Long-Range Phonon Spin Transport in Ferromagnet–Nonmagnetic Insulator Heterostructures

We investigate phonon spin transport in an insulating ferromagnet–nonmagnet–ferromagnet heterostructure. We show that the magnetoelastic interaction between the spins and the phonons leads to nonlocal spin transfer between the magnets. This transfer is mediated by a local phonon spin current and accompanied by a phonon spin accumulation. The spin conductance depends nontrivially on the system size, and decays over millimeter length scales for realistic material parameters, far exceeding the decay lengths of magnonic spin currents.

Figure 1

Heterostructure consisting of a nonmagnetic insulator of length $L$ sandwiched between two identical magnetic insulators of length $d$. The left and right spins interact via the phonons in the nonmagnetic insulator that may support a finite spin current between the magnets. The magnon spin accumulations in the magnets are parametrized with magnon chemical potentials ${\mu }_{L/R}$. These chemical potentials are generated by metallic leads (not shown) that are attached to the magnets.

Figure 2

Nonlocal spin conductance (17) as a function of the lengths $d$ and $L$ of the magnetic and nonmagnetic insulators, for the parameters stated in the main text. The lengths are given in units of half wavelengths of phonons at the ferromagnetic resonance frequency in the respective material.

Figure 3

Decay of the spin conductance (17) as a function of the length $L$ of the nonmagnetic insulator in units of half-wavelengths, for $d=2\lambda /7=200\mathrm{nm}$ and the parameters stated in the main text. Solid lines denote the envelope of the conductance curve; in the shaded area the conductance oscillates rapidly between the minimal and maximal values given by the envelope, see also Fig. 2. The dashed line is an exponential fit with decay length $\approx 2650\times \tilde{\lambda }/2\approx 0.85\mathrm{mm}$.

Figure 4

(a) Phonon spin and (b) phonon spin current densities for ${\mu }_L=0={\mu }_R$ (solid lines), and ${\mu }_L=0.01\times \hslash {\omega }_{\mathrm{FM}}$ and ${\mu }_R=0$ (dashed lines) for the parameters stated in the main text. The system size is set to $d=2\lambda /7=200$ and $L=3\tilde{\lambda }/7=275.6\mathrm{nm}$. The shaded region denotes the magnets. The small jumps in $l_z$ at the interfaces are due to the different mass densities of the magnets and the nonmagnet. On the other hand, the jumps in $j_z$ are not only caused by the change of shear modulus but also by the magnetoelastic coupling, see Eq. (9) For comparison, the macrospin magnon spin densities in both magnets are $\approx -600\times 2\hslash /\lambda$.