoFeB/Au tunneling heterostructure connected to a constant-current source. The magnetization dynamics in the free CoFeB layer with weak perpendicular anisotropy is calculated by numerical integration of the Landau-Lifshitz-Gilbert-Slonczewski equation accounting for both STT and voltage controlled magnetic anisotropy associated with the CoFeB|MgO interface. It is shown that a large-angle magnetization precession, resulting from electrically induced dynamic spin reorientation transition, can be generated in a certain range of relatively low current densities. An oscillating spin current, which is pumped into the Au overlayer owing to such precession, is then evaluated together with the injected spin current. Considering both the driving spin-polarized charge current and the pumped spin current, we also describe the charge transport in the CoFeB/Au bilayer with the account of anomalous and inverse spin Hall effects. An electric potential difference between the lateral sides of the CoFeB/Au bilayer is calculated as a function of distance from the CoFeB|MgO interface. It is found that this transverse voltage signal in Au is large enough for experimental detection, which indicates significant efficiency of the proposed current-driven spin injector.

Figures

Equations

$$\begin{array}{c}\frac{d\mathbf{m}}{dt}=-\gamma {\mu }_0\mathbf{m}\times {\mathbf{H}}_{\text{eff}}+\alpha \mathbf{m}\times \frac{d\mathbf{m}}{dt}\\ +\frac{{\tau }_{\text{STT}}}{M_s}\mathbf{m}\times (\mathbf{m}\times {\mathbf{m}}_p),\end{array}$$

$$\begin{array}{c}\Delta F=K_1(m_1^2{m}_2^2+m_1^2{m}_3^2+m_2^2{m}_3^2)+K_2{m}_1^2{m}_2^2{m}_3^2\\ +\frac{K_s}{t_f}{m}_3^2-\frac{U_{\text{IEC}}}{t_f}(m_1{m}_1^p+m_2{m}_2^p+m_3{m}_3^p)\\ +\frac{1}{2}{\mu }_0{M}_s^2(N_{11}{m}_1^2+2N_{12}{m}_1{m}_2\\ +N_{22}{m}_2^2+N_{33}{m}_3^2)\\ -{\mu }_0{M}_s(H_1{m}_1+H_2{m}_2+H_3{m}_3)\end{array}$$

$$\begin{gathered}\alpha=\frac{\gamma}{\gamma_{0}}\bigg{\{}\alpha_{0}+\frac{g_{L}\mu_{B}}{4\pi M_{s}t_{f}}\textrm{Re}[g_{\uparrow\downarrow}^{r}]\bigg{\}},\\ \frac{1}{\gamma}=\frac{1}{\gamma_{0}}\bigg{\{}1+\frac{g_{L}\mu_{B}}{4\pi M_{s}t_{f}}\textrm{Im}[g_{\uparrow\downarrow}^{r}]\bigg{\}},\end{gathered}$$

$$\begin{array}{c}\mathbf{J}=e\mu n\mathbf{E}+2{\alpha }_{\text{SH}}e\mu \mathbf{E}\times \mathbf{P}+2{\alpha }_{\text{SH}}eD\nabla \times \mathbf{P},\end{array}$$

$$\begin{array}{c}\frac{J_{ik}^s}{\hslash }=-\mu E_i{P}_k-D\frac{\partial P_k}{\partial x_i}+2{\alpha }_{\text{SH}}\mu n{\epsilon }_{ikl}{E}_l,\end{array}$$

$$\begin{array}{c}\Delta V(x_3,t)=A(x_3)m_2(t)J_0+B(x_3)J_{32}^s(x_3=t_f,t),\end{array}$$

Full text

I Introduction

In conductive ferromagnetic nanolayers, magnetic dynamics can be induced by a spin-polarized charge current exerting a spin-transfer torque (STT) on the magnetization  Slonczewski1989_6995 ; Slonczewski2007_169 . The STT results from the transfer of angular momentum and provides the opportunity to excite high-frequency magnetization oscillations in nanomagnets by applied direct or alternating (microwave) current  Kiselev2003_380 -Fang2016_11259 . Furthermore, spin-polarized charge currents with sufficiently high densities lead to magnetization switching in metallic pillars  Myers1999_867 ; Katine2000_3149 and magnetic tunnel junctions (MTJs)  Huai2004_3118 -Sato2011_042501 . Such current-induced switching serves as a mechanism for data writing in magnetic random access memories utilizing the STT effect (STT-MRAMs)  Ikeda2007_991 -Ando2014_172607 , while the magnetization precession driven by direct currents in spin-torque nanoscale oscillators (STNOs) creates microwave voltages, which makes STNOs potentially useful as frequency-tunable microwave sources and detectors  Bonetti2009_102507 -Fang2016_11259 .

In ferromagnetic nanostructures comprising insulating interlayers, the electric field created in the insulator adjacent to the metallic ferromagnet may significantly affect the magnetic anisotropy of the latter. Such voltage-controlled magnetic anisotropy (VCMA) results from the penetration of electric field into an atomically thin surface layer of the ferromagnetic metal, which modifies the interfacial magnetic anisotropy  Weisheit2007_349 -Alzate2014_112410 . The presence of VCMA renders possible to induce the magnetization precession in ferromagnetic nanostructures by microwave voltages  Nozaki2012_491 -Miura2017_42511 . It is also shown that the application of dc voltage to the ferromagnetic nanolayer possessing VCMA can lead to a spin reorientation transition (SRT)  Duan2008_122905 -Pertsev2014_024426 . Moreover, precessional 180*∘* magnetization switching using electric field pulses has been demonstrated experimentally  Shiota2012_39 ; Grezes2016_012403 . In addition, the voltage dependence of the interfacial magnetic anisotropy in CoFeB/MgO/CoFeB tunnel junctions may greatly reduce the critical current density required for the STT-driven magnetization reversal  Wang2011_64 ; Pertsev2013_02757 .

Importantly, magnetization precession in a ferromagnetic layer gives rise to spin pumping into adjoining normal metal or semiconductor  Tserkovnyak2002_117601 -Ando2011_655 . In this paper, we theoretically study the magnetization dynamics driven by a direct current applied to the Co20Fe60B20/MgO/Co20Fe60B20 tunnel junction and calculate the time-dependent spin current generated in the Au overlayer. The magnetization evolution in the free Co20Fe60B20 layer is determined by solving numerically the Landau-Lifshitz-Gilbert-Slonczewski (LLGS) equation, which accounts for the STT created by a spin-polarized tunnel current and for the VCMA associated with the Co20Fe60B${}_{20}|$MgO interface. A range of current densities is revealed, within which a steady-state magnetization precession is generated in the free Co20Fe60B20 layer. For this ”precession window”, frequencies and trajectories of magnetization oscillations are determined and used to calculate the time-dependent spin current created in the Au overlayer. Our calculations are distinguished by the account of both the spin polarization of the charge current and the precession-driven spin pumping as well as the contribution of the latter to the damping of magnetization dynamics. Finally, we solve coupled drift-diffusion equations for charge and spin currents to determine the spatial distribution of the electric potential in the Co20Fe60B20/Au bilayer.

II CURRENT-DRIVEN MAGNETIZATION DYNAMICS

We consider an MTJ comprising an ultrathin free layer with the thickness $t_{f}$ smaller than the critical thickness $t_{\textrm{\tiny{SRT}}}$, below which it acquires a perpendicular magnetic anisotropy  Kanai2012_122403 ; Ikeda2010_721 . The thickness $t_{p}$ of the pinned layer is taken to be larger than $t_{\textrm{\tiny{SRT}}}$ so that the pinned magnetization Mp has an in-plane orientation (Fig. 1). Both layers are assumed to be homogeneously magnetized, and the current flowing through the tunnel barrier is regarded uniform.

To describe the dynamics of the free-layer magnetization M$(t)$, we employ the macrospin approximation, which is well-suited for magnetic layers with nanoscale in-plane dimensions. Since the magnetization magnitude $|\bf{M}|$ = $M_{s}$ at a fixed temperature much lower than the Curie temperature can be considered a constant quantity, the LLGS equation may be reformulated for the unit vector ${\bf m}={\bf M}/M_{s}$  Brataas2012_372 and written as

[TABLE]

where $\gamma>0$ is the electron’s gyromagnetic ratio, $\mu_{0}$ is the permeability of vacuum, $\alpha$ is the Gilbert damping parameter, and ${\bf{H}_{\textrm{\tiny{eff}}}}$ is the effective field acting on the magnetization. In Eq. (1), the last term takes into account the STT proportional to the current density $J$ in the free layer, whereas the field-like torque is disregarded because it does not affect the magnetic dynamics qualitatively  Fang2016_11259 ; Zhu2012_197203 . For symmetric MTJs, the theory gives $\tau_{\textrm{\tiny{STT}}}=(\gamma\hbar/2e)(J/t_{f})\eta/(1+\eta^{2}{\bf m}\cdot{\bf m}_{p})$, where $e$ is the elementary (positive) charge, $\hbar$ is the reduced Planck constant, $\eta=\sqrt{(G_{\textrm{\tiny{P}}}-G_{\textrm{\tiny{AP}}})/(G_{\textrm{\tiny{P}}}+G_{\textrm{\tiny{AP}}})}$ and $G_{\textrm{\tiny{P}}}$ and $G_{\textrm{\tiny{AP}}}$ are the MTJ conductances per unit area in the states with parallel and antiparallel electrode magnetizations, respectively  Slonczewski2007_169 . Since we consider the MTJ connected to a constant-current source, the voltage drop $V=J/G$ across the tunnel barrier depends on the junction’s conductance $G=G_{\textrm{\tiny{P}}}(1+\eta^{2}{\bf m}\cdot{\bf m}_{p})/(1+\eta^{2})$, which leads to a non-sinusoidal dependence of the STT on the angle between m and ${\bf m}_{p}$. The effective field involved in Eq. (1) is defined by the relation ${\bf H}_{\textrm{\tiny{eff}}}=-(\mu_{0}M_{s})^{-1}\partial F/\partial{\bf m}$, where $F({\bf m})$ is the Helmholtz free energy density of ferromagnetic layer. For a homogeneously magnetized unstrained free layer made of cubic ferromagnet, the magnetization-dependent part $\Delta F({\bf m})$ of the effective volumetric energy density may be approximated by the polynomial

[TABLE]

where $m_{i}$ ($i$ = 1, 2, 3) are the direction cosines of M in the crystallographic reference frame with the $x_{3}$ axis orthogonal to the layer surfaces, $K_{1}$ and $K_{2}$ are the coefficients of the fourth- and sixth-order terms defining the cubic magnetocrystalline anisotropy, $K_{s}$ is the parameter characterizing the total specific energy of two interfaces (Co20Fe60B${}_{20}|$MgO and Co20Fe60B${}_{20}|$Au in our case), $U_{\textrm{\tiny{IEC}}}$ is the energy of interlayer exchange coupling (IEC) with the pinned layer (per unit area), $N_{ij}$ are the demagnetizing factors ($N_{13}$ and $N_{23}$ are negligible at in-plane dimensions $L_{1}$, $L_{2}>>t_{f}$), and H is the average magnetic field acting on the free layer. Since the magnetic anisotropy associated with the Co20Fe60B${}_{20}|$MgO interface depends on the electric field $E_{3}$ in MgO  Kanai2012_122403 ; Alzate2014_112410 , the coefficient $K_{s}$ is a function of the current density $J$. Using a linear approximation for the dependence $K_{s}(E_{3})$ supported by first-principles calculations  Niranjan2010_222504 and experimental data  Alzate2014_112410 , we arrive at the relation $K_{s}=K_{s}^{0}+k_{s}V/t_{b}=K_{s}^{0}+k_{s}J/(Gt_{b})$, where $K_{s}^{0}=K_{s}(E_{3}=0)$, $k_{s}=\partial K_{s}/\partial E_{3}$ is the electric-field sensitivity of $K_{s}$, and $V$ is the voltage drop across the MgO layer of thickness $t_{b}$, which is caused by the tunnel current flowing through the junction with the conductance $G$ per unit area.

The numerical integration of Eq. (1) was realized with the aid of the projective Euler scheme, where the condition $|{\bf m}|=1$ is satisfied automatically. A fixed integration step $\delta t=0.5$ fs was used in our computations. The effective field ${\bf H}_{\textrm{\tiny{eff}}}$ was determined from Eq. (2) under the assumption of negligible total magnetic field H acting on the free layer, which is justified by the absence of external magnetic sources and zero mean value of the current-induced Oersted field. Since in the considered heterostructure the magnetization dynamics in the free layer leads to the spin pumping into adjacent nonmagnetic layer, the parameters $\gamma$ and $\alpha$ involved in Eq. (1) were renormalized as  Tserkovnyak2002_117601

[TABLE]

where $\gamma_{0}$ and $\alpha_{0}$ denote the bulk values of $\gamma$ and $\alpha$, $g_{L}$ is the Landé factor, $\mu_{B}$ is the Bohr magneton, and $g_{\uparrow\downarrow}^{r}$ is the complex reflection spin-mixing conductance per unit area of the ferromagnet/normal metal contact  Zwierzycki2005_064420 . The Gilbert parameter $\alpha_{0}$ was regarded as a constant quantity, because numerical estimates show that the dependence of $\alpha_{0}$ on the power of magnetization precession  Slavin2009_1875 is negligible in our case.

The numerical calculations were performed for the Co20Fe60B20/MgO/Co20Fe60B20 junction with the barrier and electrode thicknesses equal to $t_{b}=1.1$ nm, $t_{f}=1.73$ nm, and $t_{p}=5$ nm. A rectangular in-plane shape and nanoscale dimensions $L_{1}=400$ nm and $L_{2}=40$ nm were chosen for the free layer. The demagnetizing factors of such ferromagnetic layer, calculated from the available analytic formulae  Aharoni1998_3432 , were found to be $N_{11}=0.0059$, $N_{22}=0.0626$, $N_{12}=0$, and $N_{33}=0.9315$. A high in-plane aspect ratio $L_{1}/L_{2}=10$ was given to the free layer in order to make energetically more favorable the magnetization orientations in the plane perpendicular to the pinned magnetization ${\bf M}_{p}$, which enhances the STT acting on M. The pinned layer was assumed to have a large area ensuring negligible contribution of the magnetostatic interlayer interaction to the free-layer energy $\Delta F$ in comparison with that of the IEC defined by the relation $U_{\textrm{\tiny{IEC}}}\approx 5.78\textrm{exp}(-7.43\times 10^{9}t_{b}~{}\mathrm{m}^{-1})$ mJ m*-2*  Skowronski2010_093917 . The saturation magnetization $M_{s}=1.13\times 10^{6}$ A m*-1*  Lee2011_123910 and the Gilbert damping constant $\alpha_{0}=0.01$  Ikeda2010_721 were assigned to the Co20Fe60B20 free layer, while its magnetocrystalline anisotropy was described using the coefficients $K_{1}=5\times 10^{3}$ J m*-3*  Hall1960_S157 and $K_{2}=50$ J m*-3*  Zhu2012_197203 . To quantify the VCMA associated with the Co20Fe60B${}_{20}|$MgO interface, we used the measured parameters $K_{s}^{0}=-1.3\times 10^{-3}$ J m*-2*  Ikeda2010_721 and $k_{s}=37$ fJ V*-1* m*-1*  Zhu2012_197203 . The junction’s conductance $G_{\textrm{\tiny{P}}}$ at the chosen MgO thickness was taken to be $8.125\times 10^{9}$ S m*-2*  Tsunekawa2005_FB-05 , and we used typical asymmetry parameter $\eta=0.57$  Sato2011_042501 ; Alzate2014_112410 which yields the tunneling magnetoresistance ratio TMR $=(G_{\textrm{\tiny{P}}}-G_{\textrm{\tiny{AP}}})/G_{\textrm{\tiny{AP}}}\cong 96\%$.

The numerical calculations started with the determination of the equilibrium magnetization orientation in the free Co20Fe60B20 layer at zero applied current. It was found that the initial energy landscape $\Delta F(\phi_{0},\theta_{0})$ has only two minima, which correspond to almost perpendicular-to-plane directions of the free-layer magnetization M. Owing to the IEC with the in-plane magnetized pinned Co20Fe60B20 layer, the magnetization M slightly deviates from the perpendicular-to-plane orientation, tilting towards the pinned magnetization Mp oriented along the $x_{2}$ axis ($\phi_{0}=90^{\circ}$, $\theta_{0}=0.45^{\circ}$ or $179.55^{\circ}$, see Fig. 1). The energy barrier for the coherent magnetization switching at room temperature $T_{r}$ is about $60k_{B}T_{r}$, where $k_{B}$ is the Boltzmann constant. Importantly, the perpendicular magnetic anisotropy is sufficient to prevent the coexistence of metastable states with an in-plane orientation of M, which otherwise could temporarily show up due to thermal fluctuations.

The application of a small current to the MTJ modifies the equilibrium magnetization orientation because the interfacial magnetic anisotropy changes due to a voltage drop $V=J/G$ across the barrier and a nonzero $\tau_{\textrm{\tiny{STT}}}(J)$ appears in Eq. (1). The simulations showed that at $J<0$ the magnetization M progressively rotates towards the PP direction with increasing current, remaining stable up to very high densities $|J|<10^{10}$ A m*-2*. On the contrary, the deviation of M from the PP direction increases when a positive current is applied to the MTJ ($J>0$), reaching $\theta=7.54^{\circ}$ just below the critical density $J_{\textrm{\tiny{min}}}\cong 3.9\times 10^{9}$ A m*-2* at which the magnetization dynamics arises. Remarkably, the predicted value of $J_{\textrm{\tiny{min}}}$ falls into the range of lowest critical current densities $|J_{\textrm{\tiny{min}}}(t_{f})|=(1.2-5.4)\times 10^{9}$ A m*-2* measured experimentally up to date  Zeng2013_1426 . Therefore, we focus below on the magnetic dynamics induced by positive applied currents, which correspond to the tunneling of electrons from the free layer into the pinned one.

Figure 2 (a) shows the trajectory of the end of the unit vector ${\bf m}={\bf M}/M_{s}$ after the destabilization caused by the positive current with the critical density $J_{\textrm{\tiny{min}}}$. Remarkably, the free Co20Fe60B20 layer experiences a dynamic SRT, at which the static magnetic state with almost PP orientation of m transforms into large-angle magnetization precession around in-plane (IP) direction antiparallel to the pinned magnetization Mp. The appearance of such electrically driven SRT can be attributed to the proximity of the free-layer thickness $t_{f}=1.73$ nm to the critical thickness $t_{\textrm{\tiny{SRT}}}=1.745$ nm, at which the size-induced SRT should take place in the considered MTJ at $J=0$. Indeed, the change $\Delta K_{s}=k_{s}J_{\textrm{\tiny{min}}}/(Gt_{b})$ in VCMA promotes voltage-driven SRT to the IP magnetization orientation parallel to the $x_{1}$ axis, while the STT gives rise to the precession of m. The proximity to the thickness-induced SRT also explains very large precession amplitude at $J_{\textrm{\tiny{min}}}$. With increasing current density $J>J_{\textrm{\tiny{min}}}$, the frequency of steady-state magnetization precession rises, whereas its amplitude becomes smaller [Fig. 2(b)]. The precession frequency $\nu$ ranges from 0.95 GHz at $J_{\textrm{\tiny{min}}}$ to 1.54 GHz at the maximal density $J_{\textrm{\tiny{max}}}=5.4\times 10^{9}$ A m*-2* above which the precession disappears  Note1 . Owing to strong STT, the free-layer magnetization stabilizes at $J>J_{\textrm{\tiny{max}}}$ along the direction antiparallel to the magnetization of the pinned Co20Fe60B20 layer.

III SPIN AND CHARGE CURRENTS IN NORMAL-METAL OVERLAYER

The electrically induced magnetic dynamics in the free Co20Fe60B20 layer should lead to the spin pumping into the Au overlayer. The spin-current density can be specified by a tensor ${\bf J}_{s}$ characterizing both the direction of spin flow defined by the unit vector ${\bf e}_{s}$ and the orientation of spin polarization  Dyakonov1971_459 . Since the Co20Fe60B20 thickness is well above a few monolayer range, the imaginary part of the reflection spin-mixing conductance $g_{\uparrow\downarrow}^{r}$ and the transmission spin mixing conductance $g_{\uparrow\downarrow}^{t}$ are negligible. Therefore, the pumped spin-current density ${\bf J}_{\textrm{\tiny{sp}}}$ in the vicinity of the Co20Fe60B${}_{20}|$Au interface can be calculated from the approximate relation ${\bf e}_{s}\cdot{\bf J}_{\textrm{\tiny{sp}}}\cong(\hbar/4\pi)\textrm{Re}[g_{\uparrow\downarrow}^{r}]{\bf m}\times d{\bf m}/dt$  Zwierzycki2005_064420 . Adopting for the Co20Fe60B${}_{20}|$Au interface the theoretical estimate $(e^{2}/h)\textrm{Re}[g_{\uparrow\downarrow}^{r}]\approx 4.66\times 10^{14}$ $\Omega^{-1}$ m*-2* obtained for the Fe$|$Au one  Zwierzycki2005_064420 , we calculated the spin current pumped into Au during the magnetization precession in the free layer. Figure 3 shows representative time dependences of the nonzero spin-current densities $J_{3k}^{\textrm{\tiny{sp}}}(t)$ ($k=$ 1, 2, 3), which correspond to the magnetization dynamics $\textbf{m}(t)$ appearing at the critical charge-current density $J_{\textrm{\tiny{min}}}$. Interestingly, $J_{32}^{\textrm{\tiny{sp}}}$ contains significant dc and ac components, whereas $J_{31}^{\textrm{\tiny{sp}}}$ and $J_{33}^{\textrm{\tiny{sp}}}$ are dominated by the ac component. In the steady-state regime, $J_{32}^{\textrm{\tiny{sp}}}$ oscillates with the frequency $2\nu$, which is two times higher than the precession frequency $\nu$ due to similar oscillations of the direction cosine $m_{2}$.

Taking into account the spin polarization of the charge current governed by the free-layer magnetization $\textbf{M}(t)$, we calculated the total spin-current density $\textbf{J}_{s}=\textbf{J}_{\textrm{\tiny{sp}}}+\textbf{J}_{\textrm{\tiny{sc}}}$ at the Co20Fe60B${}_{20}|$Au interface. Figure 4 shows time dependences of the most interesting component $J_{32}^{s}(t)$ evaluated at critical charge-current densities $J_{\textrm{\tiny{min}}}$ and $J_{\textrm{\tiny{max}}}$. Remarkably, the contributions of spin-polarized charge current ($J_{32}^{\textrm{\tiny{sc}}}$) and precession-induced spin pumping ($J_{32}^{\textrm{\tiny{sp}}}$) have the same sign and phase. At $J=J_{\textrm{\tiny{min}}}$, both $J_{32}^{\textrm{\tiny{sc}}}$ and $J_{32}^{\textrm{\tiny{sp}}}$ exhibit non-sinusoidal time dependences, whereas at $J=J_{\textrm{\tiny{max}}}$ the contribution $J_{32}^{\textrm{\tiny{sc}}}$ assumes almost sinusoidal shape while $J_{32}^{\textrm{\tiny{sp}}}$ becomes practically constant.

The dc and ac components of the spin-current density $J_{32}^{s}(t)$ in the steady-state regime are plotted in Fig.  5 as a function of the charge-current density $J$. At $J<J_{\textrm{\tiny{min}}}$, the dc component $\left\langle J_{32}^{s}\right\rangle$ is small and negative due to the charge-current contribution $J_{32}^{\textrm{\tiny{sc}}}$, and the ac component is zero. Within the precession window $J_{\textrm{\tiny{min}}}<J<J_{\textrm{\tiny{max}}}$, $\left\langle J_{32}^{s}\right\rangle$ grows monotonically with increasing charge current owing to the significant rise of $J_{32}^{\textrm{\tiny{sc}}}$. In contrast, the amplitude of ac component becomes maximal at $J\cong 4.72\times 10^{9}$ A m -2 near the middle of the precession window.

Thus, the Co20Fe60B20/MgO/Co20Fe60B20 tunnel junction excited by a direct charge current can be employed for the generation of spin currents in normal metals  Note2 . The power dissipation $W_{\textrm{\tiny{min}}}\cong J_{\textrm{\tiny{min}}}^{2}L_{1}L_{2}\left\langle G^{-1}\right\rangle$ of such electrically driven spin injector is estimated to be below 40 $\mu$W which is a very small value for devices with sub-micrometer size  Zeng2013_1426 . To evaluate the efficiency of the proposed spin injector, we calculated the electrical potential difference $\Delta V(x_{3},t)=\phi(x_{1}=L_{1}/2,x_{3},t)-\phi(x_{1}=-L_{1}/2,x_{3},t)$ between the lateral sides of the Co20Fe60B20Au bilayer. Owing to the inverse spin Hall effect, the spin flow in the normal metal creates such transverse voltage signal, which can be used to detect this flow experimentally  Saitoh2006_182509 .

To determine the distribution of the electric potential $\phi(\textbf{r},t)$ in the Co20Fe60B20/Au bilayer, we solved the coupled drift-diffusion equations  Dyakonov1971_459 ; Takahashi2003_052409 ; Stiles2004_054408 for charge and spin currents flowing in the Co20Fe60B20 and Au films. The continuity equations for the charge-current density J and the spin-current density $\textbf{J}_{s}$ have the form of $\nabla\cdot\textbf{J}=-\partial\rho/\partial t$ and $\hbar^{-1}\nabla\cdot\textbf{J}_{s}=-\partial\textbf{P}/\partial t-\textbf{P}/\tau_{\textrm{\tiny{sf}}}$, where $\rho$ is the charge density, P is the spin polarization density, and $\tau_{\textrm{\tiny{sf}}}$ is the spin-flip relaxation time. Since spatial variations in the electron concentration n can be neglected for metals  Stiles2004_054408 , explicit expressions for the densities J and $\textbf{J}_{s}$ reduce to

[TABLE]

[TABLE]

where E is the electric field, $\mu$ is the electron mobility, D is the diffusion coefficient, $\alpha_{\textrm{\tiny{SH}}}$ is the spin Hall angle, $\varepsilon_{ikl}$ denote the components of the Levi-Civita tensor (i, k, l = 1,2,3), and the Einstein summation convention is implied. In Eq. (4), the second term describes the anomalous Hall effect characteristic of ferromagnetic metals, while the third term represents the inverse spin Hall effect. The first term in Eq. (5) gives the contribution of the spin-polarized charge current; the last term accounts for the spin Hall effect, which manifests itself in the current-induced spin accumulation near sample boundaries.

The continuity equations were supplemented by appropriate boundary conditions, which should be fulfilled at the Co20Fe60B${}_{20}|$Au interface and the outer boundaries of the Co20Fe60B20/Au bilayer connected to a constant-current source via a gold nanoplate (Fig. 1). At the MgO$|$Co20Fe60B20 interface, the projection $J_{3}$ of the charge current density J on the $x_{3}$ axis of our reference frame orthogonal to the interface was set equal to the density $J_{0}$ of the tunnel current. In addition, the vector J was taken to be parallel to the $x_{3}$ axis near the lateral faces of the Co20Fe60B20/Au bilayer and at the contact with the Au nanoplate, where J satisfies the equality $\textbf{J}=(L_{1}/d)\textbf{J}_{0}$ involving the nanoplate thickness d = 5 nm along the $x_{1}$ axis. At the Co20Fe60B${}_{20}|$Au interface, we specified the spin-current density $\textbf{J}_{s}$ via the boundary condition $\textbf{e}_{n}\cdot\textbf{J}_{s}=\textbf{e}_{n}\cdot\textbf{J}_{\textrm{\tiny{sp}}}-(J_{0}\hbar/2e)\textbf{p}_{f}$, where $\textbf{e}_{n}$ is the unit normal vector of the interface, and $\textbf{p}_{f}=(N_{\uparrow}-N_{\downarrow})/(N_{\uparrow}+N_{\downarrow})\textbf{m}$ is the spin polarization of the ferromagnetic layer defined by the densities of states of spin-up ($N_{\uparrow}$) and spin-down ($N_{\downarrow}$) electrons at the Fermi level  Huang2008_242509 . Of course, the spin-current direction $\textbf{e}_{s}$ was taken to be parallel to the lateral faces of the Co20Fe60B20/Au bilayer in the vicinity of these faces.

The sought functions $\phi(\textbf{r},t)$ and $\textbf{P}(\textbf{r},t)$ were calculated numerically by solving the system of differential continuity equations with the aid of a finite-element method. The calculations were performed in the quasistatic approximation ($\partial\rho/\partial t=\partial P/\partial t=0$), which is justified by the fact that the period $1/f\sim 1$ ns of the current-induced magnetization precession is much longer than the characteristic time of charge ($\sim$0.1 ps  Harris1953_1114 ) and spin ($\tau_{\textrm{\tiny{sf}}}<100$ ps  Elezzabi1996_3220 ) equilibration. Since the size $L_{2}$ of the Co20Fe60B20/Au bilayer along the $x_{2}$ axis is taken to be much smaller than the size $L_{1}$ along the $x_{1}$ one ($L_{2}/L_{1}=0.1$), variations of the potential $\phi$ and the spin polarization density P along the coordinate $x_{2}$ can be ignored. Therefore, we restricted our numerical calculations to the solution of a two-dimensional problem enabling us to determine the functions $\phi(x_{1},x_{3},t)$ and $\textbf{P}(x_{1},x_{3},t)$. In addition, only the component $J_{32}^{\textrm{\tiny{sp}}}$ of the pumped spin current was taken into account in the calculations, because it was found that the components $J_{31}^{\textrm{\tiny{sp}}}$ and $J_{33}^{\textrm{\tiny{sp}}}$ have a negligible effect on the sought output voltage $\Delta V(x_{3},t)$ of the device. The thickness of Au overlayer along the $x_{3}$ axis was chosen to be much larger than the Au spin-diffusion length $\lambda_{\textrm{\tiny{sd}}}=\sqrt{D\tau_{\textrm{\tiny{sf}}}}=35$ nm  Mosendz2010_214403 and set equal to 400 nm.

In the numerical calculations, the conductivity $\sigma=e\mu n$ of Co20Fe60B20 was taken to be $4.45\times 10^{5}$ S m*-1*  Fan2014_3042 , which yields the electron mobility $\mu=n^{-1}2.8\times 10^{26}$ m*-1* V*-1* s*-1*. The anomalous Hall angle $\alpha_{\textrm{\tiny{AH}}}=2\alpha_{\textrm{\tiny{SH}}}$ and the spin polarization $p_{f}$ of Co20Fe60B20 were assumed to be 0.02  Zhu2014_202404 and 0.53  Huang2008_242509 , respectively. For Au, the conductivity equals $4.5\times 10^{7}$ S m*-1*  Haynes2014 , which gives $\mu=4.81\times 10^{-3}$ m2 V*-1* s*-1* and $D=1.25\times 10^{-4}$ m2 s*-1*. The spin-flip relaxation time $\tau_{\textrm{\tiny{sf}}}$ and the spin Hall angle of Au were taken to be 9.84 ps and 0.0035  Mosendz2010_214403 . It should be noted that the spin polarization density in the free Co20Fe60B20 layer was assumed uniform to ensure consistency with the macrospin approximation used to describe the magnetization dynamics.

Using the obtained functions $\phi(x_{1},x_{3},t)$ and $\textbf{P}(x_{1},x_{3},t)$, we calculated spatial distributions of the charge-current density $\textbf{J}(x_{1},x_{3},t)$ in the Co20Fe60B20/Au bilayer and the electrical potential difference $\Delta V(x_{3},t)$ between its lateral sides. Interestingly, the charge-current distribution at any fixed moment t can be represented as a sum of the applied uniform current $\textbf{J}_{0}$ and a vortex-like contribution $\delta\textbf{J}(x_{1},x_{3},t)$ illustrated by Fig. 6. The transverse voltage signal $\Delta V(x_{3},t)$ generated by the device decreases with increasing distance $x_{3}$ from the MgO$|$Co20Fe60B20 interface, falling rapidly within the Co20Fe60B20 layer [Fig. 7(a)].

Remarkably, the analysis of the numerical results obtained for the transverse voltage reveals that $\Delta V(x_{3},t)$ can be fitted with a high accuracy by the analytical formula

[TABLE]

where the first term represents the contribution $\Delta V_{\textrm{\tiny{AHE}}}$ of the anomalous Hall effect, while the second term describes the contribution $\Delta V_{\textrm{\tiny{ISHE}}}$ resulting from the inverse spin Hall effect. Since the coefficients $A(x_{3})$ and $B(x_{3})$ involved in Eq. (6) have very different dependences on the distance $x_{3}$ (see Fig. 8), the ratio $\Delta V_{\textrm{\tiny{ISHE}}}/\Delta V_{\textrm{\tiny{AHE}}}$ changes strongly across the Co20Fe60B${}_{20}|$Au interface. Figure 9 demonstrates that this ratio is mostly very small inside the Co20Fe60B20 layer, but rises steeply near the Co20Fe60B${}_{20}|$Au interface and exceeds 5 in the Au layer. Hence, measurements of the average voltage signal created by the Co20Fe60B20 layer provide information on the anomalous Hall effect, whereas the potential difference $\Delta V(x_{3},t)$ between the faces of the Au layer measured at $x_{3}>25$ nm characterizes the inverse spin Hall effect.

Figure 10 shows how the dc and ac components of the transverse signal $\Delta V$ averaged over the thickness $t_{f}$ of the Co20Fe60B20 layer vary with the charge-current density J. It can be seen that the curves are similar to the dependences $J_{32}^{\textrm{\tiny{sc}}}(J)$ presented in Fig. 5, which describe the spin injection into Au caused by the spin-polarized charge current. In contrast, Fig. 7(b) demonstrates the dc component $\left\langle\Delta V\right\rangle(J)$ and the amplitude $\delta V_{\textrm{\tiny{amp}}}(J)$ of the GHz-frequency ac component calculated at $x_{3}=40$ nm. Importantly, both the dc and microwave signals appear to be large enough for the experimental detection within the precession window. Moreover, the dependences $\left\langle\Delta V\right\rangle(J)$ and $\delta V_{\textrm{\tiny{amp}}}(J)$ repeat the graphs shown in Fig. 5 for the total spin-current density $J_{32}^{s}$ generated at the Au$|$Co20Fe60B20 interface, differing by a constant factor of 20.61 nV $\mu$J*-1* m2 only. Hence the measurements of $\Delta V$ by nanocontacts placed at distances $\delta x_{3}\sim\lambda_{\textrm{\tiny{sd}}}$ from the boundary of ferromagnetic layer provide information on the spin injection into the normal metal.

IV SUMMARY

In summary, we presented a comprehensive theoretical study of the spin dynamics and charge transport in the Co20Fe60B20/MgO/Co20Fe60B20/Au tunneling heterostructure connected to a constant-current source. The performed numerical calculations enabled us to find the range of current densities, within which electrically driven magnetization precession appears in the free Co20Fe60B20 layer, and to determine the precession frequencies and trajectories. Remarkably, a novel, dynamic SRT has been predicted, which is caused by the joint impact of STT and VCMA and has the form of magnetization reorientation between initial static direction and final dynamic precessional state. The results obtained for the magnetization dynamics were then used to evaluate the dc and ac components of the spin current generated in the Au overlayer owing to the precession-driven spin pumping and the oscillating spin polarization of the charge current. Taking into account the inverse spin Hall effects and anomalous Hall effects, we finally calculated the charge flow and electric-potential distribution in the Co20Fe60B20/Au bilayer via the numerical solution of coupled drift-diffusion equations for charge and spin currents. It is shown that the potential difference between the lateral faces of the Au layer is large enough for experimental detection, which demonstrates significant efficiency of the described spin injector.

ACKNOWLEDGMENTS

The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ”BASIS”.