edited for some of these contexts and discussed at length in previous Courses ([VanV- 95], [Flee-95], [Wino-92], [Peas-90], [Brow-90] and several lectures of the preceding decade). III-63 MODELING THE TRAPPING AND THERMAL DETRAPPING EFFECTS The set of trapping continuity equations has to be completed to take into account detrapping or neutralization. In the following, the case of thermal detrapping will be addressed. 5.2.1. MODELING PIE IN TERMS OF ENERGY SPECTRA: PRINCIPLES AND EQUATIONS It has been recognized that the trap depth is not unique, but exhibits a trap distribution. The following figure describes a way to model this distribution into approximate trap types: exponential tail representing shallow traps close to valence band, gaussian-like levels, boxes. Dp(E) cm-2 eV-1 exp -E/ E Figure 50. Methods of modeling the distribution of deep traps in energy. For one energy level i, the detrapping term can be described as follows by adding the terms ei .nt.i and ei . Pt,i in the continuity equations. - nt,i(E) jn(x,t) nr,i(E) jn(x,t) ptj(x,t)ei.nt (95) - pt,i(E) jp(x,t) - nr,i(E) jn(x,t) Pt,j(x,t) - ej.nt, III-64 Here Vth is the carrier thermal velocity are Nc densities of states in valence and conduction band and EApi the trap depth in energy, with respect to the band edge. This set of equations has been numerically integrated for 10 levels of energy and, for medium and thick thermal oxides, the result gives very good agreements with experimental data. Details are given at length in [Cirb-96] and [Pail-98]. The resulting set of continuity equations for free carriers is therefore (written in a generalized manner): (100) along with Poisson's equation: (101) 5.2.2. RESULTS A. Medium-thin oxide We have simulated a MOS capacitor structure having a thermal oxide of 45 nm. The trap energy distribution was taken from experimental results [Flee-92c], and is shown in Fig. 51. III-65 Hard oxide, 45 nm Energy (eV) Figure 51. Energy distribution used for 45 nm thick oxides, from [Flee-92c]. Hard, 45 nm, 1Mrad@+5V TSC bias 