Spin and charge pumping in magnetic tunnel junctions with precessing magnetization: A nonequilibrium Green function approach

We study spin and charge currents pumped by precessing magnetization of a single ferromagnetic layer within $F\left|I\right|N$ or $F\left|I\right|F$ ($F$-ferromagnet; $I$-insulator; $N$-normal metal) multilayers of nanoscale thickness attached to two normal-metal electrodes with no applied bias voltage between them. Both simple one-dimensional model, consisting of a single precessing spin and a potential barrier as the “sample,” and realistic three-dimensional devices are investigated. In the rotating reference frame, where the magnetization appears to be static, these junctions are mapped onto a four-terminal dc circuit whose effectively half-metallic ferromagnetic electrodes are biased by the frequency $\hslash \omega /e$ of microwave radiation driving magnetization precession at the ferromagnetic resonance (FMR) conditions. We show that pumped spin current in $F\left|I\right|F$ junctions, diminished behind the tunnel barrier and increased in the opposite direction, is filtered into charge current by the second $F$ layer to generate dc pumping voltage of the order of $\sim 1\mu \text{V}$ (at FMR frequency $\sim 10\text{GHz}$) in an open circuit. In $F\left|I\right|N$ devices, several orders of magnitude smaller charge current and the corresponding dc voltage appear concomitantly with the pumped spin current due to barrier induced asymmetry in the transmission coefficients connecting the four electrodes in the rotating-frame picture of pumping.

Figure 1

(Color online) (a) The 1D model of spin pumping where the sample consisting of two sites, one hosting the single spin rotating with frequency $\omega$ and the other one hosting the potential barrier of height ${\epsilon }_I$, is attached to two semi-infinite tight-binding chains ($\gamma$ is the hopping parameter) playing the role of electrodes with no applied bias voltage between them. In the rotating reference frame the spin is static and the device (a) is mapped into the four-terminal dc circuit in panel (b) whose electrodes have electrochemical potential shifted by $\pm \hslash \omega /2$ with respect to the equilibrium Fermi level $E_F$ of unbiased electrodes in the laboratory frame.

Figure 2

The (a) $S_z$-spin currents and (b) charge currents pumped by a single precessing spin as a function of the potential barrier on the second site of the sample in 1D model shown in Fig. 1a. The parameters of the model are: $f=\omega /2\pi =20\text{GHz}$; $\theta =10°$, $\Delta /E_F=0.85$, and electrons in the macroscopic reservoirs to which the electrodes are attached have the Fermi energy $E_F=2\gamma$.

Figure 3

(Color online) The magnetic tunnel junction with precessing magnetization in the left $F$ layer is modeled on a simple-cubic tight-binding lattice. The thicknesses of the ferromagnetic ($F$, $F^{\prime }$) and thin insulating $\left(I\right)$ layers are measured using the number of atomic monolayers $d_F$, $d_{F^{\prime }}$, and $d_I$, respectively. The P and AP configurations of MTJ correspond to the magnetization of the right $F$ layer being parallel or antiparallel to the $z$ axis around which spatially uniform magnetization of the left $F$ layer steadily precesses with a constant cone angle $\theta$.

Figure 4

(Color online) The dc pumping voltage in $F\left|N\right|F$ multilayers attached to two semi-infinite $N$ electrodes as the function of the thickness of $F$ layer whose magnetization is precessing with cone angle $\theta =10°$ at frequency $f=\omega /2\pi =20\text{GHz}$. The parameters describing the multilayer are $E_F=4.5\text{eV}$, $\Delta /E_F=0.85$ (in both $F$ layers), and $\gamma =1.0\text{eV}$.

Figure 5

(Color online) The dc pumping voltage in [(a) and (b)] $F\left|I\right|F$ and [(c) and (d)] $F\left|I\right|N$ multilayers attached to two semi-infinite $N$ electrodes as the function of the barrier height $U_b$ (measured relative to the Fermi energy). The magnetization of the left $F$ layer is precessing with cone angle $\theta =10°$ at frequency $f=\omega /2\pi =20\text{GHz}$. The parameters describing the multilayer are $E_F=4.5\text{eV}$, $\Delta /E_F=0.85$ (in both $F$ layers for $F\left|I\right|F$ junction), and $\gamma =1.0\text{eV}$.