Anderson's rule is used for the construction of energy band diagrams of the heterojunction between two semiconductor materials. Anderson's rule states that when constructing an energy band diagram, the vacuum levels of the two semiconductors on either side of the heterojunction should be aligned (at the same energy).[1]
Band diagrams 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).
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.[4] The electron affinity (usually given by the symbol 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 ) 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.
Once the relative positions of the conduction and valence bands for both semiconductors are known, Anderson's rule allows the calculation of the band offsets of both the valence band () and the conduction band (). 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.
Example: straddling gap
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:
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:
Limitations of Anderson's rule
Further information on other models of heterojunction band alignment: Heterojunction
In real semiconductor heterojunctions, Anderson's rule fails to predict actual band offsets. 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. 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. 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: interface states which may have a very large electrical polarization and defect states, dislocations and other perturbations caused by imperfect crystal lattice matches.
To try to improve the accuracy of Anderson's rule, various models have been proposed. 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] Tersoff[5] 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. 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.[6]
References
^Borisenko, V. E. and Ossicini, S. (2004). What is What in the Nanoworld: A Handbook on Nanoscience and Nanotechnology. Germany: Wiley-VCH.
^Anderson, R. L. (1960). "Germanium-Gallium Arsenide Heterojunctions [Letter to the Editor]". IBM Journal of Research and Development. 4 (3): 283–287. doi:10.1147/rd.43.0283. ISSN 0018-8646.
^J. Tersoff (1984). "Theory of semiconductor heterojunctions: The role of quantum dipoles". Physical Review B. 30 (8): 4874. Bibcode:1984PhRvB..30.4874T. doi:10.1103/PhysRevB.30.4874.
^Debbar, N.; Biswas, Dipankar; Bhattacharya, Pallab (1989). "Conduction-band offsets in pseudomorphic InxGa1-xAs/Al0.2Ga0.8As quantum wells (0.07≤x≤0.18) measured by deep-level transient spectroscopy". Physical Review B. 40 (2): 1058. Bibcode:1989PhRvB..40.1058D. doi:10.1103/PhysRevB.40.1058.
January 04, 2023
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For the completely unrelated rule in computer science see Anderson s rule computer science Anderson s rule is used for the construction of energy band diagrams of the heterojunction between two semiconductor materials Anderson s rule states that when constructing an energy band diagram the vacuum levels of the two semiconductors on either side of the heterojunction should be aligned at the same energy 1 Band diagrams 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 It is also referred to as the electron affinity rule and is closely related to the Schottky Mott rule for metal semiconductor junctions Anderson s rule was first described by R L Anderson in 1960 2 Contents 1 Constructing energy band diagrams 1 1 Example straddling gap 2 Limitations of Anderson s rule 3 ReferencesConstructing energy band diagrams EditMaterial parameters in common semiconductors 3 Eg eV x eV GaAs 1 43 4 07AlAs 2 16 2 62GaP 2 21 4 3InAs 36 4 9InP 1 35 4 35Si 1 12 4 05Ge 66 4 0Once 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 4 The electron affinity usually given by the symbol x displaystyle chi 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 E g displaystyle E rm g 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 Once the relative positions of the conduction and valence bands for both semiconductors are known Anderson s rule allows the calculation of the band offsets of both the valence band D E v displaystyle Delta E rm v and the conduction band D E c displaystyle Delta E rm c 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 Example straddling gap Edit 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 D E c x 2 x 1 displaystyle Delta E rm c chi 2 chi 1 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 D E v x 1 E g 1 x 2 E g 2 displaystyle Delta E rm v chi rm 1 E rm g1 chi rm 2 E rm g2 Limitations of Anderson s rule EditFurther information on other models of heterojunction band alignment Heterojunction In real semiconductor heterojunctions Anderson s rule fails to predict actual band offsets 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 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 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 interface states which may have a very large electrical polarization and defect states dislocations and other perturbations caused by imperfect crystal lattice matches To try to improve the accuracy of Anderson s rule various models have been proposed 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 Tersoff 5 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 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 6 References Edit Borisenko V E and Ossicini S 2004 What is What in the Nanoworld A Handbook on Nanoscience and Nanotechnology Germany Wiley VCH Anderson R L 1960 Germanium Gallium Arsenide Heterojunctions Letter to the Editor IBM Journal of Research and Development 4 3 283 287 doi 10 1147 rd 43 0283 ISSN 0018 8646 Pallab Bhattacharya 1997 Semiconductor Optoelectronic Devices Prentice Hall ISBN 0 13 495656 7 Davies J H 1997 The Physics of Low Dimensional Semiconductors UK Cambridge University Press J Tersoff 1984 Theory of semiconductor heterojunctions The role of quantum dipoles Physical Review B 30 8 4874 Bibcode 1984PhRvB 30 4874T doi 10 1103 PhysRevB 30 4874 Debbar N Biswas Dipankar Bhattacharya Pallab 1989 Conduction band offsets in pseudomorphic InxGa1 xAs Al0 2Ga0 8As quantum wells 0 07 x 0 18 measured by deep level transient spectroscopy Physical Review B 40 2 1058 Bibcode 1989PhRvB 40 1058D doi 10 1103 PhysRevB 40 1058 Retrieved from https en wikipedia org w index php title Anderson 27s rule amp oldid 1057896672, wikipedia, wiki, book, books, library,
