Quantum spin transfer torque induced nonclassical magnetization dynamics and electron-magnetization entanglement

The standard spin transfer torque (STT)—where spin-polarized current drives the dynamics of magnetization viewed as a classical vector—requires noncollinearity between electron spins carried by the current and magnetization of a ferromagnetic layer. However, recent experiments [A. Zholud et al., Phys. Rev. Lett. 119, 257201 (2017)] observing magnetization dynamics in spin valves at cryogenic temperatures, even when electron spin is collinear to magnetization, point at overlooked quantum effects in STT that can lead to highly nonclassical magnetization states. Using quantum many-body treatment, where an electron injected as a spin-polarized wave packet interacts with local spins comprising the anisotropic quantum Heisenberg ferromagnetic chain, we define quantum STT as any time evolution of local spins due to an initial many-body quantum state not being an eigenstate of an electron+local-spins composite system. For time evolution caused by injected spin-$\downarrow$ electron scattering off local $\uparrow$-spins, entanglement between electron and local spin subsystems takes place leading to decoherence and, therefore, shrinking of the total magnetization but without rotation from its initial orientation, which is compatible with the experiments. Furthermore, the same processes—entanglement and thereby induced true decoherence—are present even in the standard noncollinear geometry, intertwined with the usual magnetization rotation. This is because STT in a quantum many-body picture is always caused by an electron spin-$\downarrow$ factor state, and the only difference between collinear and noncollinear geometries is in the relative size of the contribution of the initial separable state containing such a factor to the superposition of separable many-body quantum states generated by time evolution.

Figure 1

Schematic view of a quantum many-body system consisting of a FM layer whose $N=5$ local spins comprise the open $XXZ$ quantum Heisenberg ferromagnetic chain with anisotropic exchange interactions $J_z>J$, and they are attached to a 1D TB chain (composed of $L_x=400$ sites) where electron hops with parameter $\gamma$. The spin-polarized electron wave packet is injected along the TB chain, with its spin pointing in the $-z$ or $+x$ direction, which is collinear [a case in which the standard STT of Slonczewski [2] and Berger [3] in Eq. (1) is absent] or noncollinear, respectively, to local spins pointing initially along the $+z$ direction. The spin of the wave packet interacts with local spins via $s-d$ exchange interaction $J_{sd}$.

Figure 2

(a) Eigenspectrum of the $XXZ$ quantum Heisenberg ferromagnetic chain whose $N=5$ local spins in Fig. 1 do not interact with electron spin ($J_{sd}=0$). (b) Eigenspectrum of the many-body Hamiltonian in Eq. (3) whose local spins interact via $s-d$ interaction ($J_{sd}=0.1$ eV) with electron spin within a TB chain of length $L_x=400$. (c) Expectation value of electron spin (first column) and local spins, extracted from their subsystem density matrices via Eq. (6), in the degenerate ground state of the lowest energy in panel (a) (red arrows) or (b) (blue arrows).

Figure 3

Time dependence of the expectation value of spin (in units $\hslash /2$) obtained from spin-$\frac{1}{2}$ density matrix in Eq. (6) for (a) spin of an injected electron wave packet in Fig. 1, which at $t=0$ points in the $-z$ direction that is collinear and antiparallel to local spins pointing in the $+z$ direction; and (c) first local spin in Fig. 1 [the time dependence of the expectation value of local spins $n=2$–5 is nearly identical to (c)]. (b) Purity defined in Eq. (7) of the subsystem composed of electron degrees of freedom (orbital and spin) or of the subsystem composed of all local spins. (d) Probability in Eq. (8) to find an electron-spin+local-spins subsystem in a many-body quantum state $|{\sigma }_e;{\sigma }_1{\sigma }_2{\sigma }_3{\sigma }_4{\sigma }_5\rangle$.

Figure 4

Panels (a)–(d) plot the same information as panels (a)–(d), respectively, in Fig. 3 but for an injected electron wave packet that at $t=0$ is spin-polarized in the $+x$ direction, i.e., noncollinear to local spins pointing in the $+z$ direction.

Figure 5

Time dependence of the expectation value of spin (in units $\hslash /2$) obtained from respective density matrices for (a) spin of injected electron wave packet that at $t=0$ points in the $-z$ direction that is collinear and antiparallel to a single local spin-$\frac{5}{2}$ pointing in the $+z$ direction; and (c) local spin-$\frac{5}{2}$. (b) Purity defined in Eq. (7) for the subsystem composed of electron degrees of freedom (orbital and spin) or of the subsystem composed of all local spins. (d) Probability in Eq. (8) to find an electron-spin+local-spin subsystem in a many-body quantum state $|{\sigma }_e;S_z\rangle$.

Figure 6

Panels (a)–(d) plot the same information as panels (a)–(d), respectively, in Fig. 5 but for an injected electron wave packet that at $t=0$ is spin-polarized in the $+x$ direction, i.e., noncollinear to local spin-$\frac{5}{2}$ pointing in the $+z$ direction.