Anisotropic Spin-Acoustic Resonance in Silicon Carbide at Room Temperature
A. Hernández-Mínguez, A. V. Poshakinskiy, M. Hollenbach, P. V. Santos, G. V. Astakhov
AAnisotropic Spin-Acoustic Resonance in Silicon Carbide at Room Temperature
A. Hern´andez-M´ınguez, ∗ A. V. Poshakinskiy, M. Hollenbach,
3, 4
P. V. Santos, and G. V. Astakhov Paul-Drude-Institut f¨ur Festk¨orperelektronik, Leibniz-Institut imForschungsverbund Berlin e.V., Hausvogteiplatz 5-7, 10117 Berlin, Germany Ioffe Physical-Technical Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia Helmholtz-Zentrum Dresden-Rossendorf, Institute of Ion Beam Physics and Materials Research,Bautzner Landstrasse 400, 01328 Dresden, Germany Technische Universit¨at Dresden, 01062 Dresden, Germany (Dated: May 5, 2020)We report on acoustically driven spin resonances in atomic-scale centers in silicon carbide at roomtemperature. Specifically, we use a surface acoustic wave cavity to selectively address spin transi-tions with magnetic quantum number differences of ± ± Hybrid spin-mechanical systems are considered as apromising platform for the implementation of univer-sal quantum transducers [1] and ultra-sensitive quantumsensors [2]. Spin states can be coupled by the strain fieldsof phonons and mechanical vibrations. Coherent sensingof mechanical resonators [3], acoustic control of singlespins [4] and electromechanical stabilization of spin colorcenters [5] based on spin-optomechanical coupling havebeen demonstrated. Similarly to a magnetic field, theapplication of a static strain field leads to a shift of thespin levels, while a resonantly oscillating strain field in-duces interlevel spin transitions. Their selection rules areimprinted by the crystal symmetry or device geometry,which provide a high degree of flexibility for on-chip co-herent spin manipulation [6–8] and may support chiralspin-phonon coupling [9].Most of the systems coupling atomic-scale spins to vi-brations studied so far are based on color centers in dia-mond [3–5, 10–14]. Two characteristics of silicon carbide(SiC) make it a natural material choice for hybrid spinoptomechanics. As diamond, SiC hosts highly-coherentoptically-active spin centers, such as negatively chargedsilicon vacancies (V Si ) [15] and divacancies (VV) [16]. Inaddition, SiC is already used in commercial nano-electro-mechanical systems (NEMS) with robust performanceand ultrahigh sensitivity to vibrations [17]. Recently, themechanical tuning [18] and acoustic coherent control [19]of the VV spin S = 1 centers in SiC have been demon-strated at cryogenic temperatures. However, symmetry-dependent spin-acoustic interactions are still largely un-explored and SiC-based hybrid spin-mechanical systemsunder ambient conditions remain elusive.In this letter, we demonstrate room-temperature spin-acoustic resonance (SAR) in 4H-SiC. We exploit the in-trinsic properties of the half-integer spin S = 3 / Si qudit [21], to realize full con-trol of the spin states using high-frequency vibrations.This is fulfilled by acoustically coupling spin sublevelswith magnetic quantum numbers ( m S ) differing by both∆ m S = ± m S = ±
2. In contrast to previous SARstudies, which were restricted to S = 1 atomic-scale spins[3–5, 10–14, 18, 19], the spin S = 3 / m S = ± m S = ± B = ( B x , B y ,
0) applied in the plane perpendicu-lar to the c-axis of 4H-SiC (i.e., with B z = 0). We de-velop a model for spin-3/2 SARs using an effective spin-strain coupling Hamiltonian [2, 22], which can describeour non-trivial observations. The selective excitation oftransitions with ∆ m S = ± m S = ± Si crystallographic config-urations [23]. Figure 1(a) displays the 4H-SiC latticewith a single V Si center. The V Si centers are createdin a 10 ×
10 mm semi-insulating 4H-SiC substrate bythe irradiation with protons with an energy of 37.5 keVto a fluence of 10 cm − [24]. Figure 2(d) shows thecalculated depth distribution of the V Si centers, which a r X i v : . [ c ond - m a t . m e s - h a ll ] M a y V Si c ǁ z (a) (b)(c) (d) ± ± ± ±
70 MHzMSPLlaser
880 900 920 940-20-100 S , S ( d B ) Frequency (MHz) -60-40-200 S ( d B ) FIG. 1. Schematic presentation of (a) the 4H-SiC lattice withone V Si defect and (b) the acoustic device. Focusing IDTs ex-cite and detect SAWs propagating in the ZnO-coated SiC. Thesample is placed between the poles of an electromagnet for theapplication of a magnetic field B in the plane of the samplesurface. (c) RF scattering parameters for IDT1 and IDT2. S and S correspond to the power reflection coefficient ofIDT1 and IDT2, respectively, and S to the power trans-mission coefficient. (d) Optical pumping cycle and readoutprinciple of the V Si spin. The double vertical arrow indicatesthe spin resonance transition at 70 MHz in zero magnetic field(see Supplemental Material [26]). has a mean depth of 250 nm below the SiC surface [25].As shown in the Supplemental Material (SM) [26], thesecenters reveal the 70 MHz zero-field spin splitting charac-teristic of the V2 centers [27–30]. After irradiation, theSiC substrate is coated with a 35-nm-thick SiO layerfollowed by a 700-nm-thick ZnO piezoelectric film us-ing radio-frequency (RF) magnetron sputtering. Finally,acoustic cavities defined by a pair of focusing interdig-ital transducers (IDTs) are patterned on the surface ofthe ZnO film by electron beam lithography and metalevaporation. Figure 1(b) displays a schematic represen-tation of our acoustic device. Each IDT consists of 80 alu-minum finger pairs for excitation/detection of SAWs witha wavelength λ SAW = 6 µ m, and an additional Bragg re-flector consisting of 40 finger pairs placed on its back side(not shown in Fig. 1). The finger curvature and separa-tion between the opposite IDTs ( ≈ µ m) are designedto focus the SAW beam at the center of the cavity. Fig-ure 1(c) displays the RF scattering ( S ) parameters of theIDTs measured with a vector network analyzer. Theyshow a series of sharp dips within the resonance band ofthe IDT at a frequency f SAW ≈
916 MHz, which corre-spond to the excitation of the Rayleigh SAW modes ofthe resonator.Figure 1(d) displays a simplified energy diagram of theV2 V Si center [23, 31] together with the optical pumpingand readout scheme [32]. The V Si center has spin 3/2, which is split in zero magnetic field into two Kramer’sdoublets due to a low symmetry of the V Si center ( C v point group). The zero-field splitting between the m S = ± / m S = ± / c -axis. Optical excitationinto the excited state (ES), followed by spin-dependentrecombination via the metastable state (MS), leads to apreferential population of the m S = ± / m S = ± / m S = ± / m S = ± / µ -PL) setup, asillustrated in Fig 1(b). The SAWs are generated by ap-plying to one of the IDTs an amplitude-modulated RFsignal of appropriate frequency. The sample is excited bya Ti-Sapphire laser (at a wavelength of 780 nm) focusedonto a spot size of 10 µ m. The V Si PL band centeredaround 900 nm (see SM [26]) is collected by an objective,spectrally filtered and detected by a photodiode detectorconnected to an amplifier locked-in to the RF modula-tion frequency. The GS spin transition frequencies aretuned to the SAW resonance frequency by applying thein-plane magnetic field B .To describe the spin-acoustic interaction of the V Si center in an external magnetic field, we consider an ef-fective spin-3/2 Hamiltonian H = H B + H def , with (1) H B = gµ B S · B + D (cid:18) S z − (cid:19) . (2)Here, S = ( S x , S y , S z ) is the spin-3/2 operator, µ B isthe Bohr magneton, g ≈ g -factor, and D = 35 MHz the zero-field splitting constant. H B de-scribes the Zeeman splitting in B = ( B x , B y , B = 0, this Hamiltonian yields the GS eigenstates dis-played in Fig. 1(d). As discussed below, H def describesthe coupling of the V Si spin and elastic deformations[2, 22].Figure 2(a) shows the Zeeman shift of the GS spinsublevels calculated from H B . For B & . B direction and the spin sublevels shift linearly with themagnetic field. We note that for in-plane magnetic fields,the states with spin projection on magnetic field direction m S = ± / B = 0 case illustrated in Fig. 1(c), the PL is nowstronger for transitions between the ES and GS involvingthe m S = ± / (a) (c)(b) (d) +3/2+1/2-1/2-3/2 D m S = ± D m S = ± xy B f B SAW
FIG. 2. (a) Evolution of the spin levels in the V Si GS un-der application of the in-plane magnetic field B ⊥ c . Thevertical double arrows indicate the resonant spin transitionsat 916 MHz induced by the SAW. (b) The SAR signal as afunction of the magnetic field strength. The data are mea-sured for two magnetic field angles ( φ B ) with respect to theSAW propagation direction (see inset). The solid curves arefits to a multi-peak Lorentzial function. The data are verti-cally shifted for clarity (c) Two-dimensional color map of the∆ m S = ± S of the IDT. (d) Simulated depth pro-files for the distribution of vacancies after irradiation (whichis proportional to the density of V Si centers) and of the u xx , u zz and u xz strain amplitudes. The optically detected SAR as a function of B ispresented in Fig. 2(b). We observe two resonances at B = 16 . B = 33 . m S = ± m S = ± f SAW = 916 MHz [cf. double verti-cal arrows in Fig. 2(a)]. Both resonances are well fittedby a Lorentzian function [solid curves in Fig. 2(b)] witha full width at half maximum (FWHM) of 2.2 mT and6.0 mT. Note that the resonances are actually doublets,which are split by approximately 1 mT and 2 mT for the∆ m S = ± m S = ± Si qudit [34]. In particular,the ∆ m S = ± m S = ± B is changedor f SAW is detuned (cf. S in the same panel), thusconfirming that the spin transitions are caused by thedynamic fields of the SAW. Additional studies summa-rized in the SM [26] show that the spatial dependence ofthe SAR intensities follow the distribution of the acous-tic field within the SAW resonator. We also prove thatour experiments are performed in the linear regime forall observed SARs [26].We are now in the position to discuss the anisotropicnature of the spin-acoustic resonances, which is a furtherimportant finding of this work. We assume the refer-ence frame illustrated in Fig. 1(b) with the SAW beampropagating along the x axis. Figure 2(b) compares theoptically detected SAR signal for two angles φ B betweenthe SAW propagation direction and the in-plane mag-netic field. While the magnetic field strengths at whichthe SARs take place are independent of the in-plane ori-entation of B , the SAR intensities do depend clearly on φ B . The full anisotropic behavior with respect to the fieldorientation on the sample plane is summarized by the cir-cles in Figs. 3(c) and 3(d), which display the intensity ofthe ∆ m S = ± m S = ± φ B .To understand the unusual angular dependence of theSARs, we develop a microscopic model for the spin-acoustic interaction. In the spherical approximation, theeffect of a lattice deformation on a spin center is describedby the interaction term H def = Ξ X αβ u αβ S α S β , (3)where Ξ is the interaction constant [2]. Being quadraticin the spin operators, such an interaction can induce spintransitions with ∆ m S = ± m S = ± B k x , the spin transitions with ∆ m S = ± u xy and u xz ,while those with ∆ m S = ± u yy , u zz and u yz . Thestrain components responsible for the spin transitions forother B directions can be obtained by the correspondingrotation of the strain tensor.A plane Rayleigh SAW propagating along x is de-scribed by the strain tensor u αβ ( t, x, z ) = u αβ ( z )e ikx − iωt + u ∗ αβ ( z )e − ikx + iωt (4)with non-vanishing components u xx , u zz , and u xz =i u xz [35]. We assume a reference frame for which u xx , u zz , and u xz are purely real. The factor i = √− u xz component is shifted by π/
2, thus resulting in an elliptically polarized strain fieldin the xz plane. Figure 2(d) compares the calculateddepth profiles of the u xx , u zz and u xz strain components[36] with the simulated depth distribution of the V Si de-fects [37].In our case, spin centers are inserted in an acousticresonator and thus subject to a combination of two coun-terpropagating SAWs travelling along x and − x with in-tensities I + and I − , respectively. We use the parameter η = ( I + − I − ) / ( I + + I − ) to distinguish different situations:a standing wave ( | η | = 0), a traveling wave ( | η | = 1) or in-termediate cases (0 < | η | < W ± and W ± of the spin transitions with ∆ m S = ± m S = ± W ± ∝ φ B h u xx sin φ B + u xz + 2 ηu xx u xz sin φ B i W ± ∝ h ( u xx sin φ B − u zz ) + 4 u xz sin φ B +4 η ( u xx sin φ B − u zz ) u xz sin φ B i . (5)The transition rates in Eq. (5) were averaged along x toaccount for the finite detection spot size, which is largerthan the SAW wavelength. The angular brackets hi in-dicate averaging along z to take into account the depthdistribution of the V Si centers as well as the strain field,as presented in Fig. 2(d).Finally, we analyze the symmetry of the SARs.Figs. 3(a)-(b) present the angular dependencies of the∆ m S = ± m S = ± η . TheSARs are always symmetric with respect to the inversionof the B x component, since our system has a ( xz ) mir-ror plane. For η = 0, the SARs are also symmetric withrespect to the inversion of the B y component due to ad-ditional presence of the time-reversal symmetry. As | η | increases, the latter symmetry breaks as the strain fieldof the travelling SAW acquires an elliptical polarization.Particularly, Fig. 3(a) shows that the ∆ m S = ± B y > ◦ < φ B < ◦ ) while itremains strong for B y < ◦ < φ B < ◦ ). Suchan asymmetric angular dependence is a clear evidence ofthe broken time-reversal symmetry in the presence of atravelling SAW. Upon inversion of the SAW propagationdirection ( η <
0, not shown), the angular dependencies ofsuch chiral SAR are flipped with respect to the horizontalaxis.Having developed a microscopic model for theanisotropic SAR, we now apply it to analyze the exper-imental data given by the circles in Figs. 3(c) and 3(d).The angular dependence of the ∆ m S = ± φ B = ± ◦ and maxima when the magnetic field rotates towards φ B ≈ ± ◦ or ± ◦ . In contrast, the angular depen-dence of the ∆ m S = ± φ B = ± ◦ . This SAR does not vanish FIG. 3. (a,b) Dependence of the ∆ m S = ± m S = ± φ B between the in-planemagnetic field and the SAW propagation direction (see in-set) calculated for the cases of standing wave ( η = 0, thickthick lines), traveling wave ( η = 1, dashed thick lines), andintermediate values of η . The strain components used are h u xx i = 9 . × − , h u zz i = 7 . × − , h u xz i = 3 . × − , h u xx u zz i = 7 . × − , h u xx u xz i = 5 . × − , h u xz u zz i =5 . × − , as obtained from the distributions of Fig. 2(d).(c,d) Angular dependencies of the ∆ m S = ± m S = ± η = 0. for any direction of the in-plane magnetic field. Thesemeasured angular dependences are best reproduced byEq. (5) by assuming η = 0, which yields the solid lines inFig. 3(c) and 3(d). This result is consistent with the ex-pected standing-wave nature of the acoustic fields withina resonator. We emphasize that our model has no fittingparameters except for the overall intensity to match thereadout optical signal.In conclusion, we observe half-integer SAR in SiC atroom temperature. Using a SAW resonator patterned onthe SiC surface, we are able to address both the ∆ m S = ± m S = ± Si spin-3/2 center with all-optical readout and without requiringextra microwave electromagnetic fields. The SARs re-veal a complex behavior, which depends on the magneticfield orientation with respect to the SAW propagationdirection. Our theoretical model describes these angulardependencies without any fitting parameter and predictschiral spin-acoustic interaction for traveling SAWs. Sucha room-temperature hybrid spin-mechanical platform canbe used to implement quantum sensors [38] with on-chipSAW control instead of microwave electromagnetic fields[2] as well as to realize acoustically driven topologicalstates [39].The authors would like to thank S. Meister andS. Rauwerdink for technical support in the preparationof the samples, S. A. Tarasenko and M. Helm for dis-cussions and critical questions, and S. Flsch for a criticalreading of the manuscript. A.V.P. acknowledges the sup-port from the Russian Science Foundation (project 20-42-04405) and the Foundation “BASIS”. G.V.A. acknowl-edge the support from the German Research Founda-tion (DFG) under Grant AS 310/5-1. 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3, 4
P. V. Santos, and G. V. Astakhov Paul-Drude-Institut f¨ur Festk¨orperelektronik, Leibniz-Institut imForschungsverbund Berlin e.V., Hausvogteiplatz 5-7, 10117 Berlin, Germany Ioffe Physical-Technical Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia Helmholtz-Zentrum Dresden-Rossendorf, Institute of Ion Beam Physics and Materials Research,Bautzner Landstrasse 400, 01328 Dresden, Germany Technische Universit¨at Dresden, 01062 Dresden, Germany
This Supplemental Material contains information about: • Photoluminescence characterization • ODMR spectrum at zero magnetic field • Spatial dependence of the spin-acoustic resonance • Dependence of the spin-acoustic resonance on SAW power • Matrix elements of the SAW-induced spin transitions
PHOTOLUMINESCENCE CHARACTERIZATION
Figure S1(a) displays the typical room-temperature photoluminescence (PL) spectrum of the V Si ensemble afterproton irradiation of the 4H-SiC. Figure S1(b) shows the depth profile of the PL intensity. The V Si defects implantedat a depth of 250 nm cause the strongest PL signal. The measurements are taken in the same confocal setup as forthe spin-acoustic resonance (SAR) experiments.
800 850 900 950 1000050100150200250300 I n t en s i t y ( c oun t s ) Wavelength (nm) (a) -40 -20 0 20 40 60 800.00.10.20.30.40.50.6 I n t en s i t y ( a r b . un i t s ) Sample depth ( (cid:181)m)SiCair (b)
FIG. S1. (a) Room-temperature PL spectrum of the V Si ensemble in 4H-SiC. (b) Depth profile of the PL intensity across thesurface of the 4H-SiC. a r X i v : . [ c ond - m a t . m e s - h a ll ] M a y ODMR SPECTRUM AT ZERO MAGNETIC FIELD
Figure S2 displays the optically detected magnetic resonance (ODMR) spectrum of the V Si ensemble at zero mag-netic field. The measurements are taken at room temperature in the same confocal setup as for the SAR experiments.The microwave frequency is applied by a coplanar waveguide placed close to the sample. The 70 MHz peak correspondsto the splitting frequency between the m S = ± / m S = ± /
40 50 60 70 80 90 1000.00.51.01.52.0 D I P L /I P L ( - ) Frequency (MHz)
FIG. S2. Room-temperature ODMR spectrum at zero magnetic field of the V Si ensemble implanted at the surface of the4H-SiC. SPATIAL DEPENDENCE OF THE SPIN-ACOUSTIC RESONANCE
The colormap of Fig. S3(a) displays the intensity of the optically detected ∆ m S = ± y = 0. Figure S3(b) shows the amplitude of the resonance, defined as the area of the signal in Fig. S3(a) whenintegrated along the magnetic field, for several positions of the laser. The spatial dependence of the SAR signal iswell fitted to a Gaussian peak with a FWHM of 11 µ m. This is about three times narrower than the finger lengthat the inner part of the IDTs (35 µ m), thus confirming the focusing efficiency of our acoustic resonator and furthersupporting the acoustic nature of the SAR signal. x y B S A W -30 -20 -10 0 10 20 300.00.10.20.30.40.50.60.7-20 -10 0 10 202019181716151413 M agne t i c F i e l d ( m T ) y (µm) (a) D I PL / I PL (10 -4 ) P ea k A r ea ( a r b . un i t s ) y (µm) (b) FIG. S3. (a) Two-dimensional color map of the ∆ m S = ± m S = ± DEPENDENCE OF THE SPIN-ACOUSTIC RESONANCE ON SAW POWER
Figure S4(a) displays the dependence of the optically detected SAR signal on the nominal RF power applied to theacoustic resonator, P rf . Figure S4(b) displays the area of both resonances [obtained from Lorentzian fits to the datain Fig. S4(a)] as a function of P rf , together with their linear fits.
10 15 20 25 30 35 40012 262422 (a) D m S =–1 D I P L /I P L ( - ) Magnetic Field (mT) D m S =–2 P rf (dBm): 0 100 200 300 400012 D m S =–1 D m S =–2 P ea k A r ea ( a r b . un i t s ) P rf (mW) (b) FIG. S4. (a) Optically detected SAR signal as a function of the magnetic field strength, measured for several nominal RFpowers P rf applied to the acoustic resonator. The data are taken for an angle φ B = 30 ◦ between the in-plane magnetic fieldand SAW propagation direction. The solid curves are Lorentzian fits to the data. The data are vertically shifted for clarity.(b) Area of the SAR peaks as a function of P rf . The solid curves are linear fits to the data. MATRIX ELEMENTS OF THE SAW-INDUCED SPIN TRANSITIONS
First, we choose the new reference frame ( x y z ) where the new in-plane axes are x k B and y ⊥ B . In such frame,the matrix elements M (+)∆ m S of spin transitions with the change of the spin projection on x axis ∆ m S induced byabsorption of the SAW read M (+) ± = √ u x y ∓ i u x z ) , (S1) M (+) ± = √ Ξ( u y y − u zz ∓ u y z ) . To obtain the matrix elements for the transitions with emission of the phonon M ( − )∆ m S , one must replace u αβ with u ∗ αβ in the above equation. Expressing the strain components in the new basis via those in the original basis u xx , u zz , and u xz , we find M (+) ± = √
3Ξ cos φ B ( u xx sin φ B ∓ i u xz ) , (S2) M (+) ± = √ Ξ( u xx sin φ B − u zz ∓ u xz sin φ B ) . We note that transitions with ∆ m S > m S < M ( ± ) ± and M ( ± ) ± . We consider that the strain field is a sum of two waves of theform u αβ ( t, x, z ) = u αβ ( z )e ikx − iωt + u ∗ αβ ( z )e − ikx + iωt with wave vectors k and − k and different intensities I + and I − .Note that the u xx and u zz strain components of the right- and left-traveling waves match while the u xz componentis opposite. The spin transition rates are obtained by averaging W ± = | M ( ± ) ± | and W ± = | M ( ± ) ± | over the x and zz