Strain-induced spin resonance shifts in silicon devices
J.J. Pla, A. Bienfait, G. Pica, J. Mansir, F.A. Mohiyaddin, Z. Zeng, Y.M. Niquet, A. Morello, T. Schenkel, J.J.L. Morton, P. Bertet
SStrain-induced spin resonance shifts in silicon devices
J.J. Pla, A. Bienfait, G. Pica,
3, 4
J. Mansir, F.A. Mohiyaddin, ∗ Z. Zeng, Y.M. Niquet, A. Morello, T. Schenkel, J.J.L. Morton, and P. Bertet School of Electrical Engineering and Telecommunications,University of New South Wales, Anzac Parade, Sydney, NSW 2052, Australia Quantronics Group, SPEC, CEA, CNRS, Universit´e Paris-Saclay, CEA-Saclay, 91191 Gif-sur-Yvette, France Center for Neuroscience and Cognitive Systems @UniTn,Istituto Italiano di Tecnologia, Corso Bettini 31, 38068 Rovereto, Italy SUPA, School of Physics and Astronomy, University of St Andrews, KY16 9SS, United Kingdom London Centre for Nanotechnology, University College London,17-19 Gordon Street, London, WC1H 0AH, United Kingdom Universit´e Grenoble Alpes, CEA, INAC-MEM, 38000 Grenoble, France Accelerator Technology and Applied Physics Division,Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA (Dated: January 10, 2018)In spin-based quantum information processing devices, the presence of control and detectioncircuitry can change the local environment of a spin by introducing strain and electric fields, alteringits resonant frequencies. These resonance shifts can be large compared to intrinsic spin line-widthsand it is therefore important to study, understand and model such effects in order to better predictdevice performance. Here we investigate a sample of bismuth donor spins implanted in a silicon chip,on top of which a superconducting aluminium micro-resonator has been fabricated. The on-chipresonator provides two functions: first, it produces local strain in the silicon due to the larger thermalcontraction of the aluminium, and second, it enables sensitive electron spin resonance spectroscopyof donors close to the surface that experience this strain. Through finite-element strain simulationswe are able to reconstruct key features of our experiments, including the electron spin resonancespectra. Our results are consistent with a recently discovered mechanism for producing shifts ofthe hyperfine interaction for donors in silicon, which is linear with the hydrostatic component of anapplied strain.
I. INTRODUCTION
The spins of dopant atoms in silicon devices have beenshown to have great promise for quantum informationprocessing (QIP) [1–6]. This has, in part, been encour-aged by the extraordinarily long spin coherence timesdemonstrated, surpassing 1 second for the electron spin[7] and 3 hours for the nuclear spin [8] of the phosphorus( P) donor. Another group-V donor with considerablepromise for QIP in silicon is bismuth (
Bi). Its large nu-clear spin I = 9 / A = 1475 MHz(which describes the interaction between the electron S and nuclear I spins A S · I ) provides rich features such asdecoherence-suppressing atomic-clock transitions [9–11],where coherence times can exceed by two orders of mag-nitude those typically achieved using other donor species.The Si:Bi system also possesses a large zero-field splittingof 7.375 GHz, making it an attractive dopant for use inhybrid superconducting devices [12, 13] such as quantummemories [14–17].In donor-based QIP devices, such as quantum bits andhybrid quantum memories, the donors are located withinclose proximity of control and detection circuitry on thesurface of the silicon chip. Recent experiments on indi- ∗ Present Address: Quantum Computing Institute, Oak Ridge Na-tional Laboratory, Oak Ridge, TN 37830, USA vidual donor electron and nuclear spin qubits adjacent tonanoelectronic circuits [18] have highlighted the impor-tance of considering the effect of these structures on thelocal environment of the spin. For example, it was shownthat the spin resonance frequencies of P donors in nano-electronic devices can experience shifts from their bulk-like values up to four orders of magnitude greater thantheir intrinsic line-widths [5, 19–21]. These shifts havebeen attributed to strain and electric fields produced bysurface metallic gates in the devices.Strain is an inherent feature of metal-oxide-semiconductor (MOS) electronic devices, which oftencombine materials that have vastly different coefficientsof thermal expansion (CTE) [22, 23]. It is thereforecrucial to understand and predict the effect of intrinsicdevice strains on donors, as this can aid the design ofscalable donor-based QIP and hybrid superconductingdevice architectures, serving as a guide to the oftenexpensive and time-consuming fabrication process. Herewe study a sample of bismuth (
Bi) donors implantedfrom 50-150 nm beneath a thin-film aluminium wire(Figs. 1a and b). We observe the Si:Bi spin resonancespectra in the device to be substantially altered fromwhat is typically found in bulk experiments [10, 24].Through analyzing a range of mechanisms, we concludethat strain induced by differential thermal contractionof the silicon and the surface aluminium structure isthe most likely explanation for the non-trivial spectra.A model is developed that is able to reproduce many a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n facets of our measurements, demonstrating the abilityto predict device behavior and illustrating the impor-tance of considering strain in semiconductor micro andnanoelectronic quantum devices.The article is organized as follows: in Section II wepresent the device architecture, physical system and ex-perimental setup utilized in our study. Section III ex-amines the electron spin resonance spectra of bismuthdonors beneath an aluminium wire, revealing non-bulk-like splittings of the resonance peaks. Mechanisms po-tentially producing the splittings are discussed in Sec-tion IV and simulations of the spin resonance spectra areperformed in Section V for one of the mechanisms iden-tified. We conclude by discussing the implications of thesimulations and the broader significance of our results forQIP in Section VI. II. EXPERIMENTAL DETAILSA. Device
Our device (Fig. 1a) consists of three superconduct-ing aluminium microwave resonators patterned on thesurface of the same silicon chip via electron-beam-lithograph. The top 700 nm of silicon is an epitaxial layerof isotopically enriched 99.95% Si, grown on a ∼ µ m thick high-resistivity float-zone silicon (100) wafer.The epitaxial layer was implanted with Bi donors ac-cording to the profile depicted in Fig. 1b.The resonators are a lumped-element LC design, theycontain a central inductive wire that produces an oscil-lating microwave magnetic field B to drive and detectspin resonance. The drive field B is proportional tothe magnetic vacuum fluctuations δB in the resonator,a quantity that we can simulate directly for our de-vice. We utilize δB in the following calculations anddiscussion: it is readily determined from our simula-tions (unlike B , which requires an accurate calibra-tion of losses and other experimental parameters), andit provides us with another important measure, the spin-resonator coupling strength g . A simulation of δB isperformed knowing only the impedance of the resonator Z and its frequency ω / π , and by calculating the re-sulting vacuum current fluctuations δi = ω (cid:112) (cid:126) / (2 Z )in the wire (where (cid:126) is the reduced Planck’s constant).The current density distribution in the superconductingfilm (depicted in Fig. 2a) is evaluated using DC equa-tions adapted from Ref. [25], which are valid for thecalculation of our microwave current due to the negli-gible ohmic losses at milli-Kelvin temperatures and be-cause the typical resonator frequency ( ∼ ≈
140 GHz) [26]. The current densitydistribution is then fed to a finite-element magnetostaticsolver (COMSOL Multiphysics), with the resulting | δB | profile shown in Fig. 2b.We observe a strong spatial dependence of the δB Al Si [110][110][010][100][001] Ф XYZ B C Bi (x10 cm -3 ) D ep t h ( n m ) a bc FIG. 1. (a)
Sketch of an LC superconducting resonatormade from a 50 nm thick film of aluminium, patterned on asilicon substrate, with central inductor 5 µ m wide and 700 µ mlong. Whilst we only show one resonator here, there are three(almost identical) resonators patterned on the same chip (seepanel c). The silicon sample was cleaved along the (cid:104) (cid:105) crys-tal axes and we specify a sample frame such that X (cid:107) [110],Y (cid:107) (cid:2) (cid:3) and Z (cid:107) [001]. The static field B is oriented in theXY-plane at a variable angle φ to X. (b) Bismuth donor dop-ing profile. The blue dashed curve shows the result of a sec-ondary ion mass spectrometry (SIMS) measurement, whilstthe red curve is the concentration of neutral donors obtainedfrom a finite-element simulation performed using the SIMSprofile, that takes into account donor ionization from theSchottky junction between aluminium and silicon (see Sec-tion IV A). (c)
Three-dimensional copper microwave cavitysample holder. The silicon chip is mounted on a sapphireholder (pictured in white) and is probed via the cavity inputand output antennas. orientation at the donor implantation depth (Fig. 2c).Underneath the wire, the Y component of the field δB dominates, whilst to the side δB is the largest. We uti-lize this trait later in order to study spins in different spa-tial regions through orientation-dependent electron spinresonance (ESR) spectroscopy [12, 27]. B. Physical System
At cryogenic temperatures, the bismuth donors bindan additional valence electron compared to the siliconatoms of the host crystal, providing a coupled electron( S = 1 /
2) and nuclear ( I = 9 /
2) spin system that isdescribed by the Hamiltonian: H /h = γ e B · S − γ n B · I + A S · I (1)where γ e = 28 GHz/T ( γ n = 6 .
963 MHz/T) is the elec-tron (nuclear) gyromagnetic ratio and B is a static mag-netic field applied in the plane of the aluminium res-onators – with a variable angle φ relative to the inductivewire (see Fig. 1a) – that allows us to fine-tune the spintransition frequencies of the Bi donors.At values of the magnetic field where the electron Zee-man frequency E z /h = γ e B (cid:46) A , the eigenstates be-come strongly mixed in the electron-nuclear spin basisand are best described by the total spin F = I + S andits projection onto B , m F [9]. We chose the frequen-cies of the resonators to be close to the Si:Bi zero-fieldsplitting of 7.375 GHz in order to minimize field-inducedlosses in the superconducting films, achieving ω / π =7 .
305 GHz for resonator A, ω / π = 7 .
246 GHz for res-onator B and ω / π = 7 .
143 GHz for resonator C. Wetherefore operate in the regime where F and m F are goodquantum numbers and we describe states in the | F, m F (cid:105) basis. In the following analysis and discussion we focuson resonators A and B – those with frequencies closer tothe zero-field splitting which we were able to study themost extensively. Table I presents important parametersthat characterize the low-field ( B < C. Sample Mounting
The device is fixed to a sapphire wafer with a smallamount of vacuum grease (this serves to minimize samplestrains produced through mounting) and the sapphire isthen clamped between the halves of a rectangular coppermicrowave cavity (Fig. 1c), which acts as a sample en-closure and permits high quality-factors of the supercon-ducting resonators by supressing radiation losses. Thecopper cavity is attached to the cold-finger of a dilutionrefrigerator and cooled to a base temperature of 20 mK,where we are able to detect the spin echo signals pro-duced by the small number of shallow-implanted donorsunderneath each wire (estimated at ∼ ) by utilizing aquantum-noise-limited ESR setup, as described in Refs.[12, 28, 29]. We direct readers to the Supplementary Ma-terial of Ref. 12 for a full schematic of the experimentalsetup. III. SPIN RESONANCE SPECTRAA. Echo-Detected Field Sweep
In this section we provide a detailed discussion of theSi:Bi ESR spectra, first reported in Refs. 12, 13. We ob-serve the ESR spectrum for resonator B by performingan echo-detected magnetic field sweep on the lowest-fieldspin resonance line (indicated by the arrow in Fig. 3a),corresponding to transition 1B, i.e. between the states | , − (cid:105) ↔ | , − (cid:105) (see Table I). We integrate the echo sig-nal A e from a Hahn echo sequence [30] (over the dashedregion depicted in the pulse protocol of Fig. 3b) and step -5 -4 -3 -2 -1 0 1 2 3 4 5 Y ( μ m) δ B δ B Z ( n m ) -5000500 | δB | (nT)250-250 ○ □ ∆ ◊ δ J ( Y ) ( n A / μ m ) δ B Y , Z ( n T ) Al abc Si FIG. 2. (a)
Calculation of the current density vacuum fluctu-ations in the inductor. Equations describing the current den-sity profile were adapted from Ref. [25]. The only inputs tothis calculation are the impedance of the resonator Z = 44 Ωand its frequency ω / π ≈ . (b) A COMSOL Multiphysics finite-elementsimulation of the spatial dependence of the magnetic fieldvacuum fluctuations δB magnitude produced by the currentdensity in panel a. The symbols beneath the white dashed lineidentify regions that will be referred to in following sections. (c) Components of δB along the Y and Z axes at a depthof 75 nm (corresponding to the peak donor concentration),marked by the white dashed line in panel b. the magnetic field B . The sweep is first performed with B (cid:107) X ( φ = 0 ◦ ), and then repeated with the orthogonalorientation B (cid:107) Y ( φ = 90 ◦ ); the resulting traces areshown in Fig. 3c. The doped silicon sample investigatedin this study has also been characterized using a standard“bulk” ESR spectrometer at X-band and with no planaron-chip resonator [24]. The grey-solid curve in Fig. 3crepresents the spin resonance spectrum from this studyextrapolated to the spin transition and frequency utilizedin our experiment (see Appendix B for further details).Instead of measuring a single peak with a line-width of ∼ µ T (as expected from the X-band measurement),we observe that the resonance is split into two peaks.Each peak has a line-width of ∼ µ T, representing atotal broadening of over an order of magnitude.Varying the amplitude of the refocusing π -pulse in theecho sequence reveals a series of Rabi oscillations ( A e ismaximized whenever the refocusing pulse equals an odd-multiple of π ) and the frequency of these oscillations isobserved to depend strongly on the magnetic field B (Fig. 3b) across these two peaks. The traces in Fig. 3cwere recorded in a “compensated” manner, ensuring that Resonator A, ω / π = 7 .
305 GHzTransition ∆ F ∆ m F B (mT) M ( df/dB ) /γ e df/dA df/dg (MHz) df/dQ | , − (cid:105) ↔ | , − (cid:105) -1 2.86 0.47 -0.90 5.00 -36.0 2.452A | , − (cid:105) ↔ | , − (cid:105) | , − (cid:105) ↔ | , − (cid:105) | , − (cid:105) ↔ | , − (cid:105) -1 3.69 0.42 -0.69 5.00 -35.8 6.145A | , − (cid:105) ↔ | , − (cid:105) | , − (cid:105) ↔ | , − (cid:105) | , − (cid:105) ↔ | , − (cid:105) -1 5.22 0.37 -0.49 5.00 -35.5 7.428A | , − (cid:105) ↔ | , − (cid:105) ω / π = 7 .
246 GHzTransition ∆ F ∆ m F B (mT) M ( df/dB ) /γ e df/dA df/dg (MHz) df/dQ | , − (cid:105) ↔ | , − (cid:105) -1 5.20 0.47 -0.90 5.00 -65.4 2.492B | , − (cid:105) ↔ | , − (cid:105) | , − (cid:105) ↔ | , − (cid:105) | , − (cid:105) ↔ | , − (cid:105) -1 6.75 0.42 -0.69 5.00 -64.9 6.27TABLE I. Numerical calculations of the spin transition parameters for the Si:Bi system at the LC resonator frequencieslisted. Parameters include: resonance field ( B ), transition matrix element ( M = |(cid:104) F, m F | S X , Z | F (cid:48) , m (cid:48) F (cid:105)| for | ∆ F ∆ m F | = 1 , df/dB ), electron g -factor ( df/dg ), hyperfine interaction( df/dA ) and quadrupole interaction ( df/dQ ). at each value of B the pulse amplitude was chosen toprovide well-calibrated π and π/ π/ t π/ = 2 . µ s and an excitation band-width of ∼
500 kHz. This pulse is heavily filtered by theresonator, reducing its bandwidth to a value determinedby the resonator line-width κ = ω /Q ≈ π ×
25 kHz(where Q = 3 . × is the quality factor of resonatorB). Thus, only spins with resonant frequencies that lie in-side the resonator bandwidth contribute to the measure-ment. In addition, these spins experience a relaxationrate which is three orders of magnitude greater than theintrinsic value, due to the Purcell effect [13]. This en-hanced relaxation is suppressed quadratically with thespin-resonator frequency detuning, such that off-resonantspins display substantially longer energy relaxation times T and quickly become saturated under the 0.2 Hz rep-etition rate of the experiment. Each B in Figs. 3b and3c therefore corresponds to a highly-selective measure-ment on a small sub-ensemble of spins with a resolution∆ B = κ/ ( df /dB ) = 1 µ T, where df /dB is the transi-tion frequency field sensitivity (listed in Table I).Comparing the echo-detected spectra for the differ-ent orientations of B (red and blue circles in Fig. 3c)provides strong evidence that the splitting and inhomo-geneous broadening of the ESR transition results fromthe presence of the on-chip LC resonator. We find thatthe low-field peak vanishes for B (cid:107) Y ( φ = 90 ◦ ) whilethe high-field peak remains relatively unchanged. Thiscan be understood by referring to Fig. 2c and notingthat the spin transition probed here (1B, see Table I) obeys the selection rule | ∆ m F | = 1 and is therefore ex-cited only when δB ⊥ B . For B (cid:107) Y, the condition δB ⊥ B is only met for spins to the side of the wire(which experience a δB field along Z). Spins underneaththe wire (where δB field almost entirely along Y) are notmeasured in this scan. For the spectrum recorded with B (cid:107) X ( φ = 0 ◦ ), spins underneath the wire as well asthose to the side observe B ⊥ δB and thus contributeto the echo signal. Thus the low-field (vanishing) peaklikely corresponds to the spins below the wire whilst thehigh-field peak is produced by spins to its side, indicatingthat the presence of the inductive wire is the source of thesplitting. In Section IV we discuss a number of potentialmechanisms (e.g. electric field, Meissner-induced mag-netic field inhomogeneity and strain) through which thiscould occur. The spin resonance frequency of the donorstherefore depends on their location relative to the wire.By measuring only a small fraction of the large inhomo-geneously broadened transition at each B field (1 µ Tagainst ∼ µ T) in Fig. 3c, we are effectively probingsub-ensembles of donors residing in specific locations inthe device.We now return to the B dependence of the Rabi os-cillations (Fig. 3b) and demonstrate that the picture de-scribed above is in good agreement with this data. Thecoupling strength between each spin and the resonator isgiven by g = γ e M | δB ⊥ | , where M is the ESR tran-sition matrix element (see Table I) and | δB ⊥ | is themagnitude of the δB component felt by the spin thatis perpendicular to B . The Rabi frequency Ω R then hasa linear dependence on the δB field through the rela-tion Ω R = 2 g √ n , where n is the mean intra-cavity pho-ton number (proportional to the input microwave power).For the high-field peak in the ESR spectra (originatingfrom spins located to the side of the wire), the sharp tran-sition at the low-field edge likely corresponds to spins farfrom the wire that are bulk-like in their behavior. Being τ τ Vary θ θ P u l s e a m p li t ude ( n W / ) A e (a.u.)1234 0 B (mT) A e ( a . u . ) B || X B || Y0 4.9 5.0 5.1 5.2 5.3 B (mT) B (mT) E i g . E ne r g y / h ( G H z ) -505 4, -4 ↔
5, -5 ○ □ ∆ ◊ bca FIG. 3. (a)
Eigenstate frequencies of the Si:Bi system.The purple states and arrow identify the | , − (cid:105) ↔ | , − (cid:105) transition (1B) probed in panels b and c. (b) Rabi oscillationsas a function of B for transition 1B. The amplitude of therefocusing pulse in a Hahn echo sequence (shown above) isvaried to reveal oscillations in the integrated echo signal A e (marked by the black dashed box in the sequence). Symbolsidentify spectral regions that are generated by spins at specificlocations in the device (see Fig. 2b). (c) A compensatedecho-detected field sweep, taken using the calibrated π -pulseamplitudes of panel b (yellow dashed line). The grey-filledcurve depicts the expected ESR spectrum, whilst the solidcircles show the measured spectra (averaged over 8 sequenceswith a repetition rate of 0.2 Hz) for different field orientations.A 2% correction was applied to B for the measured data(within the magnet calibration error) so that the high-fieldpeak aligns with the theoretical transition field. The samecorrection was applied to all experimental data in this study. F r equen cy ( G H z ) E c ho a m p li t ude , A e ( a . u . ) B (mT) bca FIG. 4. (a)
ESR transition frequencies of the Si:Bi sys-tem for B < m F = ± δB ⊥ B )whilst the dashed lines show ∆ m F = 0 transitions ( δB (cid:107) B ).The purple solid line indicates the frequency of resonator A( ω / π = 7 .
305 GHz). (b)
Compensated echo-detected fieldsweeps of the ESR transitions below 7 mT of resonator A and (c) resonator B. The theoretical spin transition frequenciesare identified by the black solid and dashed lines. far from the wire, these spins also experience a reduced δB (see Fig. 2c) and thus Rabi frequency, observed aslonger-period oscillations in Fig. 3b. Moving closer tothe wire increases the spin resonance shifts (i.e. throughlarger electric or strain fields) as well as the magnitudeof the δB field felt by the spins. We thus anticipatethe tail regions of the lines to have an enhanced Rabifrequency, and this is indeed the case. The symbols over-laid on Fig. 3b summarize the above discussion by corre-lating the different spectral regions with spins from spe-cific locations in the device (see corresponding symbolsin Fig. 2b). B. Extended Spectra
To help identify the mechanism behind the wire-induced peak splitting and broadening, we probe addi-tional spin resonance transitions (listed in Table I) usingresonators A and B, which display different sensitivitiesto the various Hamiltonian parameters. In Fig. 4a weplot the calculated low-field ESR transition frequenciesand their crossing with resonator A. Transitions obeyingthe usual spin selection rule ∆ m F = ± B ⊥ δB , aswas the case for the previous measurement on transition1B. Such transitions are typically referred to as beingof “ S X ” type, since it is primarily the S X operator thatdrives spin flips between the states. Transitions obeyingthe selection rule ∆ m F = 0 (blue lines in Fig. 4a) – ofso-called “ S Z ” type – are probed in the experiment withthe alignment B (cid:107) δB (i.e. B (cid:107) Y ). We refer thereader to Appendix A for a detailed discussion of thesetwo types of spin resonance transitions.A measurement of the first transitions with B (cid:107) X(∆ m F = ±
1) is shown in the compensated echo-detectedfield sweep of Fig. 4b (red trace) for resonator A. Alsopresented here is the spectrum recorded with B (cid:107) Y(blue trace), which is composed of both ∆ m F = 0 reso-nances from spins underneath the wire (where B (cid:107) δB )and ∆ m F = ± B ⊥ δB ). The ∆ m F = 0 transitions areobserved to lack a splitting, this is further evidence thatthey originate from spins located predominantly under-neath the wire (the only region with B (cid:107) Y). The exper-iments are repeated for resonator B and displayed in thelower traces of Fig. 4c. We display the extracted peaksplittings of the recorded transitions in Table II.
IV. PEAK SPLITTING MECHANISMS
We now turn to the analysis of possible mechanismsthrough which the presence of the aluminium wire mayinduce a splitting and broadening of the observed ESRspectra.
A. Built-in Voltage
The aluminium/silicon interface formed beneath theresonator constitutes a Schottky junction. Band bend-ing at the interface results from the difference in workfunctions of the aluminium and silicon (or from Fermilevel pinning to surface states) [31]. The band bendingcauses ionization of bismuth donors within an area known
Resonator A, ω / π = 7 .
305 GHzTransition B (mT) ∆ B (mT)1A 2.87 0.114A 3.71 0.157A 5.28 0.20Resonator B, ω / π = 7 .
246 GHzTransition B (mT) ∆ B (mT)1B 5.24 0.114B 6.84 0.14TABLE II. Experimental center fields ( B ) and peak split-tings (∆ B ) extracted from the measured ESR transitionsfor resonator A and B. ∆ F ∆ m F = 0 transitions do not dis-play a splitting and are therefore not included. Although the∆ F ∆ m F = ± F ∆ m F = − M . We do not at-tempt to extract line-widths of the peaks due to their highlyasymetrical shapes. Z (nm) E l e c t r i c F i e l d , E z ( M V / m ) -2-1.5-1-0.500.51 ρ sc / C B i Neutral BiDepletion edge 024610 C B i ( x c m - ) FIG. 5. Fraction of ionized donors, in-built electric fieldand doping profile versus depth beneath the aluminium res-onator. The implanted Bi profile (as determined from a sec-ondary ion mass spectrometry measurement) is shown in or-ange. The green curve provides the space charge density ρ sc (which represents the density of ionized donors) divided bythe donor concentration C Bi . Calculations were performedwith the finite-element electrostatic solver ISE-TCAD, witha simulation temperature of 5 K (the minimum temperatureat which convergence was achieved) and an assumed back-ground boron doping density of 10 cm − . The grey dashedbox highlights the edge of the depletion region, this roll-offis expected to get steeper at the experimental temperatureof 20 mK. At ∼
275 nm depth the bismuth donors are ion-ized once again (this time to the background boron accep-tors present in the sample), before the space charge densitybecomes negative, indicating the presence of ionized borondopants. as the depletion region. Donor ionization continues intothe semiconductor until a sufficient space-charge has beenaccumulated to counter the band bending. Immediatelyoutside of the depletion region, the total electric field isreduced to zero. At finite temperatures, however, theedge of the depletion region is broadened according toFermi-Dirac statistics, and a small fraction of neutraldonors can experience large electric fields. Such donorswould display a Stark shift of the hyperfine interaction[20] or electron g -factor through the electric field, alter-ing their resonant frequencies from those to the side ofthe wire away from the depletion region.We have performed finite-element simulations with thecommercial software ISE-TCAD, which solves the Pois-son equation self-consistently to extract the electric fieldsand ionized bismuth concentration underneath the wire,the results of which are shown in Fig. 5. This plot demon-strates that the broadening of the depletion region edgeis small relative to the width of the implantation pro-file, even at the elevated simulation temperature of 5 K— the minimum temperature at which convergence wasachieved. Donors at depths less than 50 nm are mostlyionized, whilst donors deeper than this are neutral andexperience negligible electric fields ( <
50 kV/m, with ex-pected Stark shifts below 1 kHz [32]). At the experimen-tal temperature of 20 mK, we expect an even sharperdepletion region boundary. We therefore discount thismechanism as the cause for the spectral broadening andremove the shallow donors ( <
50 nm) beneath the wirefrom the spectra simulations in the following sections.
B. Magnetic Field Inhomogeneity
It is conceivable that the superconducting resonatorcould perturb the static magnetic field in a manner thatproduces differing magnetic field profiles beneath thewire and to its side. For example, this might result fromthe component of a misaligned B field perpendicular tothe aluminium film, concentrating above or below thewire due to the Meissner effect [33]. The strength ofany such inhomogeneity increases in proportion with themagnitude of B , and as the resonators are fabricatedwithin 2 mm of one another on the same silicon chip, theinhomogeneity would be nearly identical for each of theresonators. We can rule this mechanism out due to thefact that we observe the same splitting and line-widthof the first spin transition for resonators A and B (seeTable II) and also C (see Appendix C), despite the tran-sition for resonator C occurring at almost twice the fieldof resonator B and three times that of resonator A. C. Strain
Strain can alter the spin transition frequencies ofdonors in silicon through several mechanisms. It has beenshown that the nuclear magnetic resonance (NMR) fre-quencies of donors with nuclear spin
I > / A [36] orthe electron g -factor g e , both resulting in shifts of the spinresonance frequencies. Here we will analyze all three ofthese mechanisms (QI, A and g e ) to determine if they arecapable of accounting for the ESR spectra presented inSection III.In order to aid in our discussion, we first explain theorigin of strain in our device and provide an estimateof its magnitude and spatial distribution through sim-ulations. The aluminium resonator is deposited on thesilicon substrate by electron-beam evaporation at roomtemperature, where the device is assumed to be strain-free [37, 38]. Whilst the evaporation temperature maybe above room temperature in practice, it is assumedto be only a fraction of the total temperature range ex-plored in our experiments (∆ T ≈
300 K). As the deviceis cooled to 20 mK, the approximate ten-fold difference inthe CTEs of silicon and aluminium produces substantialdevice strains through differential thermal contraction.We perform finite-element simulations of these strains us-ing the software package COMSOL Multiphysics, wherewe include temperature-dependent CTEs of the materials
Y ( μ m) Z ( μ m ) e xx e yy e zz -2-1012(x 10 -4 ) FIG. 6. Finite-element COMSOL simulations of the straintensor components along the principle crystal axes x (cid:107) [100],y (cid:107) [010] and z (cid:107) [001] and their variation as a function ofposition in the device. A cross-section of the aluminium wire(drawn to scale) is represented by the grey gradient-filled boxabove the silicon substrate (which bounds the strain data).Only half of the wire is displayed here due to it being sym-metric about its center. Here the xyz crystal axes are relatedto the sample frame XYZ (used to describe δB and the ori-entation of B ) by a 45 ◦ rotation about Z (see Fig. 1a). Weshow the result for the wire running parallel to the [110] (orX) axis, the direction in which the sample was cleaved. [39–41] and the anisotropic stiffness coefficients for sili-con [42]. Three of the six independent strain tensor com-ponents (those along the (cid:104) (cid:105) crystal axes) have beenplotted in Fig. 6 as a function of position. The full straintensor and its spatial dependence can be found in Ap-pendix D.
1. Quadrupole Interaction
There have been several recent studies that report onquadrupole interactions of group-V donors in silicon, gen-erated by strain [34, 35] or interface defects [43]. Nucleiwith a spin
I > / Q [44]. This charge distribution has an axis ofsymmetry that aligns with the nuclear angular momen-tum and interacts with an electric field gradient (EFG) V αβ (where α and β are principal axes in the local crystalcoordinate system) produced by external charges, such asthe donor-bound electron. The interaction is describedby the following quadrupole Hamiltonian: H Q /h = γ e Q V zz I (2 I − h (cid:2) I − I + η (cid:0) I − I (cid:1)(cid:3) (2)where γ is a multiplicative scaling factor (resulting fromthe Sternheimer anti-shielding effect [44]), e is the elec-tron charge, h is Planck’s constant, I is the nuclearspin operator with components I α , I in the denomina-tor is the scalar value of the nuclear spin ( I = 9 /
2) and η = ( V xx − V yy ) /V zz is an asymmetry parameter. It isevident from Eq. 2 that the existence of an EFG V αβ produces a frequency shift between states with differentnuclear spin projections m I . In the case of the Si:Bi spinsystem, quadrupole shifts in the ESR spectra are evidentat low magnetic fields because the electron and nuclearspin states are strongly mixed by the hyperfine interac-tion.In Table I we list the sensitivities of the transitions tothe quadrupole coefficient Q zz = γe Q V zz / [4 I (2 I − h ](the prefactor in the quadrupole Hamiltonian H Q ). Bycomparing the sensitivities df /dQ to the extended ESRspectra (Fig. 4) and observed peak splittings (Table II),it becomes apparent that the quadrupole interactionis unlikely to be the origin of the non-trivial spectrashape. The peak splittings ∆ B of different transitionsfor the same resonator approximately follows their mag-netic field sensitivities df /dB (see Table I), implying anunderlying mechanism with a constant frequency distri-bution across all transitions. This is clearly not the casefor the quadrupole interaction, where df /dQ increaseswith transition number. Furthermore, the ∆ F ∆ m F =0 transitions have sensitivities of opposite sign to the∆ F ∆ m F = − F ∆ m F = − df /dQ is strongly dependent on the transi-tion, we note that df /dA is constant (see Table I) so thata strain-induced inhomogeneous hyperfine interaction islikely to have the desired properties for the comparisonof different transitions.
2. Hyperfine Interaction
Silicon has a conduction band minimum that is six-folddegenerate along the (cid:104) (cid:105) equivalent crystallographic di-rections — commonly referred to as “valleys” [45]. Thedegeneracy of these valleys is broken by the confining po-tential of the donor, resulting in a singlet A ground stateand doublet E and triplet T excited states [46]. For adonor in a bulk silicon crystal (in the absence of strainand electric fields) the electron is perfectly described bythe singlet ground state | ψ (cid:105) = | A (cid:105) . The E and T statewavefunctions have vanishing probabilities at the nucleus(i.e. | ψ (0) | = 0) and consequently do not exhibit a hy-perfine interaction ( A = 0). Applying strain to a valleyshifts its energy relative to the conduction band mini-mum, resulting in a rearrangement of the relative popu-lations of each valley which can be described as a mixingof the donor A and E states. The degree of mixingcan be calculated using the “valley repopulation” model(VRM) [47], which predicts a quadratic shift of the hy-perfine interaction with an applied strain [48]:∆ A ( (cid:15) ) A (0) = − Ξ E [( (cid:15) xx − (cid:15) yy ) + ( (cid:15) xx − (cid:15) zz ) +( (cid:15) yy − (cid:15) zz ) ] (3) Y ( μ m) Z ( n m ) -2020 Δ A (MHz) ab Z ( n m ) -0.200.2 FIG. 7. Calculation of the hyperfine interaction reductionas a result of the simulated device strain. (a)
Calculated hy-perfine shift ∆ A according to the valley repopulation model,which predicts a quadratic dependence on strain. (b) Calcu-lation performed using the second-order strain model of Eq. 4.The second-order model predicts shifts an order of magnitudelarger than the VRM does, as well as displays bipolar frequen-cies due to its strong linear dependence. with Ξ u ≈ . E the energy splitting between the A andE states and (cid:15) is a general strain tensor with principalcomponents (cid:15) αα (where α are the cubic axes x (cid:107) [100],y (cid:107) [010] and z (cid:107) [001]). This expression is valid in thelimit of small (cid:15) ( | (cid:15) αα | (cid:46) × − ) and is applicable for therange of strain produced in our device. In Fig.7a we plotthe hyperfine shift ∆ A ( (cid:15) ) close to the inductive wire, cal-culated using Eq. 3. The quadratic dependence of A ( (cid:15) )on strain implies that it is only reduced from A (0), theun-strained value. It is apparent that such a distributioncould not explain the spectra of Section III, which wouldrequire both positive and negative frequency componentsin order to split the resonance peak in the manner ob-served. In addition, the VRM predicts ∆ A ≈
100 kHzfor strains of order 10 − , equating to a resonance shift of∆ A × ( df /dA ) / ( df /dB ) = 20 µ T, an order of magnitudesmaller than our observed peak broadening.Very recently, it was found that the hyperfine interac-tion of donors in silicon is also sensitive to the hydrostaticcomponent of strain [48]. This result is surprising, asthe VRM predicts no hyperfine reduction for strains thatshift all of the valleys by the same energy. A second-orderstrain model for the hyperfine shift was proposed:∆ A ( (cid:15) ) A (0) = K (cid:15) xx + (cid:15) yy + (cid:15) zz ) + L (cid:15) xx − (cid:15) yy ) +( (cid:15) xx − (cid:15) zz ) + ( (cid:15) yy − (cid:15) zz ) ] + N ( (cid:15) + (cid:15) + (cid:15) ) (4)with K = 29, L = − N = −
225 the model co-efficients for
Bi calculated using tight-binding theoryand K = 17 . | (cid:15) | (cid:46) × − the model predicts that the linearhydrostatic strain dominates the hyperfine shift. It issuggested that this term is due primarily to strain effectson the central-cell potential, inducing a coupling betweenthe 1s A state and higher donor orbital states with thesame symmetry. Experiments confirmed the existence ofthe linear term and the extracted coefficient K = 19 . − , we expect ∆ A ≈ ∼ µ T. In addition,the sensitivity of the resonance frequency to the hyper-fine interaction is constant across all spin transitions (seeTable I), in agreement with the peak splittings extractedin Table II. This mechanism provides bipolar resonanceshifts of the correct magnitude and thus constitutes alikely explanation for the spectra of Fig. 3c. It should benoted that such a mechanism is not unique to bismuth,a linear hyperfine tuning with strain was observed for allof the group-V donors in silicon [48].
3. g-Factor
The final mechanism we consider is a strain-inducedshift of the electron g-factor g e . Strain modifies g e directly (by admixing higher-lying energy bands) andthrough the valley repopulation effect [47]. This al-ters the gyromagnetic ratio γ e = g e µ B /h (where µ B is the Bohr magneton), shifting the spin resonance fre-quency through the electron Zeeman interaction γ e B · S .The g-factor shift for donors in silicon has been pre-dicted and measured to be several orders of magnitudesmaller than that of the hyperfine interaction [47, 49].In addition, the electron Zeeman energy for the range offields applied in our study ( B < E z /h = γ e B <
300 MHz, thus providing a proportion-ally lower contribution to the transition frequency thanthe hyperfine interaction A = 1457 MHz. We quantifythis with the transition parameter data in Table I. For thesame relative change, the hyperfine interaction shifts theresonant frequency by a factor ( A × df /dA ) / ( g e × df /dg )greater than does the electron g -factor, which rangesfrom 50-100 for the spin transitions explored here. Fi-nally, comparing the g-factor sensitivity df /dg for thetransitions of resonator A to those of resonator B, we ex-pect the splittings and broadenings to be a factor ∼ A e ( a . u . ) B (mT) g ( H z ) B || wire B wire MeasurementSimulation SimulationMeasurement B (mT) E c ho a m p li t ude , A e ( a . u . ) bc a FIG. 8. (a)
Compensated echo-detected field sweeps of tran-sitions 1A-8A (resonator A). The bottom traces are measureddata from Fig. 4 (plotted again here for ease of comparisonwith theory), whilst the top traces have been offset inten-tionally and are the results of our theoretical modeling. (b)
Compensated echo-detected field sweeps (measurement andsimulation) of transitions 1B-4B (resonator B). (c)
The singlespin-resonator coupling strength g as a function of field B ,extracted from transition 4B (marked by the black dashedbox in panel b). The red open circles are derived from mea-surements of the Rabi frequency. Quantitative agreement isobserved with the simulated data (red solid line). V. ESR SPECTRA SIMULATIONS
In this section we assess whether the hydrostatic hy-perfine shift can reproduce the measurement data by per-forming a full simulation of the extended ESR spectra ofFig. 4. The upper offset traces of Figs. 8a and 8b arethe results of a numerical model incorporating the finite-element simulation of δB and the hyperfine shift calcu-lations (found by applying Eq. 4 to the strain simulationsof Fig. 6). For every pixel in the device where dopants arepresent, we use the pre-determined ∆ A ( (cid:15) ) (with the ex-perimental value of K ) and calculate the spin transitionparameters by solving the modified Hamiltonian:0 H/h = H /h + ∆ A ( (cid:15) ) S · I (5)At each B we calculate the spectral overlap of all al-lowed transitions (∆ m F = ± m F = 0) with theresonator and weight the resulting spectrum from eachpixel with the corresponding donor concentration andthe appropriate component of the magnetic field vac-uum fluctuations (Fig. 2), summing over all pixels toachieve the spectra in Figs. 8a and b. We note that thedonor doping profile used in this model (red solid curve inFig. 1b) is the output of a TCAD simulation (discussedin Section IV) that takes into account the ionization ofdonors in the depletion region of the Schottky junctionformed between the aluminium wire and the silicon sub-strate. The simulation strikingly reproduces many fea-tures in the experimental data, including peak splittings,peak-height asymmetries and field orientation φ depen-dence, without a single free parameter in the model.Having successfully reproduced key features of the ESRspectra, we investigate whether our model can also cap-ture the correlation of the magnetic field vacuum fluctu-ations δB and spin resonance frequency, as discussed inSection III. As noted previously, the Rabi frequency canbe expressed in terms of the single spin-resonator cou-pling strength g = γ e M | δB ⊥ | and the mean intra-cavityphoton number n through the relation Ω R = 2 g √ n . Inthe compensated sweeps Ω R is held constant as we passover the transitions. We extract g as a function of B fortransition 1B (identified by a black dashed box in Fig. 8b)by estimating n at each field using the experimental in-put power and a calibration of the loss in our setup [13].In Fig. 8c we plot the result of the experiment (red opencircles) overlaid on the simulated spectra (grey dashedline). The data quantifies the qualitative description of-fered earlier: the coupling strength (or equivalently thevacuum fluctuations δB ) increases for the spins that arefurther detuned (those close to the edge of the wire) andreduces towards the center of the transition, reaching thelowest couplings at the inner-edge of the high-field peak(the spins farthest from the wire). Next, we use ourmodel to simulate the expected g versus B dependence,the result (red solid line in Fig. 8c) is an almost quanti-tative match to the experimental data. VI. SUMMARY
We discussed a range of mechanisms capable of al-tering the resonance frequencies of donors in micro andnanoelectronic devices and found that strain resultingfrom differential thermal contraction plays a considerablepart. We presented a technique to study such strainsin silicon devices through high-sensitivity orientation-dependent ESR spectroscopy. Our results are quanti-tatively reproduced by considering the shift of the hy- perfine interaction caused by the hydrostatic componentof strain [48]. The resulting resonance frequency shiftsof ∼ ∼ − contributed to anorder-of-magnitude broadening of the ESR lines. Whilstthe measurements were performed on bismuth donors insilicon, similar effects are expected for the other group-Vdonors [48].The level of agreement demonstrated between ourmodel, which combined finite-element simulationsand experimentally-determined Hamiltonian parameters,with the measured data shows that it accurately cap-tures the underlying physics. Remarkably, the simulationquantitatively reproduces the experimental results withno free parameters in the model. This analysis couldbe adapted to other device geometries and spin systems,and may prove to be useful for spin-based device design.The results presented in this work have implications forQIP with donors and in hybrid systems such as supercon-ducting quantum memories, which require predictabilityof spin resonance frequencies and the ability to engineernarrow spin line-widths.The high sensitivity of the donor hyperfine interactionto hydrostatic strain could be used to create a sensitivelocal probe for strain in nanoelectronic devices. We esti-mate that with typical intrinsic line-widths achieved fordonors in isotopically enriched silicon of ∼ − . This could be integrated with other techniquesfor donor metrology [50] to provide valuable insight intothe spatial variation of physical system parameters innanoscale quantum devices. The large strain sensitiv-ity also opens the prospect of driving spin resonance viamechanical resonators, or coupling donors to phonons incircuit quantum electrodynamics experiments. ACKNOWLEDGMENTS
Acknowledgments
We thank B. Lovett and P. Morte-mousque for fruitful discussions. We acknowledge thesupport of the European Research Council under theEuropean Community’s Seventh Framework Programme(FP7/2007-2013) through grant agreements No. 615767(CIRQUSS), 279781 (ASCENT) and 630070 (quRAM),and of the Agence Nationale de la Recherche (ANR)through the project QIPSE. J.J.L.M. was supported bythe Royal Society. T.S. was supported by the US Depart-ment of Energy under contract DE-AC02-05CH11231.F.A.M. and A.M. were supported by the Australian Re-search Council Discovery Project DP150101863. We ac-knowledge support from the Australian National Fabri-cation Facility.
Appendix A: Spin Resonance Transitions
The hyperfine interaction A S · I couples states in the | m S , m I (cid:105) basis that differ in the electron and nuclear spin1projections such that ∆ m S = ± m I = ∓
1. Thiscan be seen by rewriting the interaction as a product ofthe spin raising and lowering operators: A S · I = A ( S X I X + S Y I Y + S Z I Z ) (A1)= A (cid:18) S Z I Z + 12 [ S + I − + S − I + ] (cid:19) In the coupled | F, m F (cid:105) basis, these states therefore sharethe same value of m F = m S + m I . In general, we canexpand the | F, m F (cid:105) basis on the | m S , m I (cid:105) basis as: | F ± , m F (cid:105) = a ± m F | ± , m F ∓ (cid:105) + b ± m F | ∓ , m F ± (cid:105) (A2)where we use F ± to represent the higher or lower mul-tiplet F ± = I ± S (i.e. F + = 5 and F − = 4 for Bior the triplet and singlet states for P). This is true forall states aside from those with m F = ± ( I + S ) (cor-responding to | m S = ± S, m I = ± I (cid:105) ), which remain un-mixed. The mixing coefficients a ± m F ( a +m F = a − m F ) and b ± m F ( b +m F = − b − m F ) are determined by the value of m F , thehyperfine interaction strength A and the external mag-netic field B (or more precisely, the electron Zeeman energy relative to the hyperfine interaction) [9]. At highmagnetic fields (where E z /h = γ e B (cid:29) A ) a ± m F → b ± m F →
0, whilst at low magnetic fields (where γ e B (cid:46) A )strong mixing occurs.
1. “ S X ” Type When operating in the “orthogonal mode” of spin res-onance ( B ⊥ B ), the B field couples to the S X and I X spin operators. Electron spin resonance transitionsmay be driven between | F, m F (cid:105) states that contain com-ponents of the uncoupled basis that differ by ∆ m S = ± | F ± , m F (cid:105) ↔ | F ± , m F − (cid:105) and | F ± , m F (cid:105) ↔ | F ∓ , m F − (cid:105) , as can be seen from Eq. A2. The first two transitions( | F + , m F (cid:105) ↔ | F + , m F − (cid:105) and | F − , m F (cid:105) ↔ | F − , m F − (cid:105) )correspond to high-field NMR transitions (which becomeESR-allowed at low fields), whilst the third transition( | F + , m F (cid:105) ↔ | F − , m F − (cid:105) ) is a high-field ESR transitionand the fourth ( | F − , m F (cid:105) ↔ | F + , m F − (cid:105) ) is completelyforbidden at high fields – it corresponds to transitionswhere ∆ m S = ± m I = ∓ (cid:104) F ± , m F | S X + δI X | F ± , m F − (cid:105) = (cid:2) a +m F b +m F − + δ ( a +m F a +m F − + b +m F b +m F − ) (cid:3) / , (A3a) (cid:2) b − m F a − m F − + δ ( a − m F a − m F − + b − m F b − m F − ) (cid:3) / (cid:104) F ± , m F | S X + δI X | F ∓ , m F − (cid:105) = (cid:2) a +m F a − m F − + δ ( a +m F b − m F − + b +m F a − m F − ) (cid:3) / , (A3c) (cid:2) b − m F b +m F − + δ ( b − m F a +m F − + a − m F b +m F − ) (cid:3) / δ = γ n /γ e is the ratio of the nuclear and electronspin gyromagnetic ratios, which is typically of order 10 − for group-V donors in silicon. At low and intermediatefields ( γ e B (cid:46) A ), the first term in the matrix elementsof Eqs. A3a-d dominate over the components generatedby the nuclear spin (those multiplied by δ ). At highmagnetic fields, the nuclear spin component of the matrixelement is negligible for Eq. A3c but is the dominantterm in Eqs. A3a,b (the high-field NMR transitions). Itshould be noted that, in general, the matrix elementsabove are non-zero at low-fields, with the exception ofidentical particles ( S = I and δ = 1) where the singletstate ( F = 0) is ESR inactive. The singlet state becomesESR active (for example, in the case of phosphorus S = I = 1 /
2) due to the differing gyromagnetic ratios of theelectron and nuclear spins.
2. “ S Z ” Type In the “parallel mode” of spin resonance ( B (cid:107) B ),the B field couples to the S Z and I Z spin operators. Electron spin resonance transitions may be driven be-tween | F, m F (cid:105) states that contain identical componentsof the uncoupled basis, i.e. | F ± , m F (cid:105) ↔ | F ∓ , m F (cid:105) (seeEq. A2). These correspond to high-field flip-flop transi-tions (∆ m S = ± m I = ∓ (cid:104) F + , m F | S Z + δI Z | F − , m F (cid:105) = ( a +m F b − m F − a − m F b +m F ) / δ [ a +m F b − m F ( m F − /
2) + a − m F b +m F ( m F + 1 / − a +m F b +m F + δa +m F b +m F (A4)where we have used the symmetry of the mixing coeffi-cients ( a +m F = a − m F and b +m F = − b − m F ) to arrive at the finalform of Eq. A4. Note, for identical gyromagnetic ratios( δ = 1) the components of the matrix element cancel ex-actly, and driving in the parallel mode is forbidden. Fur-thermore, at high fields (where a ± m F → b ± m F → δ ≈ − ) at low magnetic fields, the “ S Z ”transitions are appreciable, comparable in strength to the“ S X ” type.2 Appendix B: Predicted ESR Line Shape
Previous studies of the sample utilized in this work[24], performed using a bulk ESR spectrometer (i.e. with-out the on-chip resonator) revealed a Gaussian line-shapewith a peak-to-peak width of σ B = 12 µ T for the high-field m I = − / | , − (cid:105) ↔ | , (cid:105) in the | F, m F (cid:105) basis), at a frequency of ω/ π = 9 .
53 GHz.This transition displays a df /dB = 0 . γ e and thus anequivalent σ f = σ B × df /dB = 200 kHz broadeningin the frequency domain. This value agrees well withother studies of bismuth-doped isotopically enriched sili-con [10], where a line-width of 270 kHz was measured andfound to be constant in the frequency domain. For the | , − (cid:105) ↔ | , − (cid:105) transition studied in this work (with ω/ π ≈ . df /dB = 0 . γ e (see Table I) and weexpect a σ B = σ f / ( df /dB ) = 8 µ T, providing a full-width-at-half-maximum (FWHM) of ∼ µ T. This issubstantially lower than the broadening we observe inour measurements using the on-chip micro-resonator, asdepicted in Fig. 3.
Appendix C: Magnetic Field Inhomogeneity -0.6 -0.4 -0.2 0 0.2 0.4 0.6 B - B (mT) E c ho a m p li t ude , A e ( a . u . ) Res. A (7.305 GHz)Res. B(7.246 GHz)Res. C (7.144 GHz)
FIG. 9. Compensated echo-detected magnetic field sweep(see Section III for measurement details) recorded for threeresonators (frequencies are listed in the figure legend) over thefirst spin transition | , − (cid:105) ↔ | , − (cid:105) . The horizontal axisdisplays the difference with the transition center fields B (listed in Table II). The transition frequency has a magneticfield sensitivity of df/dB = − . γ e for all three resonators. A magnetic field inhomogeneity, for example producedby Meissner screening of the static magnetic field B inthe vicinity of the superconducting wire, is not suspectedto contribute to the splitting and broadening of the elec-tron spin resonance (ESR) peaks observed in our exper- iment (Figs. 3 and 4). We rule this mechanism out bycomparing measurements of the first spin resonance tran-sition | , − (cid:105) ↔ | , − (cid:105) (see Table I) for each of the threeresonators A, B and C (see Fig. 9). The width and split-ting of these peaks are of similar size for each resonator,despite the transition for resonator C ( B = 9 .
29 mT, df /dB = − . γ e ) occurring at twice the field of res-onator B and three times the field of resonator A. Abroadening resulting from an inhomogeneous magneticfield would increase in proportion to the strength of thefield. Appendix D: Strain Tensor Simulation
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