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| '''Anderson's rule''' is used for the construction of [[energy band diagram]]s of the [[heterojunction]] between two [[semiconductor material]]s. It is also referred to as the [[electron affinity]] rule, and is closely related to the [[Schottky-Mott rule]] for [[metal-semiconductor junction]]s. Anderson's rule was first described by R. L. Anderson in 1960.<ref>Anderson, R. L., (1960). Germanium-gallium arsenide heterojunction, ''IBM J. Res. Dev.'' '''4'''(3), pp. 283–287</ref>
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| Anderson's rule states that when constructing an energy band diagram, the [[vacuum level]]s of the two semiconductors on either side of the heterojunction should be aligned (at the same energy).<ref>Borisenko, V. E. and Ossicini, S. (2004). ''What is What in the Nanoworld: A Handbook on Nanoscience and Nanotechnology''. Germany: Wiley-VCH.</ref>
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| In the field of [[computer security]], Anderson's rule refers to a principle formulated by [[Ross J. Anderson]]: by their nature large [[database]]s will never be free of abuse by breaches of security. If a large system is designed for ease of access it becomes insecure; if made watertight it becomes impossible to use.<ref>[http://www.guardian.co.uk/commentisfree/henryporter/2009/aug/10/id-card-database-breach Guardian newspaper article on a security breach, in which Anderson's Rule is formulated]</ref>
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| ==Using Anderson's rule to construct energy band diagrams==
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| {| align="left" class="wikitable"
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| |+ Material parameters in common semiconductors<ref name="pallab">Pallab, Bhattacharya (1997), Semiconductor Optoelectronic Devices, Prentice Hall, ISBN 0-13-495656-7</ref>
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| |-
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| !
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| ! [[band gap|''E''<sub>g</sub>]] (eV)
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| ! [[electron affinity|''χ'']] (eV)
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| |-
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| | [[GaAs]]
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| | 1.43
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| | 4.07
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| |-
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| | [[AlAs]]
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| | 2.16
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| | 2.62
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| |-
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| | [[GaP]]
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| | 2.21
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| | 4.3
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| |-
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| | [[InAs]]
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| | .36
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| | 4.9
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| |-
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| | [[InP]]
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| | 1.35
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| | 4.35
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| |-
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| | [[silicon|Si]]
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| | 1.12
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| | 4.05
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| |-
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| | [[germanium|Ge]]
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| | .66
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| | 4.0
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| |}
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| Once the vacuum levels are aligned it is possible to use the [[electron affinity]] and [[band gap]] values for each semiconductor to calculate the [[conduction band]] and [[valence band]] offsets.<ref>Davies, J. H., (1997). ''The Physics of Low-Dimensional Semiconductors''. UK: [[Cambridge University Press]].</ref> The electron affinity (usually given the symbol <math>\chi</math> in [[solid state physics]]) gives the energy difference between the lower edge of the conduction band and the [[vacuum level]] of the semiconductor. The band gap (usually given the symbol <math>E_{\rm g}</math>) gives the energy difference between the lower edge of the conduction band and the upper edge of the valence band. Each semiconductor has different electron affinity and band gap values. For semiconductor [[alloys]] it may be necessary to use [[Vegard's law]] to calculate these values.
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| Once the relative positions of the conduction and valence bands for both semiconductors are known, Anderson's rule allows the calculation of the [[band offset]]s of both the valence band (<math>\Delta E_{\rm v}</math>) and the conduction band (<math>\Delta E_{\rm c}</math>).
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| After applying Anderson's rule and discovering the bands' alignment at the junction, [[Poisson’s equation]] can then be used to calculate the shape of the [[band bending]] in the two semiconductors.
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| === Example: straddling gap ===
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| [[Image:Heterojunction variables in equilibrium.png|300px|thumb|right|[[Band diagram]]s for a straddling-gap type heterojunction, as understood by Anderson's rule. The junction alignment at equilibrium (bottom) is predicted based on a hypothetical flat-vacuum alignment (top).]]
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| Consider a heterojunction between semiconductor 1 and semiconductor 2. Suppose the conduction band of semiconductor 2 is closer to the vacuum level than that of semiconductor 1. The conduction band offset would then be given by the difference in electron affinity (energy from upper conducting band to vacuum level) of the two semiconductors:
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| :<math>\Delta E_{\rm c} = \chi_{2} - \chi_{1}\,</math>
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| Next, suppose that the band gap of semiconductor 2 is large enough that the valence band of semiconductor 1 lies at a higher energy than that of semiconductor 2. Then the valence band offset is given by:
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| :<math>\Delta E_{\rm v} = (\chi_{\rm 1} + E_{\rm g1}) - (\chi_{\rm 2} + E_{\rm g2})\,</math>
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| ==Limitations of Anderson's rule==
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| {{details|heterojunction|other models of heterojunction band alignment}}
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| In real semiconductor heterojunctions, Anderson's rule fails to predict actual band offsets.
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| In Anderson's idealized model the materials are assumed to behave as they would in the limit of a large vacuum separation, yet where the vacuum separation is taken to zero.
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| It is that assumption that involves the use of the vacuum [[electron affinity]] parameter, even in a solidly filled junction where there is no vacuum.
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| Much like with the [[Schottky-Mott rule]], Anderson's rule ignores the real [[chemical bonding]] effects that occur with a small or nonexistent vacuum separation:
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| * Interface states which may have a very large electrical [[polarization density|polarization]].
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| * Defect states, dislocations and other perturbations caused by imperfect crystal lattice matches.
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| To try to improve the accuracy of Anderson's rule, various models have been proposed.
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| The ''common anion rule'' guesses that, since the valence band is related to anionic states, materials with the same anions should have very small valence band offsets.{{citation needed|date=June 2013}}
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| Tersoff<ref>{{cite journal|author=J. Tersoff|doi=10.1103/PhysRevB.30.4874|title=Theory of semiconductor heterojunctions: The role of quantum dipoles|year=1984|journal=Physical Review B|volume=30|issue=8|pages=4874|bibcode = 1984PhRvB..30.4874T }}</ref> proposed the presence of a [[dipole]] layer due to induced gap states, by analogy to the [[metal-induced gap states]] in a [[metal-semiconductor junction]].
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| Practically, heuristic corrections to Anderson's rule have found success in specific systems, such as the ''60:40 rule'' used for the GaAs/AlGaAs system.<ref name="Debbar">{{cite doi|10.1103/PhysRevB.40.1058}}</ref>
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| ==References==
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| <references/>
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| [[Category:Semiconductor structures]]
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| [[Category:Electronic band structures]]
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| [[Category:Rules]]
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The author is known as Irwin Wunder but it's not the most masucline title out there. Bookkeeping is her day job now. His family members lives in South Dakota but his spouse wants them to transfer. It's not a typical factor but what she likes doing is base leaping and now she is trying to earn cash with it.
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