The Urbach Energy, or Urbach Edge, is a parameter typically denoted , with dimensions of energy, used to quantify energetic disorder in the band edges of a semiconductor. It is evaluated by fitting the absorption coefficient as a function of energy to an exponential function. It is often used to describe electron transport in structurally disordered semiconductors such as hydrogenated amorphous silicon.[1]
Introduction
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/3/30/Absorption_Edge_in_Semiconductors.png/507px-Absorption_Edge_in_Semiconductors.png)
In the simplest description of a semiconductor, a single parameter is used to quantify the onset of optical absorption: the band gap, . In this description, semiconductors are described as being able to absorb photons above , but are transparent to photons below .[2] However, the density of states in 3 dimensional semiconductors increases further from the band gap (this is not generally true in lower dimensional semiconductors however). For this reason, the absorption coefficient, , increases with energy. The Urbach Energy quantifies the steepness of the onset of absorption near the band edge, and hence the broadness of the density of states. A sharper onset of absorption represents a lower Urbach Energy.
History and name
[edit]The Urbach Energy is defined by an exponential increase in absorbance with energy. While an exponential dependence of absorbance had been observed previously in photographic materials,[3] it was Franz Urbach that evaluated this property systematically in crystals. He used silver bromide for his study while working at the Kodak Company in 1953.[4]
Definition
[edit]Absorption in semiconductors is known to increase exponentially near the onset of absorption, spanning several orders of magnitude.[5][6] Absorption as a function of energy can be described by the following equation:[1][7]
where and are fitting parameters with dimensions of inverse length and energy, respectively, and is the Urbach Energy. This equation is only valid when . The Urbach Energy is temperature-dependent.[7][8]
Room temperature values of for hydrogenated amorphous silicon are typically between 50 meV and 150 meV.[9]
Relationship to charge transport
[edit]The Urbach Energy is often evaluated to make statements on the energetic disorder of band edges in structurally disordered semiconductors.[1] The Urbach Energy has been shown to increase with dangling bond density in hydrogenated amorphous silicon[9] and has been shown to be strongly correlated with the slope of band tails evaluated using transistor measurements.[10] For this reason, it can be used as a proxy for activation energy, , in semiconductors governed by multiple trapping and release. It is important to state that is not the same as , since describes the disorder associated with one band, not both.
Measurement
[edit]To evaluate the Urbach Energy, the absorption coefficient needs to be measured over several orders of magnitude. For this reason, high precision techniques such as the constant photocurrent method (CPM)[11] or photothermal deflection spectroscopy are used.
References
[edit]- ^ a b c Brotherton, S. D. (2013). Introduction to Thin Film Transistors: Physics and Technology of TFTs. Springer International Publishing. ISBN 978-3-319-00001-5.
- ^ Hook, J. R.; Hall, H. E. (1991-09-05). Solid State Physics. Wiley. ISBN 978-0-471-92804-1.
- ^ Eggert, John; Biltz, Martin (1938-01-01). "The spectral sensitivity of photographic layers". Transactions of the Faraday Society. 34: 892–901. doi:10.1039/TF9383400892. ISSN 0014-7672.
- ^ Urbach, Franz (1953-12-01). "The Long-Wavelength Edge of Photographic Sensitivity and of the Electronic Absorption of Solids". Physical Review. 92 (5): 1324. Bibcode:1953PhRv...92.1324U. doi:10.1103/physrev.92.1324. ISSN 0031-899X.
- ^ Tauc, J. (1970-08-01). "Absorption edge and internal electric fields in amorphous semiconductors". Materials Research Bulletin. 5 (8): 721–729. doi:10.1016/0025-5408(70)90112-1. ISSN 0025-5408.
- ^ Wronski, C.R.; Abeles, B.; Tiedje, T.; Cody, G.D. (1982-12-01). "Recombination centers in phosphorous doped hydrogenated amorphous silicon". Solid State Communications. 44 (10): 1423–1426. Bibcode:1982SSCom..44.1423W. doi:10.1016/0038-1098(82)90023-0. ISSN 0038-1098.
- ^ a b Cody, G. D.; Tiedje, T.; Abeles, B.; Brooks, B.; Goldstein, Y. (1981-11-16). "Disorder and the Optical-Absorption Edge of Hydrogenated Amorphous Silicon". Physical Review Letters. 47 (20): 1480–1483. Bibcode:1981PhRvL..47.1480C. doi:10.1103/physrevlett.47.1480. ISSN 0031-9007.
- ^ Kurik, M. V. (1971). "Urbach rule". Physica Status Solidi A. 8 (1): 9–45. Bibcode:1971PSSAR...8....9K. doi:10.1002/pssa.2210080102. ISSN 1521-396X. S2CID 244517318.
- ^ a b Stutzmann, M. (1989-10-01). "The defect density in amorphous silicon". Philosophical Magazine B. 60 (4): 531–546. Bibcode:1989PMagB..60..531S. doi:10.1080/13642818908205926. ISSN 1364-2812.
- ^ Sherman, S.; Wagner, S.; Gottscho, R. A. (1998-06-04). "Correlation between the valence- and conduction-band-tail energies in hydrogenated amorphous silicon". Applied Physics Letters. 69 (21): 3242. doi:10.1063/1.118023. ISSN 0003-6951.
- ^ Vaněček, M.; Kočka, J.; Stuchlík, J.; Kožíšek, Z.; Štika, O.; Tříska, A. (1983-03-01). "Density of the gap states in undoped and doped glow discharge a-Si:H". Solar Energy Materials. 8 (4): 411–423. Bibcode:1983SoEnM...8..411V. doi:10.1016/0165-1633(83)90006-0. ISSN 0165-1633.