On the modeling of amplitude-sensitive ESR detection using VCO-based ESR-on-a-chip detectors

In this paper, we present an in-depth analysis of a voltage-controlled oscillator (VCO) based sensing method for electron spin resonance (ESR) spectroscopy, which greatly simplifies the experimental setup compared to conventional detection schemes. In contrast to our previous oscillator-based ESR detectors, where the ESR signal was encoded in the oscillation frequency, in the amplitude-sensitive method, the ESR signal is sensed as a change of the oscillation amplitude of the VCO. Therefore, using a VCO architecture with a built-in amplitude demodulation scheme, the experimental setup reduces to a sin5 gle permanent magnet in combination with a few inexpensive electronic components. We present a theoretical analysis of the achievable limit of detection, which uses a perturbation theory based VCO-modeling for the signal and applies a stochastic averaging approach to obtain a closed-form expression for the noise floor. Additionally, the paper also introduces a numerical model suitable for simulating oscillator-based ESR experiments in a conventional circuit simulator environment. This model can, e.g., be used to optimize sensor performance early on in the design phase. Finally, all presented models are verified against 10 measured results from a prototype VCO operating at 14 GHz inside a 0.5 T magnetic field.


Introduction
Electron spin resonance (ESR) is a very powerful spectroscopic method which is used extensively in a large variety of disciplines including chemistry, material science and the life sciences (Twahir et al., 2015(Twahir et al., , 2016Kopani et al., 2006;Azarkh et al., 2013;Qi et al., 2014;Qin and Warncke, 2015;Fehr et al., 2011Fehr et al., , 2012 (Twahir et al., 2015(Twahir et al., , 2016Azarkh et al., 2013;Qi et al., 2014;Qin and Warncke, 2015; 15 Fehr et al., 2011Fehr et al., , 2012. At its basis, ESR spectroscopy uses the spin of an electron as a very sensitive nanoscopic probe of its magnetic and electronic environment inside a molecule or a solid to provide important information, which are often difficult to obtain by other spectroscopy techniques. However, due to the small energy gap between the electron's spin energy states at room temperature and therefore almost equal Boltzmann population, ESR suffers from an intrinsically low sensitivity compared to other spectroscopy methodsSince ESR detects exclusively paramagnetic species, it is ideally suited for the detection of free radicals, which are related to premature cell aging (Kopani 20 et al., 2006), food degradation (Elias et al., 2009;Ottaviani et al., 2001) or for the detection of paramagnetic defects in semiconductor materials (Fehr et al., 2011). To overcome this problem the problem of limited sensitivity in conventional ESR, miniaturized detectors have been suggested, which improve the achievable spin sensitivity thanks to their larger B u -field and, in this way, room temperature spin sensitivities between 10 7 spins/(G · √ Hz) and 10 9 spins/(G · √ Hz) at various B 0 -field strengths have been reported in the literature, cf. Anders et al. (2012a); Twig et al. (2013); Gualco et al. (2014); Matheoud et al. (2017Matheoud et al. ( , 2018Anders et al. 25 (2012a); Twig et al. (2013); Gualco et al. (2014); Matheoud et al. (2017Matheoud et al. ( , 2018; Dayan et al. (2018); Abhyankar et al. (2020); Zhang and Niknejad (2021). Apart from the poor sensitivity associated with inductive ESR detectors, conventional ESR setups also suffer from a relatively large complexity. As a partial solution to this problem, an oscillator-based ESR detection method was presented by Anders et al. (2012a) and Yalcin and Boero (2008) which detects the ESR effect by monitoring the sample-induced inductance variation as a change in the oscillation frequency. By using integrated LC tank oscillators, this 30 approach removes the need for expensive external B 1 -field sources and also benefits from the great scaling potential of modern nanometer-scaled CMOS technologies and their very high maximum operating frequencies. Exploiting these advantages and utilizing a 45 µm detection coil inside an LC tank oscillator operating around 146 GHz, the design presented by Matheoud et al. (2017) achieves a spin sensitivity of about 2 × 10 7 spins/(G · √ Hz). The oscillator-based detection concept was then extended to the use of voltage-controlled oscillators (VCOs) by Handwerker et al. (2016), which allows for a great simplification of the 35 experimental setup, thereby, for the first time, enabling the design of battery-operated, portable ESR spectrometers. Since ESR spectroscopy presents the gold standard for the detection of free radicals (Kopani et al., 2006), which play a crucial role in many diseases (Lin and Beal, 2006;Piskounova et al., 2015), such Such portable ESR spectrometers can have a tremendous impact on emerging potentially have a very large impact on (emerging) disciplines such as personalized medicinethe analysis of irradiated food (Chauhan et al., 2009), the study of wine oxidation (Elias et al., 2009), the prevention of the formation of free radicals in vegetable oils (Ottaviani et al., 2001), on-site radia-40 tion dosimetry (Romanyukha et al., 2014), point-of-care transcutaneous oxygen monitoring (Wolfson et al., 2014;Cristea et al., 2021) or measurements of skin antioxidant capability (Haag et al., 2011).
While in the reports by Anders et al. (2012a), Matheoud et al. (2017) and Yalcin and Boero (2008) only the frequencysensitive detection option of an LC tank oscillator was discussed, a second mode of detection is available in oscillator-based ESR detectors because the oscillation amplitude is also affected by the ESR signal. This concept was first originally published 45 by Matheoud et al. (2018) using an Chu et al. (2018) using CMOS LC-tank VCO and by Matheoud et al. (2018) using a high electron mobility transistor based LC Colpitts oscillator. In both of those reports, amplitude-sensitive detection is mentioned beside frequency-sensitive detection and sensitivity calculations are performed only for the latter. In this paper, we will extend the state-of-the-art by providing both analytical and numerical models for the amplitude-sensitive detection mode. Using our analysis, we will show that a CMOS LC tank oscillator can also be used for amplitude-sensitive ESR detection with show that the amplitude and frequency-50 sensitive detection modes display the same theoretically achievable sensitivity as the frequency-sensitive approach spin sensitivity but with the potential for additional simplifications in the experimental setup a simplified experimental setup for the amplitude-sensitive detection mode. These simplifications canin turn , in turn, be used for further reductions in the size and cost of future generations of portable ESR spectrometers.
The paper is organized as follows. In section 2, we will explain the experimental setup of an amplitude-sensitive VCO-55 based ESR experiment. In section 3 and 4 we then derive analytical expressions for the ESR-induced amplitude variations in  an LC tank oscillator before we also provide analytical expressions for the amplitude noise of LC tank VCOs in section 5 to estimate the achievable limit of detection (LOD) in section 6. Next, in section 7 we provide a model suitable for simulating ESR spectroscopy experiments in conventional circuit simulators. Then, in sections 8 and 9, we compare the analytical model against these circuit simulations and validate all models using measured results from a VCO prototype operating around 14 GHz in a 60 0.5 T magnetic field. The paper is concluded with a short discussion and an outlook on future work in section 10.
2 Performing amplitude-sensitive ESR experiments using LC tank VCOs A conventional setup for ESR experiments is shown in Fig. 1a. The ESR sample is placed inside a microwave resonator which is situated inside a variable field magnet. An ESR experiment is performed by irradiating the sample with a microwave at a constant frequency through a circulator and monitoring the reflected powerwhile sweeping the static . The external magnetic field 65 B 0 in and out of resonance is swept through the resonance condition. In order to improve the achievable sensitivity, frequently lock-in detection is introduced by modulating the static magnetic field with an AC magnetic field with amplitude B m using a pair of modulation coils. The building blocks highlighted in red in Fig. 1a are those that prevent a miniaturization of the experimental setup (electromagnet), an energy efficient operation (electromagnet and modulation coils) and an integration into CMOS technology of the spectrometer electronics (circulator). In contrast, in the amplitude-sensitive detection scheme 70 incorporating VCOs shown in Fig. 1b, all required electronic components can be easily integrated into CMOS technology and the power-hungry and bulky electromagnet is replaced by a permanent magnet. The replacement of the variable field by a permanent magnet is possible because in the proposed setup, an ESR spectrum is recorded at a fixed static magnetic field B 0 while sweeping the frequency of the excitation signal (i.e. the magnetic field produced by the coil of the integrated LC tank oscillator) in and out of resonance to induce the ESR transition. Using a VCO, this frequency sweep can conveniently 75 be carried out by applying a voltage ramp to the VCO control voltage using a digital-to-analog converter (DAC). The VCO control voltage both defines the new excitation frequency, and, at the same time, tunes the LC tank inside the VCO to this frequency. This is because, in a VCO, the oscillation frequency and the resonance frequency of the LC tank are identical at all times. This is in contrast to a conventional resonator-based scheme, in which the resonance frequency and the excitation frequency can be independently defined. Moreover, the same DAC output signal can be used to produce 80 a frequency modulation at every sweep point, which allows to replace the field modulation using external modulation coils by a much more power saving frequency modulation with the same positive effect on the achievable SNR when using subsequent lock-in detection. At this point, it is important to note that the aforementioned simplifications of the experimental setup were already achieved using the frequency-sensitive VCO-based detection setup presented by Handwerker et al. (2016). However, the amplitude-sensitive ESR setup of Fig. 1b provides the additional advantage of an implicit demodulation of the ESR signal.

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More specifically, when using current biasing for the LC tank oscillator according to Fig. 2a, the voltage at the center tap of the differential tank inductor (node X in the figure) contains a demodulated version of the oscillation amplitude (Kinget, 2006). This implicit AM demodulation feature of the VCO not only removes the necessity of an external AM demodulation block but also minimizes the number of high frequency components because the lock-in amplifier can directly be connected to the inductor center tap voltage, cf. In order to be able to derive an analytical expression for the ESR induced amplitude changes in the oscillation amplitude of an LC tank VCO according to Fig. 2a, we will first derive closed-form expressions for the oscillation amplitude and frequency.
As the starting point for our analysis we will use the equivalent electrical model of the schematic of Fig. 2a shown in Fig. 2b, is the equivalent tank conductance, Q coil being the coil quality factor, and v n,R and i n,T are noise sources modeling the noise generated in the coil resistance and the cross-coupled transistor pair, respectively. For the following 100 deterministic analysis, the noise source will be set to zero and they will only be considered for the noisy case discussed in section 5. To obtain an I/V-characteristic of the static nonlinearity of Fig. 2b, which models the cross-coupled transistor pair, we have followed the approach proposed by Anders et al. (2012c), resulting in: where G m0 = βI BIAS /n is the gate transconductance (Enz and Vittoz, 2006) of a single transistor in the cross-coupled 105 differential pair for v d = 0, n ≈ 1.3 is the slope-factor transistor slope factor (Enz and Vittoz, 2006) and I BIAS is the oscillator bias current. Then, using the differential tank voltage as state variable x = v d and applying Kirchhoff's current law to node 1 , we obtain the following ordinary differential equation describing the oscillator behavior: where ε = 3 G 3 m0 / 16 n 2 , G m0 being the gate transconductance and n ≈ 1.3 the slope factor, G t is the equivalent tank con-110 ductance, I BIAS is the bias current and ω LC = 1/ √ L coil C is the resonance frequency of the LC tank. Starting from eq. (2) we can use the so-called Lindtstedt method Jordan and Smith (2007) (Jordan and Smith, 2007) to obtain first order estimates of the oscillation amplitude and frequency according to: is the overdrive parameter, which needs to be chosen greater than one to ensure a stable oscillation and all other parameters are defined as above.

ESR-induced amplitude shifts
As explained by Yalcin and Boero (2008), the effect of ESR on the spin ensemble can be modeled by means of a complex susceptibility χ = χ − j χ according to: where ∆ω = ω osc −ω L , ω osc being the oscillation frequency and ω L = −γ B 0 being the electron Larmor frequency (Schweiger and Jeschke, 2001), where γ and B 0 are the gyromagnetic ratio of electrons 1 and the applied static magnetic field strength, T 1 and T 2 are the longitudinal and transverse relaxation times, respectively, and χ 0 is the static electron susceptibility. Using the 125 complex susceptibility, the effective tank coil impedance in the presence of a resonant electron spin ensemble can be written where η is the so-called filling factor (Yalcin and Boero, 2008), which indicates how much of the sensitive coil volume is effectively filled by the ESR active material. Therefore, the effective coil inductance and coil resistance in the presence of ESR, L χ and R χ , can be written according to respectively, where Q coil is the coil quality factor. In order to obtain the oscillation voltage and frequency including the effect 130 of ESR, we can start from eq. (3) and replace the original coil inductance and resistance (i.e. in the absence of ESR), L coil and R coil , by their effective values in the presence of ESR, i.e. R χ and L χ , respectively. Since in this paper, we are only interested in the ESR-induced amplitude changes, in the following, we will only consider the effect of ESR on the oscillation amplitude described by eq. (3a), yielding:

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where G t,χ is the equivalent tank conductance in the presence of ESR. Since the The equivalent tank conductance depends on both the coil resistance R coil and -via the coil quality factor Q coilalso on the coil inductance L coil , according to G t,χ = C R coil,χ /L coil,χ , eq. (5) can be rewritten according to: Eq. (6) can be further simplified by noting that the ESR-induced inductance changes are much smaller than the original coil 140 inductance, that is η χ 1. Consequently, the term 1/(1+η χ ) can be developed into a Taylor series in η around η = 0, which can be stopped after the linear term and eq. (6) simplifies to: where we have neglected the quadratic term in η 2 , which originates from the product (1 + Q coil η χ ) · (1 − η χ ). To arrive at a closed-form expression for the ESR-induced amplitude changes, we can further develop the right hand side of eq. (7) into a first order Taylors series in η around η = 0. Then, the ESR-induced amplitude change ∆A osc,χ A osc,χ − A osc,0 can be written as: According to eq. (8), the ESR-induced amplitude change depends on both the real part of the complex susceptibility, χ , and its imaginary part, χ . However, for moderate coil quality factors with Q coil 1, the term Q coil χ largely dominates, and the ESR induced amplitude changes mostly depend on the imaginary part of the complex susceptibility according to: 5 Model of amplitude noise in LC tank VCOs Due to the great importance of timing uncertainties on the overall system of modern communication systems, oscillator phase noise is probably one of the most-discussed topics in RF circuit theory and a wide variety of models with different degrees of complexity exist in the literature ranging from simple linear time-invariant over linear time-varying to more complicated non-155 linear models (Kaertner, 1990;Hajimiri and Lee, 1998;Demir, 2002;Nallatamby et al., 2003;Magierowski and Zukotynski, 2004;Andreani et al., 2005;Sancho et al., 2007;Jahanbakht and Farzaneh, 2010;Murphy et al., 2010). One fundamental problem associated with oscillator noise modeling is related to the fact that an oscillator is a nonlinear system far away from thermal equilibrium. This leads to a situation where even the most sophisticated models available today, which rely on modeling using stochastic differential equations (SDEs), can be considered as heuristics only. This is because the Langevin approach of intro-160 ducing additional additive noise sources into the system in general fails for nonlinear dynamical systems, leading to physical inconsistencies (Thiessen and Mathis, 2010). Here, the problem essentially arises from the coupling between the different moments of the stochastic process described by the SDE, which results in a situation where the stochastically averaged SDE is in general not identical to the deterministic system to which the noise sources have been added. Therefore, due to the heuristic nature of even the most advanced models proposed in the literature, a validation against simulations and -even more impor-165 tantly -against measured data is crucial. While for oscillator phase noise such experimentally verified heuristic models exist, the field of oscillator amplitude noise is by far less explored and there is only a very small set of papers which deal with this topic typically as a side note without experimental verification (Magierowski and Zukotynski, 2004;Jahanbakht and Farzaneh, 2010). This is mostly because the oscillator amplitude noise is of negligible importance for the resulting timing uncertainty and is therefore neglected in analysis papers focusing on oscillator applications in RF systems. However, in sensor systems, 170 which use the oscillator to measure a physical quantity as an amplitude change of the oscillator output voltage, the oscillator amplitude noise determines the achievable limit of detection and its accurate modeling is of utmost importance. This includes the amplitude-sensitive ESR detection mode discussed in this paper but also eddy-current crack detection sensors for nondestructive testing (NDT) (García-Martín et al., 2011). Due to the lack of existing models on oscillator amplitude noise in the literature, in this section, we will present a model based on the stochastic averaging method proposed by Stratonovich (1963), which takes into account the nonlinearity of the oscillator but still produces closed-form expression for the autocorrelation and power spectral density of the resulting amplitude noise process.
We have already applied the stochastic averaging method to an LC tank oscillator to obtain analytical expressions for the phase and frequency noise of such circuits and verified its accuracy using measured data (Anders et al., 2012b). Following the method outlined by Anders et al. (2012b) and Anders (2011), one can derive the following SDE governing the behavior of the 180 amplitude noise, δA, of the current starved LC tank oscillator of Fig. 2: where v n,R and i n,T are the noise sources modeling the noise introduced by the coil resistance and the active cross-coupled transistor pair in Fig. 2. Then, introducing the stochastic process ξ n (t) = ω LC v n (t) − 1/C i n (t) into eq. (10), one obtains:

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Eq. (11) defines a time-dependent Ornstein-Uhlenbeck process Gardiner (2009) (Gardiner, 2009) and its solution, assuming a vanishing initial condition at t → −∞, is therefore given by: Assuming that v n,R and i n,T are Gaussian random processes with zero mean, δA will also be Gaussian with vanishing mean.
Consequently, the autocorrelation of δA, R δAδA (t, τ ), is sufficient to completely characterize the statistics of the amplitude 190 noise. This autocorrelation is given by: where it was further assumed that the initial phase ϕ 0 is a random variable uniformly distributed in the interval [0, 2 π]. The double integral of eq. (13) can be solved in closed form if one assumes that ξ n is white, i.e. R ξξ (t 1 , t 1 +τ ) = R ξξ (τ ) = α 2 n δ(τ ), using a variable transformation, cf. Stratonovich (1963), according to σ = t − t + τ and s = (t + t )/2, yielding: where the noise scaling coefficient α 2 n of the process ξ n was calculated in Anders et al. (2012b) as α 2 n = k T R coil ω 2 LC (1 + α od γnD), by Anders et al. (2012b) as where k is Boltzmann's constant, T is absolute temperature, and γ nD is the thermal noise excess factor (Enz and Vittoz, 2006) 200 of a MOSFET transistor with γ nD = 2/3 n ≈ 1 for a transistor in strong inversion and saturation. The corresponding power spectral density, which is centered around ω osc , is then given by: where ∆ω = ω − ω osc and ω c = λ is the resulting corner frequency.
6 Limit of detection 205 In this section, the results from the previous two sections will be combined to obtain the limit of detection (LOD), i.e. the minimum number of spins detectable with an SNR of three in one second of measuring time, of an amplitude-sensitive VCObased ESR detector. In order to make the results comparable with previously published resonator-based and frequency-sensitive oscillator-based ESR experiments, we will introduce the standard ESR terminology into the LOD expression. To this end, one can recast the result of eq. (9) by noting that the oscillation amplitude and the B 1 -field, i.e. the magnetic field produced by the 210 oscillation current in the tank inductor, are related according to A osc,0 ≈ ω osc L coilÎcoil = ω osc (2 B 1 B u V det )/µ 0 , where B u is the unitary magnetic field of the detection coil, V det is the sensitive detector volume and µ 0 is the vacuum permeability 2 . Then, substituting A osc,0 in eq. (9) by the above expression, we find: where we have used the notation χ (∆ω) to emphasize the fact that the imaginary part of the complex susceptibility is a 215 function of the frequency offset ∆ω = ω osc −ω L between the oscillation frequency ω osc and the Larmor frequency ω L = −γ B 0 of the electron spins at the static magnetic field strength B 0 . Using the analytical expression for the oscillator amplitude noise of eq. (16) evaluated at ∆ω = 0 and assuming a detection bandwidth of f BW , we can write the SNR of amplitude-sensitive ESR experiments as a function of both ω osc and B 1 according to:

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To find the maximum SNR, we can substitute the imaginary part of the complex susceptibility by the term including saturation in eq. (4b), then take the partial derivatives of eq. (18) and simultaneously with respect to B 1 and ω osc , equate them to zero, resulting in and find the following optimum B 1 field strength and oscillation frequency ω osc , respectively: 225 2 The coil inductance L coil can be computed from the unitary field according to L coil = 1/µ 0 · |Bu| 2 dV ≈ 1/µ 0 · B 2 u · V det . Moreover, according to standard ESR terminology, the B 1 -field is the circularly polarized field produced by the coil current in resonance with the spin ensemble, i.e. B 1 ≈ µ 0 /d coil ·Î coil /2 = Bu ·Î coil /2.
Substituting these values for B 1 and ω osc into eq. (18) we find the following expression for the maximally achievable SNR: Since the longitudinal relaxation time T 1 is always greater than or equal to half the transversal relaxation time T 2 , i.e. 2 T 1 ≥ T 2 , the SNR of eq. (21) is maximized for 2 T 1 = T 2 . Then defining the limit of detection, which is also called Using the optimum achievable SNR in eq. (21), we can define the spin sensitivity N min , according to: where N spins is the number of spins in the sample and SNRopt is that produces the optimum SNRachievable with said number of spins , SNR opt , for a filling factor of η = 1 with a detection bandwidth of f BW = 1 Hzand noting . Noting that the static electron susceptibility χ 0 can be expressed as χ 0 = µ 0 N γ 2 2 /(4kT ) (Yalcin and Boero, 2008), where N is the spin density of the sample, is the reduced Planck constant, µ 0 is the vacuum permeability, γ is the gyromagnetic ratio, k is Boltzmann's constant 235 and T is absolute temperature, the theoretical spin sensitivity of the proposed amplitude-sensitive ESR detection method can be expressed as: Since the spin sensitivity given by eq. (23) is identical to the one given by Yalcin and Boero (2008) (up to the factor √ 1 + α od γnD except for the factor 2(1 + α od γ nD ) which accounts for the different condition of T 1 = T 2 used by Yalcin and Boero 240 (2008), and the noise originating in the cross-coupled transistor pair , which was not considered by Yalcin and Boero (2008)there), the theoretically achievable spin sensitivity of an LC tank oscillator is identical for the amplitude and the frequency-sensitive detection modes and also identical to that of a conventional resonator-based ESR detector.

Simulating ESR experiments using circuit simulators
To design CMOS VCO-based ESR detectors with optimum performance, it is important to be able to accurately simulate 245 the achievable sensitivity including all transistor nonidealities. To this end, in this section, we will provide a model which is suitable for simulating the effect of ESR on the frequency and the amplitude of CMOS LC tank VCOs in conventional circuit simulators. The utilized model was first proposed by Boero (2000) in the context of conventional resonator based nuclear magnetic resonance (NMR) experiments but can also be applied to continuous-wave VCO-based ESR experiments in the nonsaturated case, i.e. for (γ B 1 ) 2 T 1 T 2 1, where the expressions for the complex magnetic susceptibility of eq. (4) simplify 250 to:  Then, by comparing the impedance of a coil filled with spins, Z χ = L coil (1 + ηχ), whose susceptibility χ behaves according to eq. (24) with that of the equivalent tank impedance Z eq of Fig. 3, one finds that the analytical and the circuit simulator model 255 are equivalent if the following relations hold: where it should be noted that according to the conventions used in this paper, γ is a negative number. According to eq. (25), 260 there are four parameters (L spin , C spin , R spin and K spin , with K spin being the coupling coefficient between the tank inductor of the VCO and the LC resonator modeling the spins, cf. Fig. 3), which model the spin ensemble in the circuit of Fig. 3 but only three parameters in the physical model without saturation (ω L = −γB 0 , T 2 and ηχ 0 ), and therefore one parameter can be chosen at will. Here, one natural choice could be to choose L spin = L coil , which always results in reasonable values for both L spin and C spin .  Inset: an enlarged view of the amplitude dip in both models.
Balance) harmonic balance solver was used, defining the static magnetic field B 0 as a sweep variable to compute the fieldsweep ESR spectrum numerically. To ensure a smooth transition between adjacent sweep points the corresponding flag was 270 enabled in the CR analysis. The result of one such simulation is shown in Fig. 4. The corresponding simulation parameters are listed in the figure caption. These parameters correspond to the prototype realization discussed in section 9. According to the figure, there is an excellent agreement between the analytical model and the circuit simulation. As highlighted by the arrows in the figure, there is a small asymmetry in the line shape, which reflects the fact that the amplitude ESR signal is both sensitive to the real part of the complex susceptibility, which displays a dispersive behavior, and the imaginary part of said the susceptibility 275 with its absorption characteristic. However, since the imaginary part is amplified by the (unloaded) coil quality factor (see eq. (8), Q coil = 10.2 for the simulation and the prototype of section 9)), the simulated ESR spectrum is mostly absorptive in nature with the small, but visible asymmetry introduced by the real part of χ. Importantly, both the analytical model and the simulation accurately predict this behavior. The peak-to-peak amplitude of the spectrum is virtually unaffected by the real part of χ, justifying the simplified expression of eq. (9), which was used to derive the LOD in section 6.

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In order to validate the analytical noise model of eq. (16), we have compared it against noise simulations performed using Keysight's GoldenGate simulator (CR analysis with noise enabled). Using the same simulation settings as for the simulations of Fig. 4 except for the bias current, which was varied as a parameter to analyze the range over which the proposed model

Measurements
In this section, we will compare the circuit simulator model of section 7 against measured data from a prototype realization of the proposed amplitude-sensitive VCO-based ESR sensor. To this end, we have used the ASIC, which was already presented  there is, in principle, a one-to-one correspondence between field and frequency sweeps and we use V TUNE to introduce a 300 frequency modulation via a DAC, i.e. additional noise via the tuning port is considered in the experiments, these field sweeps display the same SNRs but make the comparison with the model much simpler. This being said, we also validated the proposed simplified setup of Fig. 1b experimentally. we have taken into account the demodulation sensitivity from the oscillator output voltage to node X in Fig. 2a (simulated to be 1/67.9 V/V) and the simulated lock-in detection spectra were computed from the direct detection spectra, cf. Fig. 4, in MATLAB using the same modulation amplitude (frequency modulation with an amplitude of 51.5 mV rms , which together with the VCO slope of 0.8 GHz/V corresponds to an equivalent peak-to-peak field modulation of 270120 µT) that was used in the measurement. According to the figure, when taking into account the modeling uncertainty due to difficulties in precisely de-315 termining the sample volume, there is an excellent agreement between the proposed circuit simulator model and the measured data.
NextTo estimate the spin sensitivity of our system, we have also used the measured data of presented in Fig.7to determine the spin sensitivity of the utilized detector. Here, using eq. and the spin density of DPPH (N = 2 × 10 27 , (Yalcin and Boero, 2008)), . With the calculation detailed in Appendix A, we have estimated a spin sensitivity of approximately Nmin = 2 × 10 10 N min = 8.9 × 10 10 spins/(G · √ Hz), which 320 is about two orders of magnitude 445 times worse than the theoretically predicted value of N min = 2 × 10 8 spins/(G · √ Hz). To investigate this large discrepancy This is in part due to the B 1 used in the measurements B 1,meas ≈ 16.8 µT being approximately 8 times less than B 1,opt ≈ 140 µT to avoid any line broadening, leading to a reduction of approximately 2.5x in N min , according to eq. (18). Here B 1,opt is calculated from eq. (19) using T 1 = T 2 ≈ 41 ns as extracted from the measured peak-to-peak linewidth of 1.4 G (cf. Fig. 7, equivalent to an FWHM of about 3 G), while I BIAS and α od used in the measurement and 325 in the optimum SNR condition are extracted from simulations, to be respectively 1.7mA, 1.25 and 27.2mA, 5. This large α od,opt of 5 leads to a higher amplitude noise in the optimum SNR condition (cf. eq. (16)), explaining the N min reduction of 2.5x mentioned above. To investigate the remaining discrepancy of 445/2.5 ≈ 180, we have also simulated the amplitude noise at node X in Fig. 2a, which is depicted together with the simulated AM noise in the oscillator output voltage and the analytical noise model in Fig. 8. According to the figure, although the demodulation sensitivity between the oscillator 330 output voltage and node X is less than unity, the noise floor is significantly larger leading to a greatly reduced SNR in the demodulated output. More quantitatively, the demodulation sensitivity of 1/67.9 V/V together with the 2730 dB (a 7increased noise floor (an 8 dB increase due to the 1/f-noise, which is not accounted for in eq. (23), and another 2022 dB increase in the demodulated output at node X ) increased noise floor accurately explain the factor of 100 predict an factor of 250 difference between the theoretically predicted value of N min and the measured value. This agrees well with the discrepancy of 180 mentioned 335 above. Moreover, the corner frequency between the 1/f-and the white noise parts of the spectrum occurs at significantly larger frequencies, effectively preventing an operating in the white noise region because at such high modulation rates rapid scan effects that perturb the spectra (Tseitlin et al., 2011) would already become visible. To verify the accuracy of the GoldenGate noise simulations, we have used the simulated noise floor at node X of -129.5 dBc/Hz and the simulated oscillator amplitude of approximately 820 mV to predict the rms noise in the measured data of Fig. 7. Taking into account the lock-in gain of 100 340 and the lock-in bandwidth of 102.5 Hz, the simulated noise floor predicts an rms noise of 0.13.5 µV rms , which corresponds precisely approximately to the measured rms noise of 2.3 µV rms 4 .
Finally, we have also performed frequency scan ESR experiments where instead of sweeping the static magnetic field B 0 , the tuning voltage V TUNE is ramped in and out of resonance. An example spectrum of a DPPH sample with a volume of approximately 3 pl obtained using this method is shown in Fig. 9. In these frequency sweep experiments, we have achieved 345 the same sensitivity as in the field sweep experiments. Similarly to previous experiments, the VCO gain was 0.8 GHz/V and a noise floor of ≈ 0.1 mV rms can be observed, in accordance with our models. Performing frequency sweeps allows for the use of the simplified experimental setup of Fig. 1b, which is ideally suited for future point-of-care ESR spectrometers, without performance degradation. This being said, care has to be taken that no additional noise is introduced into the system via the tuning voltage input V TUNE . At this point, it is convenient that the amplitude-sensitive detection setup is much more immune 350 against such additional noise from V TUNE than the frequency-sensitive setup, where the well-known AM-to-PM conversion in the varactor makes the setup much more prone to an increase in the noise floor due to DAC noise in V TUNE .  Figure 9. Measured spectrum of a DPPH sample of a volume of approximately 3 pl. The spectrum was acquired using the setup of Fig. 1b by sweeping the tuning voltage of the VCO through the resonance frequency and simultaneously applying a small sinusoidal signal for a subsequent lock-in detection.

Conclusions
In this paper, we have introduced a new ESR detection method, which senses the ESR effect as changes of the amplitude of an integrated VCO. Together with the proposed setup, we have presented analytical models for both the ESR-induced 355 amplitude changes and the AM noise floor of the VCO-based detector. The analytical models where were then used to predict the limit of detection of the proposed method, which was shown to be identical to that of the previously presented frequencysensitive VCO-detection approach presented by Handwerker et al. (2016). The analytical models were then verified against circuit simulations including an RLC tank model for the ESR effect. Finally, we have validated the circuit simulator model against measured data obtained from a VCO prototype operating around 14 GHz. When taking into account the increased 360 noise floor at the intrinsic AM demodulation point inside the VCO, we have achieved a very good agreement between model and measured data, clearly showing that the circuit model can be used to optimize the detector performance already early on in the design phase, thereby removing the need for costly and time-consuming hardware iterations. Moreover, the presented results serve as proof of principle that with the proposed approach good sensitivities can be achieved already at moderate ESR frequencies. Since the proposed method scales very advantageously with frequency, it can fully benefit from the current ESR 365 trend of going to higher and higher fields (and therefore also operating frequencies) to further improve sensitivity. With its very simple experimental setup, cf. Fig. 1b, and the availability of permanent magnets with field strengths up to approximately 2 T, the proposed approach is ideally suited for the design of future generation of portable , point-of-care ESR spectrometers, which can play a crucial role in emerging fields such as on-site food quality control, manufacturing process control or potentially personalized medicine and home diagnostics. experiments, derived the analytical models, designed the measurement setup and conducted experiments. All authors contributed to the manuscript.
Competing interests. No competing interests declared.