Charge pumping by magnetization dynamics in magnetic and semimagnetic tunnel junctions with interfacial Rashba or bulk extrinsic spin-orbit coupling

We develop a time-dependent nonequilibrium Green function (NEGF) approach to the problem of spin pumping by precessing magnetization in one of the ferromagnetic layers within F$|$I$|$F magnetic tunnel junctions (MTJs) or F$|$I$|$N semi-MTJs in the presence of intrinsic Rashba spin-orbit coupling (SOC) at the F$|$I interface or the extrinsic SOC in the bulk of F layers of finite thickness (F, ferromagnet; N, normal metal; I, insulating barrier). To express the time-averaged pumped charge current, or the corresponding dc voltage signal in an open circuit, we construct a novel solution to double-time-Fourier-transformed NEGF equations. The two energy arguments of NEGFs in this representation are connected by the Floquet theorem describing multiphoton emission and absorption processes. Within this fully quantum-mechanical treatment of the conduction electrons, we find that (i) only in the presence of the interfacial Rashba SOC, the nonzero dc pumping voltage $V_{\mathrm{pump}}$ in F$|$I$|$N junctions can emerge at the adiabatic level (i.e., proportional to the microwave frequency), which could explain recent experiments on microwave-driven semi-MTJs [T. Moriyama et al., Phys. Rev. Lett. 100, 067602 (2008)]; (ii) a unique signature of this charge pumping phenomenon, where the Rashba SOC within the precessing F layer participates in the pumping process, is a $V_{\mathrm{pump}}$ that changes sign as the function of the precession cone angle; (iii) unlike conventional spin pumping in MTJs in the absence of any SOC, where one emitted or absorbed microwave photon is sufficient to match the exact solution in the frame rotating with the magnetization, the presence of the Rashba SOC requires taking into account up to 10 photons in order to reach the asymptotic value of pumped charge current; (iv) the disorder within F$|$I$|$F MTJs can enhance $V_{\mathrm{pump}}$ in the quasiballistic transport regime; and (v) the extrinsic SOC in F$|$I$|$F MTJs causes spin relaxation and eventually the decay of $V_{\mathrm{pump}}$, which becomes negligible when the ratio of F layer thickness to the spin-diffusion length is around five. Our formalism can also be applied to charge and spin propagation in any noninteracting open quantum system that is brought out of equilibrium by time-dependent periodic external fields of arbitrary strength or frequency.

Figure 1

(a) F$|$I$|$F MTJ and (b) F$|$I$|$N semi-MTJ with precessing magnetization of a single F layer are modeled on a simple cubic finite-size tight-binding lattice attached to semi-infinite ideal (disorder and interaction free) N leads. The thicknesses of the ferromagnetic layers and thin insulating barrier is measured using the number of atomic monolayers $d_F$ and $d_I$, respectively. For example, $d_F=8$ and $d_I=4$ in the illustration, while in the actual calculations we use $d_F=50$ and $d_I=5$ monolayers of cross section $20\times 20$ lattice sites. The interfacial Rashba SOC due to structural inversion asymmetry of the junction is included in the last monolayer of the F slab that is in direct contact with the tunnel barrier I. The F layers can also include disorder modeled as a random on-site potential and the corresponding extrinsic SOC, while binary alloy disorder in the I layer models AlO${}_x$-type tunnel barrier.

Figure 2

The Fano factor of the zero-temperature and zero-frequency shot noise vs. the disorder strength $W$ in transport through N slab with static disorder or F slab with both static disorder and the corresponding extrinsic SOC of strength ${\lambda }_{\mathrm{ESO}}=0.04\gamma$. Both slabs consists of 50 monolayers (containing $20\times 20$ atoms per cross section) that are connected to two semi-infinite ideal N leads.

Figure 3

The decay of current polarization along the diffusive F layer with static disorder of strength $W=3\gamma$ and the extrinsic SOC of strength ${\lambda }_{\mathrm{ESO}}=0.1\gamma$. The F layer is attached to two semi-infinite ideal N leads where charge current which is 100% spin-polarized along the $z$ axis, ${\mathbf{P}}^{\mathrm{in}}=(0,0,1)$, is injected from the left N lead and $P_z^{\mathrm{out}}$ is computed in the right N lead for F layer of thickness $d_F$. The unit vector of the magnetization in F is either parallel (solid line) or antiparallel (dashed line) to the $z$ axis. The inset shows spin-diffusion length as a function of ${\lambda }_{\mathrm{ESO}}$ when the F layer is replaced by a diffusive N layer with different strengths of extrinsic SOC, where each value of $L_{\mathrm{sf}}$ is extracted by fitting exponentially decaying function to $P_z^{\mathrm{out}}$ vs. $d_N$ curves.

Figure 4

(a) The setup for the measurement of the out-of-plane TAMR in F$|$I$|$N semi-MTJ, defined by Eq. (6), as a function of the angle $\phi$ between the static magnetization of the F layer and the transport direction (the $x$ axis). In panel (b), the Rashba SOC at the F monolayer in contact with the tunnel barrier I is fixed at ${\gamma }_{\mathrm{RSO}}=0.5\gamma$, while panel (c) shows $\mathrm{TAMR}(\phi ={90}^{\circ })$ for different values of ${\gamma }_{\mathrm{RSO}}$. (d) The setup for the measurement of the out-of-plane TAMR in F$|$I$|$F MTJ, defined by Eq. (7), as a function of the magnetization orientation in each of the two F layers with respect to the transport direction. The TAMR depends on the absolute magnetization directions ${\mathbf{m}}_1$ and ${\mathbf{m}}_2$. In panel (e), the Rashba SOC of strength ${\gamma }_{\mathrm{RSO}}=0.5\gamma$ is present at both F monolayers in contact with the tunnel barrier I. The F layers in both semi-MTJ and MTJ have finite thickness $d_F=50$.

Figure 5

The magnitude $|{\mathbf{P}}^{\mathrm{out}}|$ of the spin-polarization vector of outgoing charge current in the right N lead after fully $|{\mathbf{P}}^{\mathrm{in}}|=1$ spin-polarized current is injected from the left N lead traversing a monolayer with the Rashba SOC of strength ${\gamma }_{\mathrm{RSO}}$. The spin-polarization vector ${\mathbf{P}}^{\mathrm{in}}$ can point along three different axes of the coordinate system in Fig. 1, where ${\mathbf{P}}^{\mathrm{in}}=(0,1,0)$ and ${\mathbf{P}}^{\mathrm{in}}=(0,0,1)$ are parallel to the Rashba monolayer while ${\mathbf{P}}^{\mathrm{in}}=(1,0,0)$ is orthogonal to the Rashba monolayer. The direction of ${\mathbf{P}}^{\mathrm{out}}$ remains collinear with ${\mathbf{P}}^{\mathrm{in}}$, as illustrated in the inset.

Figure 6

The comparison of the dc pumping voltage in a clean N$|$F$|$I$|$F$|$N junction with finite thickness F layers ($d_F=50$) and a clean F$|$I$|$F junction whose F layers are semi-infinite.[16] The two curves can be computed using either the adiabatic NEGF formula in the rotating frame Eq. (40) or the adiabatic scattering formula Eq. (43). The parameters of these junction are chosen as $E_F=-2\gamma$, $\Delta =2\gamma$, and $U_b=9\gamma$.

Figure 7

The dc pumping voltage in a clean F$|$I$|$F MTJ with finite thickness F layers ($d_F=50$) in the absence of any SOCs computed using (i) the exact solution Eq. (40) obtained via the rotating frame approach or, equivalently, full time-dependent solution Eq. (30) with one photon processes taken into account $N_{\mathrm{ph}}=1$ and (ii) truncated (to $n=\pm 1$) continued fractions solution to double-time-Fourier-transformed NEGF equations which gives pumped charge current via Eq. (46). The shaded area marks the interval of precession cone angles $\theta \lesssim {10}^{\circ }$ beyond which the continued fractions solution is not applicable anymore.

Figure 8

The dc pumping voltage in clean F$|$I$|$N semi-MTJ [panels (a), (c), and (d)] and F$|$I$|$F MTJ [panel (b)] with finite thickness ($d_F=50$) F layers and nonzero interfacial Rashba SOC. The Rashba SOC is located within the last monolayer of the precessing F layer that is in contact with the tunnel barrier I in (a), (c), and (d) [as illustrated in Fig. 1a], or such edge monolayers are present in the left or both F layers [as illustrated in Fig. 1b] in panel (b). The data in panels (a)–(c) are computed by considering only one microwave photon exchange process, while in panel (d) we show the correction to this result when up to nine microwave photons are taken into account in Eq. (30) applied to F$|$I$|$N semi-MTJ.

Figure 9

The dc pumping voltage in F$|$I$|$F MTJs of finite thickness F layers ($d_F=50$) with (a) static disorder of strength $W$ within F layers and (b) static disorder of strength $W=3\gamma$ ensuring diffusive transport regime (see Fig. 2) and the extrinsic SOC of strength ${\lambda }_{\mathrm{ESO}}$ determined by such disorder via Eq. (2). The tunnel barrier I in both panels contains binary alloy disorder $\delta U_b=0.5\gamma$. The spin-diffusion length corresponding to the values of ${\lambda }_{\mathrm{ESO}}$ is shown in the inset of Fig. 3.