Effective Densities of States in Conduction and Valence Bands in a-Si
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EFFECTIVE DENSITIES OF STATES IN CONDUCTION AND VALENCE BANDS IN a-Si
JOZE FURLAN, FRANC SMOLE, PAVLE POPOVIC
Faculty of Electrical and Computer Engineering, University of Ljubljana, Tr.aska 25, Ljubljana, Slovenia ABSTRACT Effective densities of states in conduction and valence band, Nc and Nv, are usually set to a fixed value of 1019 to 1021 cm 3 in all computer simulations of a-Si structures. In this contribution the densities Nc and Nv are analytically expressed for different selected extended states distributions. The derivatives d(lnNc)/dx and d(lnNv)/dx, representing specific terms in transport equations, are expressed as function of position dependent band structure. The effect of an increased disorder in heterojunction is simulated by Gaussian distribution of states in a linearly graded heterojunction region. INTRODUCTION An accurate analysis of a-Si solar cell requires adequate models representing locally variable material properties of amorphous semiconductor layers forming a rather complex solar cell structure. The basic semiconductor equations solved by numerical methods incorporate material and transport parameters which include also effective densities of states at conduction and valence band-edge, Nc and Nv. In present numerical a-Si cell simulations, the values of Nc and Nv are set to a fixed value between 1019 and 10' cm-3, even in heterojunction structures where N, depends on electron affinity profile, and Nv on both, position dependent electron affinity and mobility gap. The aim of this work is to model effective densities Nc and Nv and to illuminate the effects contributing to these densities. ANALYSIS OF Nc AND Nv FOR VARIOUS CASES OF BAND STATES DISTRIBUTION The effective densities of states are given by the expressions1
Sw
Ncfgc(W)e
kTdW
(1)
0
and ®
w_
Nv gy(W)eU dWY
(2)
0
where W.=E-Ec and Wp=Ev-E represent kinetic energies of electrons in conduction band and of holes in valence band, respectively. The densities Nc and Nv can be calculated once gc(Wn) and gv(Wp) are specified. Jackson et a12 and Cody' have shown that global similarity exists between c-Si and a-Si so that both materials can be characterized by parabolic energy dependence of states in Mat. Res. Soc. Symp. Proc. Vol. 297. ©1993 Materials Research Society
364
the conduction and valence bands. However, an accurate quantitative description of extended states distribution in the range of a few kT above and below the mobility gap in a-Si is not known. Our approach is therefore based on data available for localized states in the gap and by continuing the distribution of tail states into conduction and valence bands. As shown schematically in Fig.1, the tail states in the gap near band-edges will be approximated by a pair of exponential distributions, consisting of acceptorlike and donorlike states E-Ec
g9-A=Ae
WC
Er-E
g9D=Ae Wv
each characterized by its value at the corresponding band-edge, At and A1, respectively, and by characteristic energy slopes of exponential distributions, Wc and Wv. gV(d) c,(X),
cd (c)
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