Energy-efficient spin injector into semiconductors driven by elastic waves

The generation of significant spin imbalance in nonmagnetic semiconductors is crucial for the functioning of many spintronic devices, such as magnetic diodes and transistors, spin-based logic gates, and spin-polarized lasers. An attractive design of spin injectors into semiconductors is based on spin pumping from a precessing ferromagnet, but the classical excitation of magnetization precession by a microwave magnetic field leads to the high-power consumption of the device. Here, we describe theoretically a spin injector with greatly reduced energy losses, in which the magnetic dynamics is excited by an elastic wave generated in a ferromagnet-semiconductor heterostructure by an attached piezoelectric transducer. To demonstrate the efficient functioning of such an injector, we first perform micromagnetoelastic simulations of the coupled elastic and magnetic dynamics in $\mathrm{Ni}$ films and $\mathrm{Ni}/\mathrm{GaAs}$ bilayers traversed by plane longitudinal and shear waves. For thick $\mathrm{Ni}$ films, it is shown that a monochromatic acoustic wave generates a spin wave with the same frequency and wavelength, which propagates together with the driving wave over distances of several micrometers at excitation frequencies $\nu \approx 10$ GHz close to the frequency of ferromagnetic resonance. The simulations of $\mathrm{Ni}/\mathrm{GaAs}$ bilayers with $\mathrm{Ni}$ thicknesses comparable to the wavelength of the injected acoustic wave demonstrate the development of a steady-state magnetization precession at the $\mathrm{Ni}|\mathrm{GaAs}$ interface. The amplitude of such a precession has a maximum at $\mathrm{Ni}$ thickness amounting to three quarters of the wavelength of the elastic wave, which is explained by an analytical model. Using simulation data obtained for the magnetization precession at the $\mathrm{Ni}|\mathrm{GaAs}$ interface, we evaluate the spin current pumped into $\mathrm{GaAs}$ and calculate the spin accumulation in the semiconducting layer by solving the spin diffusion equation. Then the electrical signals resulting from the spin flow and the inverse spin Hall effect are determined via the numerical solution of the Laplace's equation. It is shown that amplitudes of these ac signals near the interface are large enough for experimental measurement, which indicates an efficient acoustically driven spin pumping into $\mathrm{GaAs}$ and a rather high spin accumulation in this semiconductor.

Figure 1

Ferromagnet-semiconductor heterostructure comprising $\mathrm{Ni}$ and $\mathrm{GaAs}$ layers. An elastic wave (longitudinal or transverse) with the wave vector $\mathbf{k}$ is injected into the $\mathrm{Ni}$ layer by the attached piezoelectric transducer. The thicknesses of the $\mathrm{Ni}$ and $\mathrm{GaAs}$ layers are denoted by $t_{\text{F}}$ and $t_{\text{N}}$, respectively. Precessing magnetization in Ni creates a spin imbalance in GaAs, which then produces a measurable voltage $V_{\text{s}}$ between an attached iron probe and a nonmagnetic contact.

Figure 2

Spatial distributions of strains in the driving (a) longitudinal and (b) transverse elastic waves and strain-induced variations of the magnetization direction cosines $m_i$ in the $2\text{−}\mu \mathrm{m}$-thick $\mathrm{Ni}$ film. Excitation frequency $\nu$ is equal to the resonance frequency ${\nu }_{\text{res}}=9.6$ GHz, and snapshots are taken at (a) 0.37 ns and (b) 0.52 ns.

Figure 3

Secondary elastic waves generated by the magnetization precession induced by the primary (a) longitudinal and (b) transverse waves in the 2-$\mu \mathrm{m}$-thick $\mathrm{Ni}$ film. Snapshots are taken at 0.37 ns.

Figure 4

Time dependence of the magnetization precession at the $\mathrm{Ni}|\mathrm{GaAs}$ interface excited by (a) longitudinal or (b) transverse elastic waves with the frequency $\nu ={\nu }_{\text{res}}=9.6$ GHz. Thickness $t_{\text{F}}$ of the $\mathrm{Ni}$ film in each case is equal to a corresponding wavelength ${\lambda }_L$ or ${\lambda }_T$. The first 2 ns are shown; after that the magnetization continues to precess in a stationary mode with a constant amplitude.

Figure 5

Amplitude of the magnetization precession at the $\mathrm{Ni}|\mathrm{GaAs}$ interface as a function of the $\mathrm{Ni}$ thickness $t_{\text{F}}$ normalized by the wavelength $\lambda$ of the driving longitudinal or transverse elastic wave. The plots show the maximal change $\mathrm{\Delta }{m}_x$ of the out-of-plane direction cosine $m_x(x=t_{\text{F}},t)$ normalized by the largest value of $\mathrm{\Delta }{m}_x$ in the studied thickness range. The excitation frequency equals ${\nu }_{\text{res}}=9.6$ GHz.

Figure 6

Expected thickness dependences of the strain amplitudes at the interface between $\mathrm{Ni}$ and $\mathrm{GaAs}$ layers and ${\mathrm{CoFe}}_2{\mathrm{O}}_4$ and Pt layers. The amplitude of each strain is normalized by its maximal value. The thickness $t_{\text{F}}$ of the ferromagnetic layer is normalized by the wavelength $\lambda$ of the excited longitudinal or transverse elastic wave. The excitation frequency equals 9.6 GHz ($\mathrm{Ni}/\mathrm{GaAs}$) and 11 GHz (${\mathrm{CoFe}}_2{\mathrm{O}}_4/\mathrm{Pt}$).

Figure 7

Dependence of the amplitude of the magnetization precession at the $\mathrm{Ni}|\mathrm{GaAs}$ interface on the excitation frequency $\nu$. The points show the maximal deviation $\mathrm{\Delta }{m}_x\left(\nu \right)$ of the out-of-plane magnetization direction cosine $m_x$ from its equilibrium value. The thickness of the $\mathrm{Ni}$ layer equals three quarters of the wavelength of the driving longitudinal or transverse elastic wave.

Figure 8

Time dependences of the spin-current densities $J_{xj}^{\mathrm{SP}}$ pumped into $\mathrm{GaAs}$ by the (a) longitudinal and (b) transverse elastic waves with the frequencies 9.9 and 10 GHz, respectively. The time period shown in the figure corresponds to the steady-state regime of the magnetization precession at the $\mathrm{Ni}|\mathrm{GaAs}$ interface. The thickness of the $\mathrm{Ni}$ layer equals three quarters of the wavelength of the driving longitudinal or transverse elastic wave.

Figure 9

Amplitude $\delta V_z^{\mathrm{ISHE}}\left(x\right)$ of the ac voltage between the lateral sides of the $\mathrm{Ni}/\mathrm{GaAs}$ bilayer excited by (a) longitudinal and (b) transverse elastic waves with frequencies 9.9 and 10 GHz, respectively. The thickness of the $\mathrm{GaAs}$ layer equals $5\mu \mathrm{m}$, and its width $w_{\mathrm{N}}$ is indicated in the figure.

Figure 10

Amplitude $\delta V_s$ and phase ${\varphi }_s$ (inset) of the ac spin voltage between the lateral Fe probe and a normal-metal contact at the $\mathrm{GaAs}$ free surface plotted as a function of the distance $x-t_{\mathrm{F}}$ from the $\mathrm{Ni}|\mathrm{GaAs}$ interface. The $\mathrm{Ni}/\mathrm{GaAs}$ bilayer is excited either by the longitudinal wave with frequency 9.9 GHz or by the transverse wave with frequency 10 GHz. The thickness of the $\mathrm{GaAs}$ layer equals $5\mu \mathrm{m}$.