Magnon-mediated spin current noise in ferromagnet $|$ nonmagnetic conductor hybrids

The quantum excitations of the collective magnetization dynamics in a ferromagnet (F)—magnons—enable spin transport without an associated charge current. This pure spin current can be transferred to electrons in an adjacent nonmagnetic conductor (N). We evaluate the finite temperature noise of the magnon-mediated spin current injected into N by an adjacent F driven by a coherent microwave field. We find that the dipolar interaction leads to squeezing of the magnon modes, giving them wave-vector-dependent nonintegral spin, which directly manifests itself in the shot noise. For temperatures higher than the magnon gap, the thermal noise is dominated by large-wave-vector magnons which exhibit negligible squeezing. The noise spectrum is white up to the frequency corresponding to the maximum of the temperature or the magnon gap. At larger frequencies, the noise is dominated by vacuum fluctuations. The shot noise is found to be much larger than its thermal counterpart over a broad temperature range, making the former easier to be measured experimentally.

Figure 1

System schematic. An applied static magnetic field $\left(H_0\widehat{z}\right)$ saturates the magnetization of the ferromagnet (F) along the $z$ direction. An oscillating magnetic field $\left(h_{0}cos\omega t\widehat{x}\right)$ creates nonequilibrium, in addition to thermal, magnonic excitations in F, which annihilate at the latter's interface with a nonmagnetic conductor (N), creating new excitations and transferring spin current.

Figure 2

The eigenfrequency $\left({\omega }_0/2\pi \right)$ and the squeezing-mediated relative change in the effective spin $\left(\delta \right)$ of the uniform s-magnon mode vs the external plus effective anisotropy field ${\mu }_0{H}_{za}={\omega }_{za}/\left|\gamma \right|$. The degree of squeezing is larger for iron $(M_s=1.7\times {10}^6$ A/m) film as compared to the YIG $(M_s=1.4\times {10}^5$ A/m) film due to the former's larger saturation magnetization and, hence, stronger dipolar interaction. $\left|\gamma \right|\approx 1.8\times {10}^{11}$ Hz/T for both materials.

Figure 3

Normalized spin current shot noise power spectra [Eq. (51)]. The system considered is a $\mathrm{YIG}|\mathrm{Pt}$ bilayer driven with a coherent microwave drive at ferromagnetic resonance, i.e., $\omega ={\omega }_0$. $l_x$ denotes the thickness of the YIG layer.

Figure 4

Normalized zero-frequency noise power vs temperature for $\mathrm{YIG}|\mathrm{Pt}$ bilayers. The numerically evaluated results, depicted by solid lines with symbols, are compared with the analytical expressions, depicted by dashed lines. The YIG thicknesses considered are (a) 10 nm and (b) $1\mu \mathrm{m}$. The former corresponds to a quasi-2D continuum and the latter to a quasi-three-dimensional (3D) continuum.

Figure 5

Noise power spectra normalized to their respective zero-frequency values for $\mathrm{YIG}|\mathrm{Pt}$ bilayers at $T=1$ K. The numerically evaluated results, depicted by solid lines with symbols, are compared with the analytical expressions, depicted by dashed lines. The YIG thicknesses considered are (a) 10 nm and (b) $1\mu \mathrm{m}$. The former corresponds to a quasi-2D continuum and the latter to a quasi-3D continuum.