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PHYSICAL REVIEW RESEARCH 2, 043200 (2020)

Nuclear surface acoustic 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 (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 -to- 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 L in an in- ing spin that leads to a detectable back onto ertial frame of reference acquires an extra −L · ω in a the SAW device. noninertial frame rotating with an angular 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 -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 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 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 shift gyromagnetic ratios of nuclei are orders of magnitude smaller of nuclear magnetic resonance (NMR) [5] and extra nuclear than that of , 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 (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 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 . elastic medium carrying the 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 must maintain attribution to the author(s) SAW with a λSAW and an 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 of the Barnett field changes with x,the the volume z < 0, whereas the region z > 0 is . 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 -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 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 -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 oscillator, respectively. Here, m is the magnetic having 2 decaying exponentially with z. Since the angular velocity ω is units of kilogram × (meter2/tesla ), with which the magnetic κ κ = ω2 constant m is expressed as m m 0 [16]. From the equipartition theorem, the Sff of the magnetic Langevin noise f (t )isgivenby 1 Sff = 4 κmkBT, (6) ω0Q

where kB is the Boltzmann constant and T is the tempera- ture of the magnetic oscillator. With the oscillating transverse magnetic moment, My(t ) = M0ey cos ω0t,wehavethefollow- ing rms SNR [16]: M √0 ω Q = √ 2 = M 0 , (7) y rms 0 κ ν FIG. 2. A density plot of the component of the normalized Sff ν 8 mkBT vortex field (Re[])y = (Re[∇ × u˙])y accompanying with the same SAW as that shown in Fig. 1. where ν is the measurement band width.

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To evaluate the SNR, all we need is to deduce the magnetic spring constant κm in Eq. (7). To this end, let us consider the Hamiltonian describing the SAW oscillator with a mass μ, U (t ), and momentum P(t )as 1 1 H = P(t )2 + μω2U (t )2 + H . (8) SAW 2μ 2 0 I Here, P(t ) is obtained by equating the kinetic energy of the oscillator to the integral of the kinetic energy density of the SAW over the volume containing a two-dimensional (2D) 1 2 = Gaussian SAW mode (see Appendix A2b), i.e., 2μ P(t ) ∗ ρ ρ FIG. 3. A mode profile of a displacement field Re[ux]atz = 0 cavity u˙u˙ dv, with being the mass density of the elastic and t = 0 within a 2D Gaussian-SAW cavity mode made of LiNbO3 medium. The last term HI provides the nuclear-spin–oscillator = λ coupling, which can, from Eq. (B6), be read as (see with the beam waist w0 SAW. Here, the of the SAW cavity can be considered as A = 2λ × 40λ with λ = 40 μm, Appendix B2) SAW SAW SAW thus only the central region of it is depicted. Superimposed is a 1 sample with its lateral dimension of 2λ × 4λ (markedbythe H = H˜ dv =− m · dv SAW SAW I I 2γ rectangle). cavity cavity k P(t ) =− e · M t , γ ζμ y ( ) (9) 2 The NSAR can be detected either electronically, or opti- where M(t ) is the nuclear magnetic moment within the spec- cally, through the SAW-cavity response. As for the electronic imen put on the SAW cavity. The cavity effectively removes signal transduction an inter-digitated converts the the spatial degree of freedom of the SAW and makes it pos- acoustic signal into an electric signal, which is in ampli- sible to model the SAW mode as a whole as if it were a fied electronically. The optical transduction can be carried out single oscillator [17]. Here, the dimensionless constant ζ is a through the acousto-optic effect such as the moving-boundary geometrical factor ranging from 0.1 to 10 (see Appendix B3), effect, the photoelastic effect, and electro-optic effect [18]. and ζ 2 can be interpreted as an effective mass coefficient of Note here that as for the latter optical scheme, the signal the oscillator [17], which, in the current context, is determined carried by SAW can, in principle, be faithfully transferred by the overlap between the SAW cavity mode and the shape of into an optical signal with the minimum quantum noise added the sample (see Appendix B2). The thickness of the sample [19]. The similar idea of rf signal-to-optical signal transduc- on the SAW is assumed to be comparable or shorter than tion schemes based on electro-mechanical system for NMR λSAW. We thus have the oscillating Barnett field associated detection have been pursued [20–24]. = = kP(t ) with the SAW in the cavity as Bω(t ) Bω(t )ey ( γζμ)ey. As concrete examples, let us examine three different spin- 2 1 1 13 31 To make the connection to the standard equation of nuclei, H, C, and P. As for the SAW material, we 2 ρ motion, Eq. (5), we need to perform a unitary transfor- take lithium niobate (LiNbO3) with the mass density of . / 3 λ ∼ μ mation [16] to change the canonical variables {U (t ), P(t )} 4 65 g cm . Targeting the SAW wavelength SAW of 40 m, k ω / π { ω , } ω =  = we have the resonance frequency 0 2 of 88 MHz, which into B (t ) (t ) , namely, B (t ) 2γζμP(t ) and (t ) γζμ is determined by the relation [7]. Since the gyro- − 2 U (t ). Consequently, the Hamiltonian H in Eq. (8) k SAW magnetic ratios are γ = 268 × 106,67.3 × 106, and 108 × becomes − − 106 rad s 1 T 1 for 1H, 13C, and 31P, respectively, we need 1 2γ 2 = . = ζ 2μ 2 B0 2 1, 8.2, and 5.1 T to bring the nuclear-spin-resonance HB Bω(t ) ω = 2 k to 0. The volume of the SAW cavity, Vc Aλ , can be made far smaller than that of the pickup ω2 SAW 1 0 2 coil used for the standard inductive detection of NMR. A + (t ) − Bω(t )ey · M(t ), (10) 2γ 2 2 ζ 2μ SAW cavity having the area A = 2λSAW × 40λSAW should be k feasible [15]. Figure 3 displays a 2D Gaussian-SAW cavity and the equation of motion for Bω(t ) becomes the standard mode within the xy plane with the beam waist w = λ . κ 0 SAW form given by Eq. (5). Here, the magnetic spring constant m, To deduce the geometric factor ζ , we consider a sample with given by its lateral dimension of 2λSAW × 4λSAW and the thickness of λ / ∼ 2γ 2 SAW 100 400 nm, which is put on the center of the SAW κ = ζ 2μ , (11) cavity as shown by the green rectangle in Fig. 3. The geo- m k metric factor ζ is then found to be ∼0.6 (see Appendix B3). is proportional to the gyromagnetic ratio γ squared. From With these parameters, the magnetic spring constant, Eq. (11) −2 −2 −2 −2 Eq. (7), rms for the proposed NSAR detection scheme grows becomes κm = 9.0 × 10 ,0.6 × 10 , and 1.6 × 10 JT / as γ 3 2, unlike the conventional nuclear induction scheme for 1H, 13C, and 31P, respectively. These values are well- 5/2 where rms ∝ γ . This difference comes from the fact that compared to the one for the inductive method with a small the former NSAR scheme addresses the angular momentum micro-coil [16]. The quality factor of the SAW cavity could of nuclei, while the latter conventional NMR addresses the reach 104 [15], which is 102 better than those of magnetic moment of nuclei. the good electromagnetic LC (Q ∼ 102 [16]), we

043200-3 KOJI USAMI AND KAZUYUKI TAKEDA PHYSICAL REVIEW RESEARCH 2, 043200 (2020) expect the SNR rms in Eq. (7) of the NSAR could be sig- Yuichi Ohnuma, Ryo Sasaki, Toshiya Ideue, and Yoshi- nificantly increased compared with those of the conventional hiro Iwasa for useful discussion. We acknowledge financial NMR with the inductive method. support from JST ERATO (Grant No. JPMJER1601), JST There are several other unconventional ways to detect CREST (Grant No. JPMJCR1873), and JSPS KAKENHI NMR from small number of spins. They include the resistively (Grant No. 19H05602). measured scheme with nanoscale point contact in fractional quantum Hall regime [25] and the magnetic resonance APPENDIX A: TWO-DIMENSIONAL GAUSSIAN microscopy (MRFM) [26,27], as well as the scheme based SURFACE ACOUSTIC MODES on nitrogen-vacancy spin magnetometers [28,29]. The first scheme is applicable only for internal nuclei and sample- We describe here two-dimensional (2D) Gaussian surface specific. The latter two methods are tuned to detect spins in acoustic wave (SAW) modes. Among these modes, we are the limited volume and good for three-dimensional imaging. interested in the focused beamlike fundamental mode, which The proposed NSAR detection scheme is unique in that it is can have a small beam radius at the beam waist of the order of particularly suitable for 2D thin samples. tens of micrometers. A cavity that supports the fundamental mode thus has a very small mode volume, which is instru- mental in realizing large spin-rotation coupling. We begin by IV. PROSPECTS recapitulating the basic wave equations to equip us with the The question we now ask is to what extent can a sample notations and all that for the discussion of the 2D Gaussian be thin? By way of illustration we shall estimate the expected SAW modes. 77 SNR for the particular case of Se contained in WSe2, one of 2D semiconductors called transition metal dichalcogenides 1. Rudimentary information (TMDs) [30], for which the optically pumped dynamic nu- clear polarization (DNP) technique [31] is expected to work In an elastic medium with a mass density ρ, K, and shear modulus μ , the wave equation is given by [7] [32–34]. Take a monolayer flake of WSe2 with the lateral s size of 160 μm × 80 μm, which matches within the central ∂2u 1 region of the Gaussian mode of the 88-MHz SAW cavity ρ = K + μ ∇(∇ · u) + μ ∇2u, (A1) ∂ 2 s s shown in Fig. 3. Obtaining such a large-area single-crystalline t 3 monolayer flake is challenging but we note the promising 77 where u represents the displacement vector field. In terms of developments [35]. With the natural abundance 7.63% of Se, the longitudinal phase the flake contains ≈2.1 × 1010 77Se nuclei, which is still four orders of magnitude smaller compared with the number 4 1/2 K + μs C = 3 of spins ever successfully detected by DNP-NMR using the L ρ (A2) conventional induction method, where the single-shot rms SNR was reported to be 0.6 [36]. Under the magnetic field and the transverse phase speed = . , 77 γ = . × 6 −1 −1 of B0 10 8T Se nuclei ( 51 0 10 rad s T )on the SAW cavity can be resonantly excited by the SAW. From μ 1/2 C = s , Eq. (7) with T = 4 K and ν = 10 kHz, we have the single- T ρ (A3) −3 shot rms SNR, rms ∼ 0.3 × 10 . Now, assuming 150-fold improvement of nuclear spin polarization by optically pumped Eq. (A1) can be rewritten as DNP, the available spins increase up to roughly 3 × 109 and ∼ . ∂2u we have rms 0 04, which suggests that 1000-time average = C2 − C2 ∇(∇ · u) + C2 ∇2u. (A4) would allow us to achieve unity SNR. We anticipate the NASR ∂t 2 L T T scheme to bring new insight that help to understand the rela- In an infinitely extended medium, the longitudinal and the tively unexplored role of nuclear spins in 2D semiconductors. transverse waves behave independently with the following respective wave equations: V. CONCLUSION ∂2u L = C2∇2u (A5) The oscillating Barnett field created by a small-volume ∂t 2 L L high-Q SAW cavity can be exploited to detect nuclear spin resonance through the spin-rotation coupling. The proposed and scheme is particularly well-suited to investigate nuclei in 2D 2 ∂ uT extended samples. The expected SNR suggests that detection = C2 ∇2u . (A6) ∂ 2 T T of an NSAR signal from a single flake of atomically thin t 2D semiconductor is feasible once combined with the DNP technique. 2. Surface acoustic waves When the medium is semi-infinite, that is to say, the medium occupies only up to z = 0 from the bottom along ACKNOWLEDGMENTS the z direction, surface acoustic waves (SAWs) emerge. The We are indebted to Yasunobu Nakamura, Ryusuke displacement vector field is obtained by solving Eq. (A4) with Hisatomi, Kotaro Taga, Rekishu Yamazaki, Shotaro Kano, the free boundary condition at the surface z = 0.

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with the relation A 2ikq 0 = L (A15) ψ 2 + 2 0 k qT imposed from the boundary conditions. FIG. 4. Semi-infinite medium and a wave vector k of a SAW. The first step to obtain the Gaussian modes is to make the scalar constants ψ0 and A0 in Eqs. (A13) and (A14) depending a. Plane-wave solutions on x and y:

Suppose now that a monochromatic surface wave infinitely ψ → ψ (x, y), (A16) extended in the y direction propagates in the x direction, as 0 0 shown in Fig. 4. In this case, the plane-wave solution of A → A (x, y). (A17) Eq. (A4)isgivenby[7] 0 0 ⎡ ⎤ ⎡ ⎤ −ω This step induces the changes ψ qL z − 2qL qT qT z i(kx 0t ) ik 0 e 2+ 2 e e ux k qT ⎡ ⎤ ⎢ ⎥ ∂ψ −ω = ⎣ ⎦ = ⎣ 0 ⎦, 0 + ψ qL z i(kx t ) u 0 (A7) ∂x ik 0 e e 2 −ω ∂ψ −ω u ψ qL z − 2k qT z i(kx 0t ) → = ⎣ 0 qL z i(kx t ) ⎦ z qL 0 e 2+ 2 e e uL uL ∂ e e (A18) k qT y qL z i(kx−ωt ) qLψ0e e where ψ0 is a scalar constant, ω0 is the angular frequency, = 2π k λ is the wave vector along x, and and SAW ⎡ ⎤ qT z i(kx−ωt ) qL = k 1 − κξ, (A8) −qT A0e e ⎣ ⎦ uT → uT = 0 , (A19) ∂ −ω q = k 1 − ξ (A9) A0 + qT z i(kx t ) T ∂x ikA0 e e are the imaginary wave vectors along z for longitudinal and in Eqs. (A13) and (A14), respectively. transverse modes, respectively, with The step is to insert the ansatz 2 − ν CT 1 2 = + κ = = (A10) u uL uT (A20) 2 − ν CL 2(1 ) into Eq. (A4) assuming that the condition Eq. (A15) would and still hold. Before doing this let us tidy up Eq. (A4). Using the C2 ω 2 ξ = R = , fact that 2 (A11) CT CT k ∂2u =−ω2 =− 2 2 0u CRk u (A21) and CR being the phase speed of the surface wave (CR < CT < ∂t 2 C ). Here, the Poisson ratio ν in Eq. (A10) is given in terms L and keep assuming the monochromaticity of the wave, that is of K and μs by − ω u ∝ e i 0t , the time dependent wave equation (A4) becomes 3K − 2μs time-independent one ν = . (A12) 2(3K + μ ) s 2 2 + 2 − 2 ∇ ∇ · + 2 ∇2 = . CRk u CL CT ( u) CT u 0 (A22) b. Gaussian modes 2 Dividing this equation by CT , we have our version of Now we consider the complications when the plane-wave Helmholtz’s equation: condition is forgone. This can be done in two steps. The plane- wave solution for u in Eq. (A7) can be read as a sum of the two 1 ξk2u + − ∇(∇ · u) +∇2u = . solutions: namely, the longitudinal plane-wave solution [the κ 1 0 (A23) plane-wave solution of Eq. (A5)], which can be obtained by qL z i(kx−ω0t ) taking a divergence of a scalar potential, ψ = ψ0e e , Inserting the ansatz u in Eq. (A20) into Helmholtz-like ψ , that is, Eq. (A23), we have a partial differential equation for 0(x y). ⎡ ⎤ ⎡ ⎤ −ω After some lengthy manipulation, the equation can be read as ψ qL z i(kx 0t ) uL;x ik 0e e ⎣ ⎦ ⎣ ⎦ qL z 2 uL = 0 = ∇ψ = 0 , (A13) ie ∂ ψ0 ∂ψ0 −ω √ √ + ψ qL z i(kx 0t ) 2ik uL;z qL 0e e 1 − ζ 1 − κζ ∂y2 ∂x and the transverse plane-wave solution (the plane-wave solu- 2iκ eqT z ∂2ψ ∂ψ + 0 + 2ik 0 = 0. (A24) tion of Eq. (A6)), which can be obtained by taking rotation of ζκ − 2 ∂y2 ∂x 0 −ω  = qT z i(kx 0t ) a vector potential, A [A0e e ], that is, ψ , 0 The function 0(x y) has to satisfy this equation for any value ⎡ ⎤ ⎡ ⎤ of z. Thus we have qT z i(kx−ω0t ) uT ;x −qT A0e e ⎣ ⎦ ⎣ ⎦ ∂2ψ (x, y) ∂ψ (x, y) uT = 0 = ∇ × A = 0 , (A14) 0 + 0 = . −ω 2ik 0 (A25) qT z i(kx 0t ) ∂ 2 ∂ uT ;z ikA0e e y x

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This is nothing but the familiar one having the Gaussian solutions.

c. Fundamental mode The solutions of Eq. (A25) form the Gaussian modes [37]. Note here that since ψ0(x, y) is two-dimensional, it makes the form a bit different from the well-known three-dimensional Gaussian modes of electromagnetic waves. Let us now ana- lyze the fundamental mode. Assume that the solution is given by the following Gaussian form [37]: FIG. 5. A mode profile of a displacement field Re[ux]in = . λ k Eq. (A36) for the case of w0 0 5 SAW (the beam waist diameter ψ (x, y) = α exp i p(x) + y2 , (A26) is λ) with λ = 40 μm. The material is assumed to be LiNbO . 0 2q(x) SAW 3 with two complex functions of x, p(x), and q(x), which are where called the complex phase shift and the complex beam parame- ter, respectively. The later parameter q(x) in particular plays a λ x θ(x) = tan−1 SAW . (A34) decisive role for the Gaussian mode. Hereafter, the amplitude π 2 w0 of the mode α is considered to be normalized (α = 1). To explicitly obtain p(x) and q(x) let us plug the form (A26)into Putting everything together, we have the following normalized form of the fundamental mode: Eq. (A25), we have two differential equations w y2 ∂q(x) ψ (x, y) = 0 exp − = 1, (A27) 0 w(x) w(x)2 ∂x θ(x) k ∂ p(x) i × − − 2 . = . (A28) exp i y (A35) ∂x 2q(x) 2 2R(x)

The real part of the ux at z = 0 and t = 0 can then be obtained The solution of Eq. (A27) can be written as by inserting this form into Eq. (A20), π 2 ∂ψ , w0 0(x y) q(x) = q + x =−i + x. (A29) Re[ux] = Re + ikψ0(x, y) 0 λ ∂x SAW 2ikq Here, q = q(0) is the value of the complex beam parameter at −q L ψ (x, y) eikx , (A36) 0 T 2 + 2 0 x = 0(beam waist), where the value becomes pure imaginary k qT π 2 w0 = q0 =−i( ) with w0 being interpreted as the beam radius which are plotted in Figs. 5 and 6 for the cases of w0 λSAW at the beam waist. This mystic statement can be revealed to 0.5 λSAW (the beam waist diameter is λSAW) and w0 = λSAW be reasonable when we decompose the complex parameter (the beam waist diameter is 2 λSAW), respectively. q(x) into two real parameters the radius of curvature of the wavefront R(x) and the beam radius w(x)as APPENDIX B: NUCLEAR-SPIN–OSCILLATOR COUPLING 1 1 λ 1. From SAW field to an oscillator = + i SAW , (A30) q(x) R(x) πw(x)2 Let us imagine a SAW cavity carrying the fundamental mode that we have considered in Sec. A 2 c. The cavity effec- where R(x) =∞at x = 0. With Eqs. (A29) and (A30), we tively removes the spatial degree of freedom of the velocity have the following useful relations: λ x 2 w(x) = w 1 + SAW , (A31) 0 πw2 0 πw2 2 R(x) = x 1 + 0 . (A32) λSAWx

With the solution for q(x) given by Eq. (A29), the solution of Eq. (A28) can be written as i λ x i w(x) p(x) = ln 1 + i SAW = ln eiθ(x) , π 2 FIG. 6. A mode profile of a displacement field Re[ux]in 2 w0 2 w0 Eq. (A36) for the case of w0 = λSAW (the beam diameter is 2 λ) with (A33) λSAW = 40 μm. The material is assumed to be LiNbO3.

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field u˙ (x, y, z; t ) and makes it possible to model the SAW assume Vc = T × W × L = λSAW × 2λSAW × 40λSAW, where mode as if it were an oscillator with a mass μ = ρVc, where ρ T , W , and L are the thickness, the width, and the length of is the mass density of the elastic medium and Vc is the cavity the SAW cavity. The Hamiltonian describing the oscillator is volume [15]. For the sake of concreteness, we shall henceforth given by

1 1 H = P(t )2 + μω2U (t )2, (B1) SAW 2μ 2 0 where U (t ) and P(t ) are the displacement and the momentum of the oscillator. Here, the cycle average of P(t )2 is obtained by equating the kinetic energy of the oscillator to the integral of the kinetic energy density of standing-wave SAW over the SAW cavity, that is, 1 1 2 = ρ + ∗ · + ∗ = ρ · ∗ P(t ) (u˙ u˙ ) (u˙ u˙ ) dv u˙ u˙ dv 2μ 2 cavity cavity 2Re[u˙ ] 2Re[u˙ ] 2 2 2 + ∂ψ 2 2 2 2 2 2q z 4k qL 2 2q z 4k qL (qL qT ) 2 (q +q )z 0 2q z = ω ρ k + q |ψ | e L + |ψ | e T − |ψ | e L T + e L dv, (B2) 0 L 0 2 + 2 0 2 + 2 0 ∂ cavity k qT k qT y where u is given by Eqs. (A18)–(A20) with ψ0(x, y) is given by Eq. (A35). Here, we neglect the contribution from the terms ∂ψ(x,y) ∂ψ0 (x,y) ψ0 (x,y) ∂ψ(x,y) ψ0 (x,y) with since ∼ and is far less than kψ(x, y), qLψ(x, y), qT ψ(x, y), and ∼ for our cavity. From ∂x ∂x 40λSAW ∂y 2λSAW Eq. (B2), we can assume

P(t ) = 2μω0Uc cos ω0t (B3) and

U (t ) = 2Uc sin ω0t, (B4) where Uc has a dimension of length and is defined by 1 2 2 2 + ∂ψ 2 2 1 2 2 2 2q z 4k qL 2 2q z 4k qL (qL qT ) 2 (q +q )z 0 2q z U = √ k + q |ψ | e L + |ψ | e T − |ψ | e L T + e L dv . (B5) c L 0 2 + 2 0 2 + 2 0 ∂ Vc cavity k qT k qT y

2. Nuclear-spin–oscillator coupling We now consider how the oscillator discussed in Sec. B 1, emerged when the SAW field has been integrated within the Gaussian SAW cavity, interacts with the nuclear spins in the sample on the SAW cavity. The Hamiltonian for the nuclear-spin– oscillator coupling is given by integrating the interaction energy density H˜ over the volume of the SAW cavity, that is, I 1 HI = H˜I dv =− m · dv, (B6) cavity 2γ cavity where m is the nuclear magnetization (the magnetic moment per unit volume) and  is the vortex field (for standing-wave SAW), which can be expressed as ⎡ ⎤ ∂ψ , 0 (x y) qT z ik ∂y e ∗ 2kqL ⎢ ⎥ −ω  = ∇ × + = ω 2 − 2 ψ , qT z i(kx 0t ) + . (u˙ u˙ ) 0 ⎣(k qT ) 0(x y)e ⎦e c.c (B7) 2 + 2 ∂ψ , k qT 0 (x y) qT z 2Re[u˙ ] qT ∂y e ∂ψ , ∂ψ , ψ , = ψ , 0 (x y) 0 (x y) ∼ Note that 0(x y) given by Eq. (A35) has the maximum at y 0, where 0(x y) ∂y . We thus approximate ∂y 0, so that the vortex field (t ) has only the y-component, namely, (t ) = (t )ey with 2 2 k − q −ω (t ) = 2ω kq T eqT z(ψ (x, y)ei(kx 0t ) + c.c.). (B8) 0 L 2 + 2 0 k qT

We note two important points here. First, the oscillating rotating component, changes linearly with x. This implies that  π vortex field (t ) along the y-axis can be considered as a sum asimple 2 pulse created by the conventional NMR coil would of a pair of rotating components in the opposite directions. not produce such transverse magnetization that is detectable For nuclei having the positive (negative) gyromagnetic ra- with the SAW device, because the phase of the signal contri- tio γ , only the clockwise (counter-clockwise) component is bution in one place on the SAW destructively interferes with relevant. Second, the phase of (t ), and thereby that of the that in another. Mathematically, naively integrating the vortex

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field over the cavity with a uniformly distributed transverse magnetization m would result in cancellation of the integrated nuclear-spin–oscillator coupling, that is, HI = 0. To have the nonzero nuclear-spin–oscillator coupling, the magnetization m has to be prepared in such a way that the profile of the excited magnetization constructively yields the nonzero total integrated spin-rotation coupling. We now consider two possible excitation schemes and then the resul- tant nuclear-spin–oscillator coupling used for detecting the nuclear surface acoustic resonance (NSAR).

a. Excitation scheme I FIG. 8. The geometric factors ζ as a function of the width w of A simple way to develop the detectable helical transverse λ SAW × × λ nuclear magnetization is to use the same local spin-rotation the sample, where the sample geometry is 100 w 4 SAW with λ = μ coupling for both the excitation and detection processes. SAW 40 m. Here, the sample is assumed to be placed on the Initially, the nuclear magnetization is assumed to be in ther- center of the SAW cavity whose mode profile is depicted in Fig. 6. ζ mal equilibrium in the polarizing static field B . Then, the The larger point corresponds to the value for the reference sample 0 = λ = μ SAW cavity is excited to switch on the nuclear-spin–oscillator thickness of w 2 SAW 80 m. coupling H˜I given in Eq. (B6), so that the individual local magnetization starts to be rotated about the axis in the xy π = at t after applying a quasi-instantaneous 2 pulse at t 0is plane, whose phase is determined by that of the local vortex given by field. When the SAW excitation is continued until the angle of rotation has reached π/2, the x-dependent local magnetization my(t ) = m0 sin ω0t, (B11) becomes m 1 where 0 is defined as m(0) = mB (0)ex | | mB max (0) m = eqT z ψ x, y eikx . 0 ψ , (2Re[ 0( ) ]) (B12) mB 2 0(0 0) = eqT z ψ x, y eikx e , ψ , (2Re[ 0( ) ]) x (B9) 2 0(0 0) The spatial profile of the vortex field is now engraved in the where mB is the initial thermal magnetization under the mag- magnetization my(t ), which will, in the end, result in netic field B0 along z axis before exciting the cavity and the nonzero total integrated spin-rotation coupling, HI in Eq. (B6). k2 − q2 max |(0)| = 4ω kq T ψ (0, 0) (B10) 0 L 2 + 2 0 k qT b. Excitation scheme II  is the maximum value of (0) within the sample volume An alternative scheme uses a pulsed field gradient Bz(x), Vs. The magnetization m(0) is along x axis and its sign is which linearly varies with x. The effect of Bz(x), applied alternating as moving along x axis. After the initial excitation immediately after the π/2 pulse for an interval τ, is to create pulse the magnetization starts precessing about z axis. When a periodic phase grating of the transverse magnetization, i.e., ignoring relaxation, the y component of the magnetization the magnetization helix in the xy plane. The condition to attain

FIG. 7. The geometric factors ζ as a function of the thickness t of FIG. 9. The geometric factors ζ as a function of the length l of λ × λ × λ SAW the sample, where the sample geometry is t 2 SAW 4 SAW with the sample, where the sample geometry is × 2λSAW × l with λ = μ 100 SAW 40 m. Here, the sample is assumed to be placed on the λSAW = 40 μm. Here, the sample is assumed to be placed on the center of the SAW cavity whose mode profile is depicted in Fig. 6. center of the SAW cavity whose mode profile is depicted in Fig. 6. The larger point corresponds to the value ζ for the reference sample The larger point corresponds to the value ζ for the reference sample λ = SAW = = λ = μ thickness of t 100 400 nm. thickness of l 4 SAW 160 m.

043200-8 NUCLEAR SURFACE ACOUSTIC RESONANCE WITH … PHYSICAL REVIEW RESEARCH 2, 043200 (2020) mode matching is given by c. Detection

Let us suppose that the transverse magnetization my(t ), ∂B γ z λ τ = 2π. (B13) represented as ∂x SAW my(t ) = m0 cos ω0t, (B14) For instance, a typical clinical MRI can produce a gradient has been prepared. This can be done by changing the phase of field of the order of 0.1 T/m. For microimaging systems, even the excitation pulse, or, considering the quasi-instantaneous π =− π 2 stronger gradient fields by an order of magnitude are feasible. pulse is applied at t ω with the excitation scheme dis- − 2 0 Then, for 1H spins with γ = 268 × 106 rad s 1 T, the width of cussed in Sec. B 2 a. With Eqs. (B8) and (B14) the interaction the gradient pulse may be adjusted to ≈590 μs. Hamiltonian, Eq. (B6), becomes

1 HI (t ) =− (t )my(t )dv 2γ sample 2 2 1 k − q −ω =− 2ω kq T eqT z(ψ (x, y)ei(kx 0t ) + c.c.)m cos ω tdv. (B15) γ 0 L 2 + 2 0 0 0 2 sample k qT

Here, we shall note two things: First, since the magnetization my(t )inEq.(B14) is nonzero only within the sample, the integral in Eq. (B15) is accordingly running only over the sample region. Second, since the magnetization is excited in such a way that its projection onto the y axis is given by my(t ) = m0 cos ω0t, we can employ the rotating-wave approximation to pickup the in-phase ikx i(kx−ω0t ) component, (2Re[ψ0(x, y)e ] cos ω0t ), from (ψ0(x, y)e + c.c.)inEq.(B15). As a result we have, with Eq. (B12), k k2 − q2 H ∼− 2ω q T eqT z(2Re[ψ (x, y)eikx] cos ω t )m cos ω tdv I γ 0 L 2 + 2 0 0 0 0 2 sample k qT k k2 − q2 1 =− 2ω q T (eqT z(2Re[ψ (x, y)eikx])2 cos ω t )m cos ω tdv γ 0 L 2 + 2 ψ , 0 0 B 0 2 sample k qT 2 0(0 0) k =− [2ω U ] cos ω t e · M(t ), (B16) 2γ 0 s 0 y where Us is defined by 1 k2 − q2 1 U = q T (2eqT zRe[ψ (x, y)eikx])2dv, (B17) s L 2 + 2 ψ , 0 Vs sample k qT 2 0(0 0) having a dimension of length, and M(t )istheuniformly oscillating magnetic moment along y, which is given by

M(t ) = M0ey cos ω0t (B18) with M0 = mBVs being the nominal magnetic moment within the sample. With Eq. (B3), the spin-rotation coupling Hamiltonian, Eq. (B16), can then be expressed in terms of P(t )as k P(t ) k P(t ) HI =− (2ω0Us cos ω0t ) ey · M(t ) =− ey · M(t ). (B19) 2γ 2μω0Uc cos ω0t 2γ ζμ Here, we have introduced a dimensionless geometric factor ζ as U ζ = c , (B20) Us where Uc and Us both have a dimension of length and are defined by Eqs. (B5) and (B17), respectively.

3. Geometrical factor ζ The geometrical factor ζ appeared in Eq. (B19) can be interpreted as the square root of the effective mass coefficient of the oscillator [17], which, in the current context, is determined by the overlap between the SAW cavity mode and the shape of the sample. Figures 7, 8, and 9 show the geometric factors ζ as we change the thickness (t), the width (w), and the length (l)of λ = × × = SAW × λ × λ λ = μ the sample from the reference sample geometry, Vs t w l 100 2 SAW 4 SAW with SAW 40 m, respectively. Here, the sample is assumed to be placed on the center of the SAW cavity whose mode profile is depicted in Fig. 6.

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