Phase Transition and an Instanton Model of Geiger-Mode Pulses in Single-Photon Avalanche Diodes

A theoretical study categorizing the modes or the "phase" of the carrier dynamics in a single-photon avalanche diode is presented. For the basic model described by a nonlinear Lienard equation, a Lagrangian, a Hamiltonian, and a Lyapunov function are identified. From the energetics studies, variation in the phases is found to be described by the profile (symmetry) change in the available energy induced by change in the quenching resistance value. Each phase is found to correspond to an operational mode of the device; a diode biased at the breakdown voltage, Impact Ionization Transit-Time Diode, and single-photon avalanche diode. Bifurcation points of the Lienard equation are regarded as phase transition points. The Geiger-mode pulse phase is found to emerge with a symmetry-breaking instability in the available energy profile beyond a threshold value of the quenching resistance. Utilizing energy conservation, high-speed transition processes of discharge and a recharge periods are, respectively, formulated as an instanton and an anti-instanton, particles highly localized in time, by quantum mechanical path integrals. A successfully quenched Geiger-mode pulse is described by an instanton-anti-instanton pair bonded by a key condition of the carrier extinction effect. An analogy of the phenomenology to superconductivity is discussed.