Nuclear surface acoustic resonance with spin-rotation coupling

We show that, under an appropriate out-of-plane static magnetic field, nuclear spins in a thin specimen on a surface acoustic wave (SAW) cavity can be resonantly excited and detected through spin-rotation coupling. Since such a SAW cavity can have the quality factor as high as ${10}^4$ and the mode volume as small as ${10}^{-2}{\mathrm{mm}}^3$ the signal-to-noise ratio in detecting the resonance is estimated to be quite high. We argue that detecting nuclear spin resonance of a single flake of an atomically thin layer of two-dimensional semiconductor, which has so far been beyond hope with the conventional inductive method, can be a realistic target with the proposed scheme.

Figure 1

A snapshot of a vectorial velocity field $\mathrm{Re}\left[\dot{\mathit{u}}\right]$ in the $zx$ plane accompanying with a plane monochromatic surface wave propagating along the $x$ axis. The values $q_L$ and $q_T$ used here are for ${\mathrm{LiNbO}}_3$ with ${\lambda }_{\mathrm{SAW}}=40\mu \mathrm{m}$.

Figure 2

A density plot of the $y$ component of the normalized vortex field ${\left(\mathrm{Re}\left[\mathit{\Omega }\right]\right)}_y={\left(\mathrm{Re}\left[\mathbf{\nabla }\times \dot{\mathit{u}}\right]\right)}_y$ accompanying with the same SAW as that shown in Fig. 1.

Figure 3

A mode profile of a displacement field $\mathrm{Re}\left[u_x\right]$ at $z=0$ and $t=0$ within a 2D Gaussian-SAW cavity mode made of ${\mathrm{LiNbO}}_3$ with the beam waist $w_0={\lambda }_{\mathrm{SAW}}$. Here, the area of the SAW cavity can be considered as $A=2{\lambda }_{\mathrm{SAW}}\times 40{\lambda }_{\mathrm{SAW}}$ with ${\lambda }_{\mathrm{SAW}}=40\mu \mathrm{m}$, thus only the central region of it is depicted. Superimposed is a sample with its lateral dimension of $2{\lambda }_{\mathrm{SAW}}\times 4{\lambda }_{\mathrm{SAW}}$ (marked by the green rectangle).

Figure 4

Semi-infinite medium and a wave vector $\mathit{k}$ of a SAW.

Figure 5

A mode profile of a displacement field $\mathrm{Re}\left[u_x\right]$ in Eq. (A36) for the case of $w_0=0.5{\lambda }_{\mathrm{SAW}}$ (the beam waist diameter is $\lambda$) with ${\lambda }_{\mathrm{SAW}}=40\mu \mathrm{m}$. The material is assumed to be ${\mathrm{LiNbO}}_3$.

Figure 6

A mode profile of a displacement field $\mathrm{Re}\left[u_x\right]$ in Eq. (A36) for the case of $w_0={\lambda }_{\mathrm{SAW}}$ (the beam diameter is 2 $\lambda$) with ${\lambda }_{\mathrm{SAW}}=40\mu \mathrm{m}$. The material is assumed to be ${\mathrm{LiNbO}}_3$.

Figure 7

The geometric factors $\zeta$ as a function of the thickness $t$ of the sample, where the sample geometry is $t\times 2{\lambda }_{\mathrm{SAW}}\times 4{\lambda }_{\mathrm{SAW}}$ with ${\lambda }_{\mathrm{SAW}}=40\mu \mathrm{m}$. Here, the sample is assumed to be placed on the center of the SAW cavity whose mode profile is depicted in Fig. 6. The larger point corresponds to the value $\zeta$ for the reference sample thickness of $t=\frac{{\lambda }_{\mathrm{SAW}}}{100}=400\mathrm{nm}$.

Figure 8

The geometric factors $\zeta$ as a function of the width $w$ of the sample, where the sample geometry is $\frac{{\lambda }_{\mathrm{SAW}}}{100}\times w\times 4{\lambda }_{\mathrm{SAW}}$ with ${\lambda }_{\mathrm{SAW}}=40\mu \mathrm{m}$. Here, the sample is assumed to be placed on the center of the SAW cavity whose mode profile is depicted in Fig. 6. The larger point corresponds to the value $\zeta$ for the reference sample thickness of $w=2{\lambda }_{\mathrm{SAW}}=80\mu \mathrm{m}$.

Figure 9

The geometric factors $\zeta$ as a function of the length $l$ of the sample, where the sample geometry is $\frac{{\lambda }_{\mathrm{SAW}}}{100}\times 2{\lambda }_{\mathrm{SAW}}\times l$ with ${\lambda }_{\mathrm{SAW}}=40\mu \mathrm{m}$. Here, the sample is assumed to be placed on the center of the SAW cavity whose mode profile is depicted in Fig. 6. The larger point corresponds to the value $\zeta$ for the reference sample thickness of $l=4{\lambda }_{\mathrm{SAW}}=160\mu \mathrm{m}$.