Nonlinear Modeling of Continuous-Wave Spin Detection Using Oscillator-Based ESR-on-a-Chip Sensors

Abstract

In this chapter, an advanced nonlinear energy-based modeling of LC tank oscillators used as sensors for ensembles of electron or nuclear spins is presented. Recently, this oscillator-based sensing principle has been gaining significant attention in the electron spin resonance community for biomedical and material science applications. Since the sensing principle relies on the coupling between a harmonic oscillator (the spin ensemble) and an intrinsically nonlinear electrical oscillator, it presents an excellent example of the high relevance of nonlinear dynamical systems modeling for practical sensing applications. In order to provide a self-contained overview, after a short general motivation that highlights the relevance of the topic, the chapter begins with a description of the experimental setup of the oscillator-based spin-detection approach, which is somewhat different from that for conventional resonator-based detection. In this description, it is shown how continuous-wave spin-detection experiments can be carried out using LC tank oscillators by monitoring the oscillation frequency when sweeping the static magnetic field \(B_0\). At this point, it is also explained how standard field modulation using a modulation field \(B_\mathrm {m}\) parallel to \(B_0\) can be used to increase the signal-to-noise ratio by means of phase-sensitive detection using a conventional lock-in amplifier. Then the interaction between the nonlinear electrical oscillator and the spin ensemble is modeled using the solution of the Bloch equation in the steady state, which models the dynamics of the spin ensemble, and the magnetic energy associated with the inductor of the LC tank oscillator. In this way, under steady-state conditions as they occur in continuous-wave ESR and NMR experiments, expressions for spin-related changes in inductance and resistance can be derived, which are in turn related to changes in the oscillation frequency and amplitude of the LC tank oscillator. To quantify the resulting change in oscillation frequency and also derive expressions for the expected noise floor, which eventually determines the achievable limit of detection, the chapter then provides a detailed discussion of the nonlinear modeling of LC tank oscillators in the presence of noise. The resulting model of the LC tank oscillator is subsequently used to find analytical expressions for the limit of detection of frequency-sensitive oscillator-based spin detectors. Finally, experimental results from a prototype realization of an oscillator-based CMOS ESR-on-a-chip detector are used to validate the accuracy of the derived signal and noise models.