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Prediction of band discontinuities in semiconductor heterojunctions: A simple model
Prediction of band discontinuities in semiconductor heterojunctions: A simple model Yiannis Karafyllidis Ag. Panteleimonos 22, Polichni, GR-565 33 Thessaloniki, Greece Paul Hagouel Optelec, Thessaloniki, Greece Epaminondas Kriezis Aristotelian University of Thessaloniki, Department of Electrical Engineering, Thessaloniki, Greece A simple model for the prediction of heterojunction valence band discontinuities is proposed in this paper. Only two properties of semiconductors are used in this model, namely the lattice constant and the energy gap.
1. Introduction A semiconductor heterojunction is defined as the interface between two different semiconductor materials. The two semiconductors involved have different energy gaps. Because of this difference, the conduction and valence band edges are discontinuous at the interface. The difference between the two energy gaps AEg is distributed to the conduction band discontinuity AEc and to the valence band discontinuity AEv AE~ = AEc + AEv
(1)
The properties of heterojunction devices, from simple photon detectors to superlattices, depend to a great extend on the band discontinuities. In spite of several years of theoretical effort the fundamental question in the heterojunction theory is still the energy-band lineup at the heterojunction interface. The first model for semiconductor heterojunctions was proposed by Anderson [1] in 1962. In this model the conduction band discontinuity is given by the difference between the electron affinities of the two semiconductors. Since then several theories and models have been presented [2-19] and some attempts have been made to classify them into categories [15, 18, 20-22]. Generally the problem of the determination of the band discontinuities separates into three distinct subproblems: (a) The determination of the energy bands, relative to the potential of the bulk, in each semiconductor. This potential is the so-called reference potential. (b) The determination of the relative alignment of the two reference potentials. (c) The determination of the contribution of the interface dipoles to the band discontinuities.
MICROELECTRONICS JOURNAL Vol. 22 Nos. 7-8 © 1991 Elsevier Science Publishers Ltd., England
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E Karafyllidis et aL
60
The models that neglect the interface dipole contribution and deal with the other two sub-problems are called linear models. The main difficulty with linear models is the choice of the potential to be used as the reference potential. Usually the average crystal potential is used as reference. The exact calculation of the average crystal potential is impossible and a decision has to be made on which approximate method will be used to calculate that potential. The interface dipoles are of essential concern in the non-linear models. The most confusing point about interface dipoles, as stated by Lambrecht et al. [17], is that there is not a clear definition of the term "'diopole'" in heterojunction theory. Kroemer [23] pointed out that there are two possible kinds of interface dipoles: dipoles arising from the formation of interface chemical bonds and dipoles caused by the interdiffusion of atoms between the two sides of the junction. Tersoff [9] proposed that there is a dipole at the interface associated with gap states induced by the band discontinuities. The interface chemical bonds which give rise to an interface dipole were carefully treated by some workers [18]. In spite of all those efforts it is still not certain what kind of dipoles exist at the interface and by what amount they contribute to the band discontinuities. In recent years several attempts have been made to measure experimentally the heterojunction band discontinuities, but the reliability of the experimental data has been questioned. Kroemer [23] selected some of the experimental data as most reliable. Margaritondo [24] criticized that selection. The attempts to determine the heterojunction band discontinuities using experimental methods have not yet clarified the situation. So far the understanding of the heterojunction band discontinuities is not satisfactory. As a result of this situation, the Anderson model is still the most widely used discontinuity model [25, 26]. The Anderson model was strongly criticized because it uses the electron affinity, which is a free surface parameter, to describe interface properties [27]. Furthermore, the values of the electron affÉnity, reported by different authors, for a given semiconductor are often discrepant from each other by several tenths of an electronvolt [20, 28], while the order of magnitude of the band discontinuities is some tenths of an electronvolt. The use of the Anderson model reflects the need for a simple model. A simple model for the prediction of valence band discontinuities is proposed in this paper. 2. The model
In the nearly-free-electron model the free electron is the zero approximation and the periodic crystal potential is treated as a perturbation [29]. At the first Brillouin zone boundary, the electron energy is given by E
(kg) = E()(ks) + V0 - V~
(2)
E +(k~) = E °(k s ) + V~)+ V~
(3)
and
where k~ is the wavevector at the zone boundary. The electron energy in the crystal at the zone boundary is either depressed by the amount V~below the free-electron
Band discontinuities in semiconductor heterojunctions
61
parabola or raised by the same a m o u n t above the flee-electron parabola [29]. F r o m eqns. (2) and (3) we have E + (kg) - E - (kg) = 2Vg = Eg
(4)
where in the case of a semiconductor crystal Eg is the energy gap. E ° (k) is the free electron energy
E ° (k) : - k2 2m
(5)
V0 is given by
Vo = (k [ V(r) l k) = f W~* (r) V(r) Vk(r) dr
(6)
where V(r) is the periodic crystal potential. In the case of a semiconductor heterojunction, for the first semiconductor crystal the edge of the valence band is given by 1 Ev, = E ° (kgl) + [I01 - - - Egl 2
(7)
where kg I is the wavevector at the zone boundary. If the direction perpendicular to the heterojunction interface is the [100] direction, then
kgl -
(8)
aj
where al is the lattice constant of the first semiconductor. For the second semiconductor crystal the edge of the valence b a n d is given by 1 Ev2 = E 0 (kg2) + V02 -
E~
_
2
(9)
where kg 2 is the wavevector at the zone boundary. If the direction perpendicular to the heterojunction interface is the [100] direction, then 7t
k~ = - a2
where a2 is the lattice constant of the second semiconductor. The valence band discontinuity is given by
(10)
62
Y. Karafyllidis et al.
1
A E = E~, - E~2 = E ° (kg,) - E ° (kg2) +
Vol -
1/o2 -
--
(Eg I -
Eg2)
(11)
2 The electron energy versus wavevector diagram for the first semiconductor is represented on the left-hand side of Fig. 1, and the same diagram for the second semiconductor is represented o n the fight-hand side o f the same figure. The dashed vertical line represents the heterojunction interface. The relative position of the energy gaps, and the valence and conduction b a n d discontinuities are also shown in Fig. 1. The total energy o f an electron moving freely from semiconductor 1 to semiconductor 2 across the interface remains constant. The potential energy of the electron is changed by an a m o u n t
E
E
/ I g2
/,
k
EglI ~\
// I Vs "n/a2
-
n/a1
1~/a 2
wT/aI
Fig. 1. Energy E vs. wavevector k for an electron in a heterojunction crystal. The dashed line represents the heterojunction interface. The relative position of the energy gaps is shown in this figure.
V~ = V0j - V02
( 12)
and the kinetic energy is changed accordingly. It is beyond the scope o f this paper to go any further. F r o m eqns. (5), (11) and (12) we have ~2 ~v
-
1 (k 2 - k~) -
(Eg, - Eg2) + V~
2m
(13)
2
One more term must be added to eqn. (13) to include the interface dipole contribution to the valence b a n d discontinuity. Let the interface dipole potential be equal to Va. After that we have ~2 /~E v -
1 (k~ -
2m
k~)
-
(Eg 1 -
Eg2) +
Vs +
Va
(14)
7
As stated in the introduction it is difficult to calculate V~ and Vd and it can be done only approximately. Instead of calculating approximately those potentials we
63
Band discontinuities in semiconductor heterojunctions
decided to ignore their contribution. The valence band discontinuity is then given approximately by ~2
1
/kE~ = ~2m (k~ - k~) - -~ (E~= - E~)
(15)
3. Application of the model The proposed model has to be compared with other models. This comparison can only be made on the basis of some experimental data. The Katnani and Margaritondo measurements were chosen to serve this basis. Katnani and Margaritondo [20] performed photoemission measurements of the valence band discontinuity on several interfaces involving group IV, III-V and II-VI semiconductor substrates and germanium or silicon overlayers. They compared [20, 22] the measured valence band discontinuities with the predictions of the Anderson [1], Frensley-Kroemer [3], Harrison [5] and Tersoff [9] models. The proposed model was applied to 21 different heterojunctions; the absolute values of the calculated valence band discontinuities are listed in Table 1. For this calculation the direction perpendicular to the heterojunction interface was the [100] direction, for both semiconductors. In the same table are listed the valence band discontinuities measured by Katnani and Margaritondo. In Table 1 the results are listed so that the value of AEv for the GaAs-Si heterojunction is found at the intersection of the GaAs row with the silicon column. If AEv is positive for the Si-Ge heterojunction, then AEv is taken to be negative for the Ge-Si heterojunction. The average magnitude of the discrepancy between the calculated values and the measured ones is 0.23 eV. To test the Anderson model Katnani and Margaritondo used the electron affinity values that best fitted their experimental data. Even after that the average accuracy TABLE 1 Experimental and calculated values of valence band discontinuities (values in electronvolts) Semiconductor
Experimental Si
Ge Si GaAs GaP GaSb InAs InP InSb CdS CdSe CdTe
Ge
I -0.17 0.05 0.95 0.05 0.15 0.57 0.00 1.55 1.20 0.75
Model predictions Si
Ge
-0.1347 0.17 0.35 0.80 0.20 0.33 0.64 0.00 1.75 1.30 0.85
0.2483 0.5795 0.0630 0.1296 0.2980 0.0948 0.8193 0.5375 0.5990
0.1347 0.3829 0.7141 0.1976 0.0510 0.4327 0.0388 0.0940 0.6722 0.7346
E Karq['vllidis et at,
64
of the Anderson model was not better than 0.25 eV [20]. The average accuracy of both the Frensley-Kroemer and Harrison models is about 0.4 eV [20]. To test the Tersoffmodel another set ofheterojunctions was considered [22]. This set includes heterojunctions between germanium or silicon and GaAs, GaSb, GaP, InAs, InP and the Si-Ge heterojunction. The average accuracy of the Tersoff model is 0.13 eV. The corresponding values for the same set of heterojunctions are 0.30 eV for the Harrison model and 0.47 for the Frensley-Kroemer model [22]. The average accuracy of the proposed model for the same set of heterojunctions is 0.13 eV.
4. Conclusions
The proposed model is simple and clear. Only two properties of semiconductors are used in this model: the lattice constant and the energy gap. For these properties accurate values exist and those values can easily be found in almost any text book. The proposed model is as accurate as the Anderson model. Consequently, this model can replace the Anderson model until something better is found. Compared with other models, the proposed model accuracy is satisfactory in spite of the omission of the potentials Vs and Va in (15). The calculation of those potentials sometimes involves considerable approximations. On the other hand, the order of magnitude of the band discontinuities is some tenths of an electronvolt. Consequently the omission of these potentials is sometimes in favour of the accuracy of the model.
5. References
[1] [2]
[3] [4]
[5] [6] [7] [8] [9]
Anderson, R.L., "Experiments on Ge-As heterojunction", Solid State Electron., vol. 5, p. 341, 1962. Frensley, W.R. and Kroemer, H., "Prediction of semiconductor heterojunction discontinuities from bulk band structures", J. Vac. Sci. Technol., vol. 13, p. 810, 1976. Frensley, W.R. and Kroemer, H., "Theory of the energy-band lineup at an abrupt semiconductor heterojunction", Phys. Rev. B, vol. 16, p. 2642, 1977. Baraff, G.A., Appelbaum, J.A. and Hamann, D.R., "Self-consistent calculation of the electronic structure at an abrupt GaAs-Ge interface", Phys. Rev. Lett., vol. 38, p. 237, 1977. Harrison, W.A., "Elementary theory of heterojunctions", J. Vac. Sci. Teehnol.,vol. 14, p. 1016, 1977. v Ross, O., "Theory of extrinsic and intrinsic heterojunctions in thermal equilibrium", Solid State Electron., vol. 23, p. 1069, 1980. Chatterjee, A. and Marshak, A.H., "Theory of abrupt heterojunctions in equilibrium", Solid State Electron., vol. 24, p. 1111, 1981. Katnani, A.D. and Margaritondo, G., "Empirical rule to predict heterojunction band discontinuities", J. Appl. Phys., vol. 54, p. 2522, 1983. Tersoff, J., "Theory of semiconductor heterojunctions: The role of quantum dipoles", Phys. Rev. B, vol. 30, p. 4874, 1984.
Band discontinuities in semiconductor heterojunctions
65
[10] Niles, D.W., Margaritondo, G., Perfetti, P., Quaresima, C. and Capozi, M., "Heterojunction band discontinuity control by ultrathin interlayers", Appl. Phys. Lett., vol. 47, p. 1092, 1985. [11] Niles, D.W., Colavita, E., Margaritondo, G., Perfetti, P., Quaresima, C. and Capozi, M., "'Understanding and controlling heterojunction band discontinuities", J. Vac. Sci. Technol. A, vol. 4, p. 962, 1985. [12] Van de Walle, C.G. and Martin, R.M., "Theoretical calculations of heterojunction discontinuities in the Si/Ge system", Phys. Rev. B, vol. 34, p. 5621, 1986. [13] Tersoff, J., "Summary abstract: Failure of the common anion rule for latticematched heterojunctions", I Vac. Sci. Teehnol. B, p. 1066, 1986. [14] Harrison, W.A. and Tersoff, J., "Tight-binding theory of heterojunction band lineups and interface dipoles", J. Vac. Sci. Technol. B, vol. 4, p. 1068, 1986. [15] Lambrecht. W.R.L. and Segall, B., "Theory of semi-conductor heterojunction valence-band offsets: from supercell band structure calculations toward a simple model", Phys. Rev. Len., vol. 61, p. 1764, 1988. [16] Munoz, A., Cherty, N. and Martin, R.M., "Modification of heterojunction band offsets by thin layers at interfaces: Role of the interface dipole", Phys. Rev. B, vol. 41, p. 2976, 1990. [17] Lambrecht, W.R.L., Segall, B. and Andersen, O.K., "Self-consistent theory of heterojunction band offsets", Phys. Rev. B, vol. 41, p. 2813, 1990. [18] Lambrecht, W.R.L. and Segall, B., "Interface bond polarity model for semiconductor heterojunction band offsets", Phys. Rev. B, vol. 41, p. 2832, 1990. [19] Lambrecht, W.R.L. and Segall, B., "Interface dependence of band offsets in lattice matched heterojunctions", Phys. Rev. B, vol. 41, p. 8353, 1990. [20] Katnani, A~D. and Margaritondo, G., "Microscopic study of semiconductor heterojunctions: Photoemission measurement of the valence band discontinuity and of the potential barriers", Phys. Rev. B, vol. 28, p. 1944, 1983. [21] Milnes, A.G., Semiconductor heterojunction topics: Introduction and overview", Solid State Electron., vol. 29, p. 99, 1986. [22] Margaritondo, G., "Controlling heterojunction band discontinuities: A surface physicist's approach", Solid State Electron., vol. 29, p. 123, 1986. [23] Kroemer, H., Proceedings NATO Advanced Study Institute on Molecular Beam Epitaxy and Heterostruetures, Erice 1983, Chang, L.L. and Ploog, K., eds., Martinus Nijhoff, The Netherlands, 1983. [24] Margaritondo, G., "Comment on 'Theory of semiconductor heterojunctions: The role of quantum dipoles' ", Phys. Rev. B, vol. 31, p. 2526, 1985. [25] Sze, S.M., High-Speed Semiconductor Devices, Wiley, 1990. [26] Alferov, Zh.I., Semiconductor Heterostructures: Physical Processes and Applications, MIR Publishers, 1989. [27] Kroemer, H., CRC Crit. Rev. Solid State Sci., vol. 5, p. 555, 1975. [28] Sharma, B.L. and Purohit, R.K., Semiconductor Heterojunctions, Pergamon Press, 1974. [29] Ziman, J.M., Principles of the Theory of Solids, Cambridge, 1972.
1. Introduction A semiconductor heterojunction is defined as the interface between two different semiconductor materials. The two semiconductors involved have different energy gaps. Because of this difference, the conduction and valence band edges are discontinuous at the interface. The difference between the two energy gaps AEg is distributed to the conduction band discontinuity AEc and to the valence band discontinuity AEv AE~ = AEc + AEv
(1)
The properties of heterojunction devices, from simple photon detectors to superlattices, depend to a great extend on the band discontinuities. In spite of several years of theoretical effort the fundamental question in the heterojunction theory is still the energy-band lineup at the heterojunction interface. The first model for semiconductor heterojunctions was proposed by Anderson [1] in 1962. In this model the conduction band discontinuity is given by the difference between the electron affinities of the two semiconductors. Since then several theories and models have been presented [2-19] and some attempts have been made to classify them into categories [15, 18, 20-22]. Generally the problem of the determination of the band discontinuities separates into three distinct subproblems: (a) The determination of the energy bands, relative to the potential of the bulk, in each semiconductor. This potential is the so-called reference potential. (b) The determination of the relative alignment of the two reference potentials. (c) The determination of the contribution of the interface dipoles to the band discontinuities.
MICROELECTRONICS JOURNAL Vol. 22 Nos. 7-8 © 1991 Elsevier Science Publishers Ltd., England
59
E Karafyllidis et aL
60
The models that neglect the interface dipole contribution and deal with the other two sub-problems are called linear models. The main difficulty with linear models is the choice of the potential to be used as the reference potential. Usually the average crystal potential is used as reference. The exact calculation of the average crystal potential is impossible and a decision has to be made on which approximate method will be used to calculate that potential. The interface dipoles are of essential concern in the non-linear models. The most confusing point about interface dipoles, as stated by Lambrecht et al. [17], is that there is not a clear definition of the term "'diopole'" in heterojunction theory. Kroemer [23] pointed out that there are two possible kinds of interface dipoles: dipoles arising from the formation of interface chemical bonds and dipoles caused by the interdiffusion of atoms between the two sides of the junction. Tersoff [9] proposed that there is a dipole at the interface associated with gap states induced by the band discontinuities. The interface chemical bonds which give rise to an interface dipole were carefully treated by some workers [18]. In spite of all those efforts it is still not certain what kind of dipoles exist at the interface and by what amount they contribute to the band discontinuities. In recent years several attempts have been made to measure experimentally the heterojunction band discontinuities, but the reliability of the experimental data has been questioned. Kroemer [23] selected some of the experimental data as most reliable. Margaritondo [24] criticized that selection. The attempts to determine the heterojunction band discontinuities using experimental methods have not yet clarified the situation. So far the understanding of the heterojunction band discontinuities is not satisfactory. As a result of this situation, the Anderson model is still the most widely used discontinuity model [25, 26]. The Anderson model was strongly criticized because it uses the electron affinity, which is a free surface parameter, to describe interface properties [27]. Furthermore, the values of the electron affÉnity, reported by different authors, for a given semiconductor are often discrepant from each other by several tenths of an electronvolt [20, 28], while the order of magnitude of the band discontinuities is some tenths of an electronvolt. The use of the Anderson model reflects the need for a simple model. A simple model for the prediction of valence band discontinuities is proposed in this paper. 2. The model
In the nearly-free-electron model the free electron is the zero approximation and the periodic crystal potential is treated as a perturbation [29]. At the first Brillouin zone boundary, the electron energy is given by E
(kg) = E()(ks) + V0 - V~
(2)
E +(k~) = E °(k s ) + V~)+ V~
(3)
and
where k~ is the wavevector at the zone boundary. The electron energy in the crystal at the zone boundary is either depressed by the amount V~below the free-electron
Band discontinuities in semiconductor heterojunctions
61
parabola or raised by the same a m o u n t above the flee-electron parabola [29]. F r o m eqns. (2) and (3) we have E + (kg) - E - (kg) = 2Vg = Eg
(4)
where in the case of a semiconductor crystal Eg is the energy gap. E ° (k) is the free electron energy
E ° (k) : - k2 2m
(5)
V0 is given by
Vo = (k [ V(r) l k) = f W~* (r) V(r) Vk(r) dr
(6)
where V(r) is the periodic crystal potential. In the case of a semiconductor heterojunction, for the first semiconductor crystal the edge of the valence band is given by 1 Ev, = E ° (kgl) + [I01 - - - Egl 2
(7)
where kg I is the wavevector at the zone boundary. If the direction perpendicular to the heterojunction interface is the [100] direction, then
kgl -
(8)
aj
where al is the lattice constant of the first semiconductor. For the second semiconductor crystal the edge of the valence b a n d is given by 1 Ev2 = E 0 (kg2) + V02 -
E~
_
2
(9)
where kg 2 is the wavevector at the zone boundary. If the direction perpendicular to the heterojunction interface is the [100] direction, then 7t
k~ = - a2
where a2 is the lattice constant of the second semiconductor. The valence band discontinuity is given by
(10)
62
Y. Karafyllidis et al.
1
A E = E~, - E~2 = E ° (kg,) - E ° (kg2) +
Vol -
1/o2 -
--
(Eg I -
Eg2)
(11)
2 The electron energy versus wavevector diagram for the first semiconductor is represented on the left-hand side of Fig. 1, and the same diagram for the second semiconductor is represented o n the fight-hand side o f the same figure. The dashed vertical line represents the heterojunction interface. The relative position of the energy gaps, and the valence and conduction b a n d discontinuities are also shown in Fig. 1. The total energy o f an electron moving freely from semiconductor 1 to semiconductor 2 across the interface remains constant. The potential energy of the electron is changed by an a m o u n t
E
E
/ I g2
/,
k
EglI ~\
// I Vs "n/a2
-
n/a1
1~/a 2
wT/aI
Fig. 1. Energy E vs. wavevector k for an electron in a heterojunction crystal. The dashed line represents the heterojunction interface. The relative position of the energy gaps is shown in this figure.
V~ = V0j - V02
( 12)
and the kinetic energy is changed accordingly. It is beyond the scope o f this paper to go any further. F r o m eqns. (5), (11) and (12) we have ~2 ~v
-
1 (k 2 - k~) -
(Eg, - Eg2) + V~
2m
(13)
2
One more term must be added to eqn. (13) to include the interface dipole contribution to the valence b a n d discontinuity. Let the interface dipole potential be equal to Va. After that we have ~2 /~E v -
1 (k~ -
2m
k~)
-
(Eg 1 -
Eg2) +
Vs +
Va
(14)
7
As stated in the introduction it is difficult to calculate V~ and Vd and it can be done only approximately. Instead of calculating approximately those potentials we
63
Band discontinuities in semiconductor heterojunctions
decided to ignore their contribution. The valence band discontinuity is then given approximately by ~2
1
/kE~ = ~2m (k~ - k~) - -~ (E~= - E~)
(15)
3. Application of the model The proposed model has to be compared with other models. This comparison can only be made on the basis of some experimental data. The Katnani and Margaritondo measurements were chosen to serve this basis. Katnani and Margaritondo [20] performed photoemission measurements of the valence band discontinuity on several interfaces involving group IV, III-V and II-VI semiconductor substrates and germanium or silicon overlayers. They compared [20, 22] the measured valence band discontinuities with the predictions of the Anderson [1], Frensley-Kroemer [3], Harrison [5] and Tersoff [9] models. The proposed model was applied to 21 different heterojunctions; the absolute values of the calculated valence band discontinuities are listed in Table 1. For this calculation the direction perpendicular to the heterojunction interface was the [100] direction, for both semiconductors. In the same table are listed the valence band discontinuities measured by Katnani and Margaritondo. In Table 1 the results are listed so that the value of AEv for the GaAs-Si heterojunction is found at the intersection of the GaAs row with the silicon column. If AEv is positive for the Si-Ge heterojunction, then AEv is taken to be negative for the Ge-Si heterojunction. The average magnitude of the discrepancy between the calculated values and the measured ones is 0.23 eV. To test the Anderson model Katnani and Margaritondo used the electron affinity values that best fitted their experimental data. Even after that the average accuracy TABLE 1 Experimental and calculated values of valence band discontinuities (values in electronvolts) Semiconductor
Experimental Si
Ge Si GaAs GaP GaSb InAs InP InSb CdS CdSe CdTe
Ge
I -0.17 0.05 0.95 0.05 0.15 0.57 0.00 1.55 1.20 0.75
Model predictions Si
Ge
-0.1347 0.17 0.35 0.80 0.20 0.33 0.64 0.00 1.75 1.30 0.85
0.2483 0.5795 0.0630 0.1296 0.2980 0.0948 0.8193 0.5375 0.5990
0.1347 0.3829 0.7141 0.1976 0.0510 0.4327 0.0388 0.0940 0.6722 0.7346
E Karq['vllidis et at,
64
of the Anderson model was not better than 0.25 eV [20]. The average accuracy of both the Frensley-Kroemer and Harrison models is about 0.4 eV [20]. To test the Tersoffmodel another set ofheterojunctions was considered [22]. This set includes heterojunctions between germanium or silicon and GaAs, GaSb, GaP, InAs, InP and the Si-Ge heterojunction. The average accuracy of the Tersoff model is 0.13 eV. The corresponding values for the same set of heterojunctions are 0.30 eV for the Harrison model and 0.47 for the Frensley-Kroemer model [22]. The average accuracy of the proposed model for the same set of heterojunctions is 0.13 eV.
4. Conclusions
The proposed model is simple and clear. Only two properties of semiconductors are used in this model: the lattice constant and the energy gap. For these properties accurate values exist and those values can easily be found in almost any text book. The proposed model is as accurate as the Anderson model. Consequently, this model can replace the Anderson model until something better is found. Compared with other models, the proposed model accuracy is satisfactory in spite of the omission of the potentials Vs and Va in (15). The calculation of those potentials sometimes involves considerable approximations. On the other hand, the order of magnitude of the band discontinuities is some tenths of an electronvolt. Consequently the omission of these potentials is sometimes in favour of the accuracy of the model.
5. References
[1] [2]
[3] [4]
[5] [6] [7] [8] [9]
Anderson, R.L., "Experiments on Ge-As heterojunction", Solid State Electron., vol. 5, p. 341, 1962. Frensley, W.R. and Kroemer, H., "Prediction of semiconductor heterojunction discontinuities from bulk band structures", J. Vac. Sci. Technol., vol. 13, p. 810, 1976. Frensley, W.R. and Kroemer, H., "Theory of the energy-band lineup at an abrupt semiconductor heterojunction", Phys. Rev. B, vol. 16, p. 2642, 1977. Baraff, G.A., Appelbaum, J.A. and Hamann, D.R., "Self-consistent calculation of the electronic structure at an abrupt GaAs-Ge interface", Phys. Rev. Lett., vol. 38, p. 237, 1977. Harrison, W.A., "Elementary theory of heterojunctions", J. Vac. Sci. Teehnol.,vol. 14, p. 1016, 1977. v Ross, O., "Theory of extrinsic and intrinsic heterojunctions in thermal equilibrium", Solid State Electron., vol. 23, p. 1069, 1980. Chatterjee, A. and Marshak, A.H., "Theory of abrupt heterojunctions in equilibrium", Solid State Electron., vol. 24, p. 1111, 1981. Katnani, A.D. and Margaritondo, G., "Empirical rule to predict heterojunction band discontinuities", J. Appl. Phys., vol. 54, p. 2522, 1983. Tersoff, J., "Theory of semiconductor heterojunctions: The role of quantum dipoles", Phys. Rev. B, vol. 30, p. 4874, 1984.
Band discontinuities in semiconductor heterojunctions
65
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