Fluctuations in the collected charge in integrating photoconductive detectors under small and large signals: the variance problem

Charge collection efficiency (CE) eta(0) under small signal conditions, corresponding to a uniform field in the detector medium, has been widely used in evaluating the performance of photoconductive detectors. The present paper answers the question, 'What is the variance of the collected charge in an integrating detector as a function of photoinjection level and what are the errors if we continue to use the small signal equations?' The variance sigma(2)(0) in eta(0) under small signals has been theoretically derived in the literature and has been a key factor in the detective quantum efficiency modeling of integrating detectors based on various semiconductors. sigma(2)(0) is a noise source and can degrade the detector performance under incomplete charge collection. The statistical variance sigma(2)(0) in the CE eta(0), under small signals and the variance sigma(2)(r) in the CE eta(r) under an arbitrary injection level r (injected charge divided by charge on the electrodes) have been studied using the Monte Carlo simulation model developed in this work to evaluate the difference between sigma(2)(r) and sigma(2 )(0)from small to large signals. Initial injection of electron and hole pairs and their subsequent transport and trapping in the presence of an electric field, which is calculated from the Poisson equation, is used to calculate the photocurrent. Each injected carrier is tracked as it moves in the semiconductor until it is either trapped or reaches the collection electrode. Trapped carriers do not contribute to the photocurrent but continue to contribute to the field through the Poisson equation. The instantaneous photocurrent i(ph)(t) is calculated from the drift of the free carriers through the Shockley-Ramo theorem. i(ph)(t) is integrated over the duration of the photocurrent to calculate the total collected charge and hence the CE eta(r) . The variance sigma(2)(r) in eta(r) is found from multiple simulations of eta(r) . The eta(r) and sigma(2)(r) have been generated over varying charge injection ratios r, the electron and hole ranges mu tau , mean photoinjection depths delta and drift mobility ratios b. At full injection, the deviation Delta sigma(2)(r) of the CE variance sigma(2)(r)( )from the uniform field case sigma(2)(0) (i.e.Delta sigma(2)(r) = sigma(2)(r) - sigma(2)(0)) may be as much as 40% larger or 20% lower than the small signal model prediction. This study provides the extent of errors involved in the variance of the CE in non-uniform fields and quantifies the increase in errors that can occur in high injection cases. In practice, typical injection ratios are less than 0.2, which means that the magnitude of percentage error Delta sigma(2)(r)/sigma(2)(0) is less than 5%.