Urbach energy

The Urbach Energy, or Urbach Edge, is a parameter typically denoted ${\displaystyle E_{0}}$, 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

While not apparent in a simple energy-position band diagram (left), the density of states (centre) in both the valance band and the conduction band increases the further from the band gap. This means the absorbance of the semiconductor will increase with energy.

In the simplest description of a semiconductor, a single parameter is used to quantify the onset of optical absorption: the band gap, ${\displaystyle E_{G}}$. In this description, semiconductors are described as being able to absorb photons above ${\displaystyle E_{G}}$, but are transparent to photons below ${\displaystyle E_{G}}$.^[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, ${\displaystyle \alpha }$, 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

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

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]

${\displaystyle \alpha (E)=\alpha _{0}\exp {\biggl (}{\frac {E-E_{1}}{E_{0}}}{\biggr )}}$

where ${\displaystyle \alpha _{0}}$ and ${\displaystyle E_{1}}$ are fitting parameters with dimensions of inverse length and energy, respectively, and ${\displaystyle E_{0}}$ is the Urbach Energy. This equation is only valid when ${\displaystyle \alpha \propto \exp(E)}$. The Urbach Energy is temperature-dependent.^[7][8]

Room temperature values of ${\displaystyle E_{0}}$ for hydrogenated amorphous silicon are typically between 50 meV and 150 meV.^[9]

Relationship to charge transport

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, ${\displaystyle E_{A}}$, in semiconductors governed by multiple trapping and release. It is important to state that ${\displaystyle E_{0}}$ is not the same as ${\displaystyle E_{A}}$, since ${\displaystyle E_{A}}$ describes the disorder associated with one band, not both.

Measurement

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

1. Brotherton, S. D. (2013). Introduction to Thin Film Transistors: Physics and Technology of TFTs. Springer International Publishing. ISBN 978-3-319-00001-5.
2. Hook, J. R.; Hall, H. E. (1991-09-05). Solid State Physics. Wiley. ISBN 978-0-471-92804-1.
3. 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.
4. 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.
5. 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.
6. 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.
7. 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.
8. 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.
9. 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.
10. 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.
11. 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.