A Laplace Transform Technique for Direct Determination of Density of Electronic States in Disordered Semiconductors from
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A Laplace Transform Technique for Direct Determination of Density of Electronic States in Disordered Semiconductors from Transient Photocurrent Data Mariana J. Gueorguieva, Charles Main and Steve Reynolds School of Science and Engineering, University of Abertay Dundee, Dundee, DD1 1HG, U.K. ABSTRACT A new technique for direct determination of the density of electronic states (DOS) in disordered semiconductors is described. It involves Laplace transformation of transient photocurrent data I(t) followed by the numerical solution of the system of linear algebraic equations obtained from the Fredholm integral of the first kind, for a DOS represented by a series of discrete levels. No approximations are used in the solution, and no prior assumptions as to the form of the DOS are made. The fidelity of this method is assessed and compared with existing techniques by application to computer-simulated I(t) data generated from single-level and continuous DOS profiles, and to experimental data. INTRODUCTION Transport in amorphous semiconductors is generally considered to occur by multiple trapping (MT), in which free carriers interact via capture into and emission from an energy distribution of localised states. If an excess carrier density is created by means of a short light pulse the subsequent transient photocurrent (TPC) decay I(t) contains useful information about these processes. A wide range of methods, employing various physical assumptions and/or mathematical approximations, has evolved to permit the extraction of the density of states (DOS) from TPC I(t) data [1]. Several require no prior assumptions as to the form of the DOS (so-called ‘spectroscopic’ methods), and involve either Fourier [2,3] or Laplace [4,5] transformations. Here we report on a further development, in which the exact solution of the MT rate equations cast in the form of a Fredholm integral in the s-domain is obtained, which we term the Exact Laplace Transform (ELT) method. It contains no mathematical simplifications and is capable in principle of extracting the DOS to a high degree of accuracy. In the following sections, the method of solution is described and tested by applying it to computer-simulated I(t) data from a variety of model DOS profiles, including both quasicontinuous distributions and discrete levels, and to experimental TPC data from hydrogenated amorphous silicon (a-Si:H). THEORY We assume that current is carried solely by carriers in extended states, which interact through trapping and release with a distribution of localised states. The basic linearised multiple trapping equations for free and trapped electrons respectively are: dn (t ) n(t ) dn(t ) = −∑ i − + n0δ (t ) ; dt dt τf i
dni (t ) = ω i n(t ) − γ i ni (t ) dt
A27.8.1
(1a, 1b)
In the above system of equations n(t ) is the free carrier density at time t, ni (t ) is the trapped carrier density at the ith localized state at time t, no is the injected free carrier density, τf is the free carrier recombination lifetime, g ( Ei ) is the density of states for the the ith
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