PHYSICAL REVIEW RESEARCH 2, 043200 (2020) Nuclear surface acoustic resonance with spin-rotation coupling Koji Usami1,* and Kazuyuki Takeda 2,† 1Research Center for Advanced Science and Technology (RCAST), The University of Tokyo, Meguro-ku, Tokyo 153-8904, Japan 2Division of Chemistry, Graduate School of Science, Kyoto University, 606-8502 Kyoto, Japan (Received 10 July 2020; accepted 16 October 2020; published 9 November 2020) 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 104 and the mode volume as small as 10−2 mm3 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. DOI: 10.1103/PhysRevResearch.2.043200 I. INTRODUCTION resonantly rotating and the counter-rotating components, and the former can cause transition between the spin states, creat- A particle with an orbital angular momentum L in an in- ing spin coherence that leads to a detectable back action onto ertial frame of reference acquires an extra energy −L · ω in a the SAW device. noninertial frame rotating with an angular velocity ω with re- The presence of the spin-rotation coupling between elec- spect to the inertial frame [1]. The same argument holds for a tron spins and SAW has been predicted [8,9] and confirmed particle with a spin angular momentum S. Due to the extra en- through generation of alternating electron-spin currents [10] ergy −S · ω emerged in the rotating frame as a consequence of and through resonant excitation of spin wave [11] in a thin the spin-rotation coupling [2], the spin system is magnetized layer of conductors deposited on the SAW device. Impor- as if it were exposed to a magnetic field Bω = ω/γ , where tantly, it is not the magnetic moment γ S of the spin but its γ is the gyromagnetic ratio of the particle. Development of angular momentum S that is involved in the spin-rotation the magnetization by rotation was first observed by Barnett coupling. It follows that, for a given angular velocity ω of in 1915 in a rotating ferromagnetic body [3]. Very recently, mechanical rotation, the spin-rotation coupling is indepen- the Barnett effect was also reported for paramagnetic electron dent of the gyromagnetic ratio. Therefore, even though the spins [4] as well as for nuclear spins, causing frequency shift gyromagnetic ratios of nuclei are orders of magnitude smaller of nuclear magnetic resonance (NMR) [5] and extra nuclear than that of electrons, the spin-rotation coupling for nuclei is polarization [6] by sample spinning at ∼10 kHz. expected to be comparable to that for electrons. In this paper, we explore the possibility of accessing nu- The proposed approach offers a new mechanism of nuclear clear spin resonance through the alternating Barnett field in surface acoustic resonance (NSAR), distinguishing itself from the presence of a static, polarizing magnetic field B .Tothis 0 well known nuclear acoustic resonance (NAR) in bulk materi- end, the Barnett field Bω has to be normal to B and be 0 als [12], where the nuclear spins interact with acoustic waves rotating around B at the frequency matched to the Larmor 0 through the dynamic electrical quadrupole coupling [13]or spin-precession frequency ω =−γ B , which can be several 0 0 dynamic Alpher-Rubin coupling [14]. The Barnett field in- tens of MHz or even higher. To realize such seemingly impos- duced by the SAW cavity can be confined in a volume far sible, rapid change of the direction of mechanical rotation and smaller than the size of the coil used in the conventional NMR thereby of the Barnett field, we propose to exploit a surface experiments [15]. Moreover, the quality factor of the state- acoustic wave (SAW) device and attach on it a thin layer of-art SAW cavities can reach 104 [15], being two orders of of the material containing the nuclear spins of interest. The magnitude higher than that of the conventional LC resonator. elastic medium carrying the surface wave undergoes ellip- The small cavity volume and the large quality factor of the tic backward rotation [7], and the resultant acoustic vortex SAW cavity potentially leads to the improved signal-to-noise field and thereby the Barnett field oscillates at the SAW fre- ratio (SNR), and thereby offering a vital tool to characterize quency. The oscillating Barnett field is a superposition of the structures and dynamics of thin samples, such as van der Waals materials. *[email protected] † [email protected] II. SAW, SPIN-ROTATION COUPLING, AND BARNETT FIELD Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further Let us consider a semi-infinite elastic medium on which a distribution of this work must maintain attribution to the author(s) SAW with a wavelength λSAW and an angular frequency ω0 and the published article’s title, journal citation, and DOI. propagates along the x axis. The surface plane is taken to be 2643-1564/2020/2(4)/043200(10) 043200-1 Published by the American Physical Society KOJI USAMI AND KAZUYUKI TAKEDA PHYSICAL REVIEW RESEARCH 2, 043200 (2020) given by /2, the individual nuclear spins in the thin layer on the surface of the elastic medium where the SAW propagates ˜ =−1 ˜ · experience the local spin-rotation coupling HI 2 I [2], where I˜ is the angular momentum density of the nuclear spins. Alternatively, using the Barnett field Bω = /(2γ ) and the nuclear magnetization m = γ I˜ (the magnetic moment per unit volume), the spin-rotation coupling is expressed in the form of the Zeeman coupling as H˜I =−m · Bω, (4) The Barnett field associated with the SAW is oscillating at the ω FIG. 1. A snapshot of a vectorial velocity field Re[u˙]inthe angular frequency 0 of the SAW, which can be far higher zx plane accompanying with a plane monochromatic surface wave than those possible with pneumatic spinning of a sample container. As a consequence, under the out-of-plane static propagating along the x axis. The values qL and qT used here are for LiNbO3 with λSAW = 40 μm. magnetic field B0 nuclear spins experience resonance when ω0 =−γ B0. lying in the xy plane at z = 0, and the elastic medium occupies Since the phase of the Barnett field changes with x,the the volume z < 0, whereas the region z > 0 is vacuum. Within transverse magnetization has to change its phase with x in the monochromatic and plane-wave approximation, the dis- the same way, in order to be detected by the SAW device. placement field u(x, z, t ) is given by a sum of the longitudinal This requirement is fulfilled either by employing the same qL z i(kx−ω0t ) component uL = ∇ψ0e e and the transverse compo- SAW mode both for excitation and detection, or by creating qT z i(kx−ω0t ) nent uT = ∇ × A0e e ey as [7] the helical transverse magnetization using the conventional radio-frequency excitation in combination with a pulsed field qL z qT z i(kx−ω0t ) ux (x, z, t ) = ikψ0e − qT A0e e , (1) gradient. −ω u (x, z, t ) = q ψ eqL z + ikA eqT z ei(kx 0t ). (2) z L 0 0 III. SIGNAL-TO-NOISE RATIO Here, the wave vector k = 2π/λ along the direction of SAW The SNR for the proposed NSAR detection can be ana- propagation (x) is real, while those along z are imaginary both lyzed by following a general formalism developed by Sidles for the longitudinal and the transverse displacements. ψ and 0 and Rugar on the SNR of any detectors comprised of a har- A are constants having units of meter2 and depend on each 0 monic oscillator coupled to the precessing magnetic moment = 2ikqL ψ other through A0 2+ 2 0. Figure 1 depicts the real part k qT [16]. Indeed, the SNR for both the conventional electrical = ∂u of the velocity field u˙ ∂t in the zx plane, where a point detection and mechanical detection of NMR has successfully particle in the field undergoes elliptic backward rotation. been described with this theoretical framework. In the present The vortex field accompanying the SAW is given by ∇ × case, the equation of motion for the Barnett field Bω(t ) = u˙. Straightforward calculation gives its dominant y component Bω(t )ey in the sample on the SAW device is y(t )as ω ¨ + 0 ˙ + ω2 = − · , 2 2 mBω(t ) m Bω(t ) m 0Bω(t ) f (t ) ey M(t ) (5) k − q −ω Q (t ) = 2kq ω T ψ eqT zei(kx 0t ). (3) y L 0 2 + 2 0 k qT where M(t ) is the time-dependent nuclear magnetic moment ω Figure 2 shows a snapshot of the real part of the oscillating and 0, Q, and f (t ) are the resonance angular frequency, vortex field y(t ), where we can observe that the field is the quality factor, and the Langevin noise of the magnetic localized in the vicinity of the surface with its amplitude oscillator, respectively. Here, m is the magnetic mass having 2 decaying exponentially with z.