On the Origin of the Urbach Rule and the Urbach Focus
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On the Origin of the Urbach Rule and the Urbach Focus J. A. Guerra1,2, L. Montañez1, F. De Zela1, A. Winnacker2 and R. Weingärtner1,2 1
Pontifical Catholic University of Peru, Sciences Department, Physics Section, Av. Universitaria 1801, Lima 32, Peru. 2 University of Erlangen-Nurnberg, Institute of Material Science 6, Martensstr. 7, 91058 Erlangen, Germany ABSTRACT A simple derivation of sub-bandgap exponential tails and fundamental absorption equations ruling the optical absorption of amorphous semiconductors are presented following the frozen phonon model. We use the Kubo-Greenwood formula to describe the average transition rate for the optical absorption process. Asymptotic analysis leads to the commonly observed exponential tail as well as the Tauc expression for the fundamental absorption. We test our theoretical results with experimental absorption coefficients of amorphous Si:H, SiC:H, AlN and SiN. The validity of the Urbach focus concept is evaluated. INTRODUCTION Crystalline wide bandgap semiconductors have been of increasing interest in the past decade due to their advantageous properties for applications in the visible and ultraviolet regions. Their amorphous counterparts share such features [1-5] and offer additional properties such as low production costs and higher doping incorporation, which makes them attractive for device applications. Thus, in order to design devices with these amorphous materials a good understanding of their optical properties is necessary. One important aspect is the wavelength dependent absorption coefficient in the fundamental and adjacent region: The Tauc region describes the extended states related transitions. The Urbach region accounts for transitions related to localized states, observed as the so called exponential tail in the absorption coefficient. Tauc derived the behavior of the fundamental absorption following a well-known theoretical approach used for crystals. Here indirect transitions are allowed for simulating energy conservation by disorder, i.e. by the “frozen phonons”. However, this method fails in the Urbach region. The Toyazawa’s group [6,7] introduced the static disorder effect assuming a Gaussian distribution for thereby random site energies into the electronic Hamiltonian, achieving the exponential behavior of the band edges. Other contributions, such as the work of Dunstant [8], Soukoulis, Sajeev, Cohen and Economou [9, 10] and later on the contribution of the group of O’Leary on understanding the electronic properties of amorphous materials [11-14], introduced the effect of thermal fluctuations in the band edge (or the band edge fluctuations due to static
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Urbach rule Eq. 5
- log(CF/0)
log(α)
16 14 12 10 8 6 a) b) 4 1.4 1.6 1.8 2.0 2.2 1.4 1.6 1.8 2.0 2.2 Photon Energy (eV)
15 10 5 0 10 12 14 16 18 20 -1 Urbach Slope β (eV ) c)
Figure 2. Global (a) and independent (b) fits of the Urbach region of the absorption coefficient of a-Si:H using the Urbach rule. log(CF/0) vs ȕ using the parameters CF/0 and obtained from i
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