Computer Modelling of Non-Equilibrium Multiple-Trapping and Hopping Transport in Amorphous Semiconductors
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A15.3.1
Computer Modelling of Non-Equilibrium Multiple-Trapping and Hopping Transport in Amorphous Semiconductors C. Main1, J. M. Marshall2 , S. Reynolds3, M.J. Rose1 and R. Brüggemann4 1
University of Dundee, Division of Electronic Engineering and Physics, Dundee DD1 4HN, U.K. University of Wales Swansea, Singleton Park, Swansea SA2 8PP, U.K. 3 Institute of Photovoltaics, Forschungszentrum Jülich, 52425 Jülich, Germany 4 Fachbereich Physik, Carl von Ossietzky Universität Oldenburg, Germany 2
ABSTRACT In this paper we demonstrate a simple computational procedure for the simulation of transport in a disordered semiconductor in which both multi-trapping and hopping processes are occurring simultaneously. We base the simulation on earlier work on hopping transport, which used a Monte-Carlo method. Using the same model concepts, we now employ a stochastic matrix approach to speed computation, and include also multi-trapping transitions between localised and extended states. We use the simulation to study the relative contributions of extended state conduction (with multi-trapping) and hopping conduction (via localised states) to transient photocurrents, for various distributions of localised gap states, and as a function of temperature. The implications of our findings for the interpretation of transient photocurrents are examined.
INTRODUCTION The non-equilibrium distribution of excess carriers in a disordered semiconductor, created by a short laser pulse, subsequently relaxes in energy toward thermal equilibrium. Relaxation and transport involve localised states either by ‘multi-trapping’ (MT) processes, or by direct inter-site tunnelling or ‘hopping’. In the former case, the transient photocurrent (TPC) arises from trap-limited band transport, whilst in the latter case, from hopping transport. Analytical studies have been made of TPC assuming one or other of these processes is dominant [1,2]. Earlier studies focussed mainly on exponential band-tails, for simplicity. More recently, computer modelling has allowed the study of TPC with arbitrary distributions of states. The present authors have developed a TPC spectroscopy which gives the energy distribution of localised states (DOS) from TPC [3], assuming only MT, while Marshall [4,5] has studied, by Monte-Carlo computer modelling, the case of transient photocurrents in the case of hopping transport. In this paper we include both MT and hopping, and we replace the Monte-Carlo solution with a matrix-based Markov chain computation, adapted to deal with extremely wide time intervals which can cause difficulties with ‘stiff’ equation systems. We outline the solution method, and apply it to several illustrative DOS cases, following the parallel progress of transient MT and hopping transport. We examine cases of a broad exponential tail, and a ‘standard’ a-Si:H DOS and examine the influence of hopping on our MT-based DOS analysis.
A15.3.2
MODELLING We represent the continuous distribution g(E) of localised states by a ‘ladder’ of states g(Ei) grouped into slices of en
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