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Electronic States of Semiconductor Compounds and Alloys
Electronic States of Semiconductor Compounds and Alloys$ JC Phillips, Rutgers University, Piscataway, NJ, USA r 2016 Elsevier Inc. All rights reserved.
1 2 3 4 5 6 7 8 9 10 11 12 13 Further Reading
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Chemical Bonding in Octet Compounds Pseudopotential Form Factors Ionicity Charge Densities Energy Bands of Octet Compounds Elastic Constants and Phonons Fundamental Optical Spectra Electronic States of Defects Surfaces and Interfaces PbS or ANB10 N Compounds Ternary Octet Compounds Electronic Structures of Alloys Applications
1 2 3 4 4 5 6 7 9 9 9 10 10 10
Chemical Bonding in Octet Compounds
Because of their tetrahedrally coordinated structures, the electronic states of octet compounds and alloys can be described approximately as superpositions of Pauling's sp3 hybrid atomic orbitals (Figure 1). In a diatomic structure there are two kinds of such hybrids, those pointing toward and away from the nearest neighbors, corresponding to bonding (valence) and antibonding (conduction) band states. When the two atoms in the unit cell are different and have different electronegativities, the covalent bonds found in elemental semiconductors (Si, Ge) become partially ionic. For many decades, the question of how to define the ionic and covalent fractions of chemical bonds was open and considered by many to be unanswerable, but this problem was solved for the octet compounds by Phillips' dielectric theory. It turns out that not only the electronic states but also almost all the physical properties of octet semiconductors are smooth functions (usually linear) of ionicity defined dielectrically, thus microscopically justifying much of Pauling's heuristic discussion of chemical bonding in molecules and solids, and explaining its many great successes. In semiconductors, there are many energy gaps between the valence and conduction band states, the most important one for transport properties being the smallest gap. However, all the gaps follow chemical trends that are best described in terms of the average gap Eg defined dielectrically εð0Þ ¼ 1 þ Aðℏωp =Eg Þ2
½1
where A is a fixed number of order unity that corrects for band dispersion, and ωp is the plasma frequency defined in terms of the valence electron density n0 by ω2p ¼ 4πn0 e2 =m
½2
Of course, in most cases, the average gap Eg is much larger than the smallest gap Emin ¼ E0. For example, in (Si, Ge) the gaps in electron volts are Eg ¼ (12.5,11.5) and Eg ¼ (1.1,0.7). All the gaps follow chemical trends as functions of ionicity similar to those followed by the average gap, which is decomposed into its covalent or homopolar (Eh) and ionic or charge transfer (C) parts according to the Cartesian relation E2g ¼ E2h þ C2
½3
The covalent gap Eh is a function only of the lattice constant, and is easily interpolated for compounds from the values of C, Si, Ge, and (gray) Sn. The ionic gap C is then obtained either from the known values of ε(0) in the octet compounds, or from the analytic systematics of Eh and C obtained from other compounds. In horizontal series such as Ge, GaAs, and ZnSe, composed of elements
☆
Change History: February 2015. J.C. Phillips added Keywords, PACS indices removed, and one new reference added.
Reference Module in Materials Science and Materials Engineering
doi:10.1016/B978-0-12-803581-8.01065-1
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Electronic States of Semiconductor Compounds and Alloys
Figure 1 (a) Directed bonding and (b) antibonding orbitals in ANB8–N semiconductors. The A atoms are represented by smaller spheres and are more electropositive, while the B atoms are represented by larger spheres and are more electronegative. The bonding orbitals have lower energy both because they are directed toward their nearest neighbors (covalency) and because they are centered predominantly on the more electronegative ion (ionic effect). These apparently qualitative effects are represented quantitatively (accuracy B1%) by the energy gaps Eh and C. Reproduced from Phillips, J.C., 1973. Bonds and Bands in Semiconductors. New York: Academic Press, with permission from Elsevier.
from the same period, where the lattice constant changes are very small, it is found that C is proportional to N – 4. Thus, C behaves as a dielectrically defined electronegativity difference.
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Pseudopotential Form Factors
The actual wave functions in semiconductors are not described accurately by the hybrid atomic orbitals shown in Figure 1, but they are described very accurately outside the atomic cores by superpositions of several hundred plane waves. The surprising aspect of these wave functions is that they can be derived from Cohen–Chelikowsky pseudopotential form factors Vp(q) that depend on only a few parameters/atoms (if they are derived empirically from observed optical spectra) or that can be calculated from atomic potentials very precisely with no adjustable parameters; the values obtained empirically or from first principles are in excellent
Electronic States of Semiconductor Compounds and Alloys
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agreement. That is why one says that pseudopotentials are ‘transparent, reliable and transferable (TRT).’ (Aside: this is a rare, virtually unprecedented, occurrence in quantum theory.) Without pseudopotentials, the knowledge and understanding of electronic states in solids would still be in a primitive, disorganized, and erratically inaccurate state. Comparing the screened pseudopotentials for Al and Si shown in Figure 2, normalized to the free-electron Fermi energy EF, it is to be noted that the form factors Vp(q) of these adjacent elements in the periodic table as functions of x¼ q/kF appear at first to be quite similar, as one would expect from the fact that their atomic cores are isoelectronic. A closer inspection shows that the nodal value q0 is smaller for Si than for Al, and the repulsive overshoot (due to exclusion of valence electrons from the atomic cores) is slightly larger. Most striking, however, are the positions of the reciprocal lattice vectors labeled Gi. In Al, with its close-packed 12fold coordinated structure, the first two G's lie in the repulsive region beyond q0, and covalent bonding is not possible. However, for Si q0 lies well inside kF and is large in magnitude and negative, favoring the formation of covalent bonds.
Figure 2 The electronic structures of solids arise from a complex interplay of electronic interactions at atoms and interatomic interactions between multiply scattered waves. Two such radically different materials as Al, a nearly free-electron metal, and Si, the semiconductor that is the basis of microelectronics, appear to have quite similar TRT pseudopotential form factors. The most striking differences arise because the reciprocal lattice vectors Gi are different for close-packed Al and tetrahedrally coordinated Si, leading to different values for V(Gi), marked by the solid circles. However, a close examination of the form factors reveals other differences as well. Note that the number of valence electrons alone does not account for the differences in physical properties: Pb, like Si, is tetravalent, but it has the same close-packed crystal structure as Al. A close examination of the TRT pseudopotential calculations shows that the G that is largely responsible for this difference is G2 ¼(200), because V(G2) reverses the sign between Si and Pb. Reproduced from Phillips, J.C., 1973. Bonds and Bands in Semiconductors. New York: Academic Press, with permission from Elsevier.
Thus far the atomic orbital and pseudopotential descriptions are quite similar. Now, however, a miracle happens. The covalent and ionic parts of the average energy gap are described quite accurately by Eh =2 ¼ Vs ðG2 Þ þ ½Vs ðG1 Þ2 =DT
½4
C=2 ¼ 2Vs ðG1 ÞVa ðG1 Þ=DT
½5
where DT is a free-electron energy difference that depends only on the lattice constant, and Vs,a(C) ¼ VA(G)7VB(G). These energy gaps are central to understanding chemical trends in the electronic states of ANB8–N compound semiconductors and their alloys. They have been derived from tabulated atomic pseudopotential form factors without elaborate calculation. Their transferability has been tested elegantly for the series Mg2X (X ¼ Si, Ge, Sn), with a triatomic cubic fluorite structure and many new reciprocal lattice vectors, with the results shown in Figure 3. TRT successes of this kind are unique to pseudopotentials.
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Ionicity
Given the covalent or homopolar (Eh) and ionic or charge transfer (C) energy gaps, one defines the bond ionicity as fi ¼ C2 =ðE2h þ C2 Þ. Almost all bulk properties of ANB8–N compound semiconductors and their alloys are linear functions of fi.
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Electronic States of Semiconductor Compounds and Alloys
Figure 3 To test the assumption that pseudopotential form factors are transferable, self-consistent electronic structures were calculated for the X elements shown, and for the compound semiconductors Mg2X. Values of the form factors derived from elemental band structures are represented by filled circles, while values for the compounds are marked by the empty circles. The overall consistency is spectacular, and validates the TRT description. Reproduced with permission from Au-Yang, M.Y., Cohen, M.L., 1969. Electronic structure and optical properties of Mg2Si, Mg2Ge, and Mg2Sn. Physical Review 178, 1358–1364; © American Physical Society.
(Strictly speaking, this statement has been tested only for 16 compounds. These are: Si, Ge, AlSb, GaP, GaAs, GaSb, InP, InAs, InSb, ZnS, ZnSe, ZnTe, CdS, CdSe, CdTe, and CuCl. High-quality samples are gradually becoming available for AlN and GaN, and these should soon join the list.) The largest database against which the bond ionicity has been tested is 70 octet compounds that are either tetrahedrally or octahedrally coordinated. Success in predicting this fundamental covalent/ionic structural dichotomy had eluded all other attempts, but the parameter-free dielectric approach is completely successful, as shown in Figure 4.
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Charge Densities
The pseudoatom charge densities calculated using several hundred plane waves are in excellent agreement with X-ray results, after subtraction of the chemically uninteresting core background. A surprising feature of the charge densities is the appearance of a bond charge between the atoms. In elemental cases (diamond, Si, Ge, gray Sn), this charge is, of course, situated halfway between two atoms, but for compounds it moves closer to the anion. The displacement is linear in the ionicity fi. Alternatively, one can integrate the excess charge density above the background. As shown in Figure 5, this excess charge extrapolates to zero as fi approaches the critical ionicity derived in Figure 4.
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Energy Bands of Octet Compounds
The energy bands of compound semiconductors have been calculated with great precision over a wide energy range (detailed comparisons with experiment cover an energy range of 10–20 eV) using the pseudopotential form factors illustrated in the next paragraph. The same form atomic factor occurs in many compounds, and the variations from one compound to another for the same atom are so small as to be scarcely perceptible. Two kinds of form factors have been used: semi-empirical and self-consistent
Electronic States of Semiconductor Compounds and Alloys
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Figure 4 Values of Eh and C for B70 octet compounds, including both tetrahedral semiconductors and octahedral insulators. The line corresponding to fi ¼ C 2 =ðEh2 þ C 2 Þ ¼ 0:785 (5) separates the two classes exactly. Reproduced from Phillips, J.C., 1973. Bonds and Bands in Semiconductors. New York: Academic Press, with permission from Elsevier.
first-principle, including in some cases corrections for many-electron correlation and exchange. To illustrate the differences between elemental band structures and compound band structures, one can compare (Figure 6) the energy bands of Ge and GaAs, which are quite similar because the main change in the crystal potential comes from the ionic potential C produced by the difference between Ga and As. The most important difference is that the smallest energy gap Emin ¼ E0 is indirect in the elemental semiconductors Si and Ge, that is, the valence and conduction band edges lie at different points in the crystal momentum space k, but it is direct in most compound semiconductors, where both edges fall at k¼ 0. Almost all optical applications, including the most important one, multiquantum well lasers, require a direct optical gap. The gap at k¼ 0 is denoted conventionally by, that is associated with states near the (1 1 1) Brillouin zone face by E1, and that is associated with states near the (2 0 0) Brillouin zone face by E2.
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Elastic Constants and Phonons
The pseudopotential theory yields phonon spectra ωi(q) for semiconductors by a convolution of the pseudopotential form factors in the conventional dynamical matrix theory, but this must be done on a case-by-case basis. More interesting are the general trends in the normalized shear modulus, shown in Figure 7; it is nearly linear in the ionicity fi, except for collapsing in the Cu halides as the ionicity approaches the critical value 0.785 (Figure 4) for transition to the rock salt structure.
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Electronic States of Semiconductor Compounds and Alloys
Figure 5 The bond charge as a function of dielectric ionicity for two isoelectronic series of octet semiconductors (third and fourth periods) clearly extrapolates to zero for the critical ionicity derived in Figure 4. Reproduced with permission from Walter, J.P., Cohen, M.L., 1971. Physical Review 4B, 1877; © American Physical Society.
Figure 6 The energy bands of Ge and GaAs over an energy range of B20 eV are quite similar; all the important differences in their physical properties stem from scarcely perceptible shifts in energies of their lowest conduction bands at the symmetry points Γ, X, and L in crystal momentum space k. These shifts, B0.3 eV, are accurately predicted by TRT pseudopotential band structures, as these are calculated with an accuracy B0.03 eV over the entire range of 20 eV. Accuracies of this kind are generally not achieved in quantum calculations even for polyvalent atomic spectra. Reproduced with permission from Chelikowsky, J. R., Cohen, M.L., 1973. Physical Review Letters 31, 1582; © American Physical Society.
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Fundamental Optical Spectra
At first sight, the optical spectra of semiconductors seem to be uninformative, because the energy bandwidth is large compared to the spacing of structures generated by crystal symmetry. However, modern TRT pseudopotential calculations yield very accurate
Electronic States of Semiconductor Compounds and Alloys
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Figure 7 Increasing ionicity in octet compound semiconductors reduces the bond charge (Figure 5) and leads to the archetypical tetrahedraloctahedral phase transition (Figure 4). It also dramatically reduces the resistance of the open tetrahedral structure to shear. As shown here, this reduction appears to extrapolate to zero at fi ¼1, but in the Cu halides, there is a dramatic collapse as fi approaches the critical value 0.785. This means that although there is a large volume change at the transition, so that it is technically first order, quantum mechanically it appears to be continuous, that is, second order, presumably because the bond charge (Figure 5) extrapolates to zero for the critical ionicity derived in Figure 4. Reproduced from Phillips, J.C., 1973. Bonds and Bands in Semiconductors. New York: Academic Press, with permission from Elsevier.
energy bands, and oscillator strengths are routinely calculated including corrections for the Coulombic final state (excitonic) interactions. The resulting optical spectra agree so well with experiment (see Figure 8 for GaAs) that often the discrepancies result merely from experimental problems with sample surfaces. Moreover, lock-in modulation methods uncover the fine structure in the reflectivity spectra (see Figure 9 for GaAs) with an improvement in resolution of at least a factor of 10, as well as doubling the number of identified symmetry transitions. The static dielectric constant ε1(0) is connected to the absorptive dielectric function ε2(ω) through the Kramers–Kronig relation as an integral over ε2(ω). Thus if the average energy gap is E2R ¼ E2h þ C2 , then all the symmetry-derived peaks in ε2(ω) should follow similar trends, using the same values of the dielectric electronegativity difference C. This is indeed the case; there are small downward shifts in peak energies relative to one-electron calculations, but most of these are already incorporated into the definition of C from ε1(0), which itself includes similar corrections.
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Electronic States of Defects
The most common defects in compound semiconductors are vacancies and antisite defects (i.e., A ions on B sites). The energies of formation and the electronic states associated with these defects depend on the position of the Fermi energy (the doping level) and strain. These energies can be estimated very accurately using pseudopotentials. The antisite defect energies are at least 10 times smaller than the vacancy energies, and are comparable to thermal energies at growth temperatures. Thus antisite defects are the most important defects in most compound semiconductors. An example is strain at quantum well interfaces. Macroscopically, one would expect that such interfaces would be most stable when strain-free. In practice, a small amount of interfacial misfit is found
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Electronic States of Semiconductor Compounds and Alloys
Figure 8 TRT pseudopotential calculations of the reflectance spectrum of GaAs (including oscillator strength redistribution to lower energies caused by interactions between excited electrons and holes) are in excellent agreement with experiment. Reproduced with permission from Rohlfing, M., Louie, S.G., 1998. Electron–hole excitations in semiconductors and insulators. Physical Review Letters 81, 2312; © American Physical Society.
Figure 9 The derivative of the reflectance spectrum shown in Figure 8 has also been calculated by one-electron TRT pseudopotentials; this derivative is less sensitive to many-electron interactions than the original spectrum, and the calculated results are in spectacularly good agreement with experiment. Reproduced from Zucca, R.L.R., Walter, J.P., Shen, Y.R., Cohen, M.L., 1970. Solid State Communications 8(8), 627–632, with permission from Elsevier.
to be optimal. This misfit may suppress diffusion of antisite defects; it is larger, for example, in InP-based cases (large antisite size difference) than in GaAs-based cases (small antisite size difference). Empirically, in manufacturing devices, the presence of these defects is reduced by the use of proprietary (confidential) methods. In other words, here science and technology are separated by important business considerations.
Electronic States of Semiconductor Compounds and Alloys
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Surfaces and Interfaces
Another technologically important topic is the nature of the electronic states at surfaces and interfaces. The chemistry of these states is different in compound semiconductors than in elemental semiconductors. In the most important case – Schottky barriers between metals and compound semiconductors such as GaAs – the interface states for a midgap Fermi energy are taken mainly from the metal and the conduction band of the semiconductor.
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PbS or ANB10N Compounds
The two extra electrons outside the sp3 octets are an extra s2 lone pair associated with the cation: other examples of these compounds are GeTe, SnSe, SnTe, PbSe, and PbTe, as well as the elemental materials As, Sb, and Bi. The crystal structure is usually rock salt (sixfold coordinated). The weakly bound s2 electrons produce strong absorption in the infrared region (B0.2 eV). The spin–orbit splittings of heavy elements such as Pb and Te are larger than the energy gaps, and produce unusual band structures calculated again very accurately by pseudopotential methods, in excellent agreement with optical data. The band structures of these compounds can be quite complex, as illustrated for the example of SnTe in Figure 10. Again TRT pseudopotential derivative reflectivity spectra are in excellent agreement with experiment, as shown for the example of PbSe in Figure 11.
Figure 10 The structure of the energy bands of SnTe near the energy gap is complex. The complexity is caused by a combination of factors, including inversion of energy levels relative to octet compounds by the presence of lone pair electrons, and very large spin–orbit splittings. Reproduced from Tung, Y.W., Cohen, M.L., 1969. Physics Letters A 29(5), 236–237, with permission from Elsevier.
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Ternary Octet Compounds
Further refinements and additional tunability (sometimes called band-structure engineering) are possible in two ways: through alloying (next paragraph) and through supercell formation (ternary compounds), the most important example being the octet chalcopyrite compounds AIIBIVCV2 (CuBIIICVI 2 ) which are pseudo III–V (II–VI) compounds with tetragonally distorted structures. Phase diagrams and accordingly sample preparation are complex in the chalcopyrites, in order for the cations to order on their respective sublattices in a unique polytype. Thus, the TRT pseudopotential theory has played a very important part in the development of these materials.
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Electronic States of Semiconductor Compounds and Alloys
Figure 11 The modulated (derivative) reflectivity spectra of ANB10N compounds calculated by one-electron TRT pseudopotentials are also in spectacularly good agreement with experiment. The example shown here is PbSe. Reproduced with permission from Kohn, S.E., Yu, P.Y., Petroff, Y., et al., 1973. Physical Review B 8, 1477–1488; © American Physical Society.
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Electronic Structures of Alloys
The most important optical materials in practice (see following section) are semiconductor III–V alloys. The energy bands and optical transitions of these alloys are given approximately by linear interpolation between the corresponding structures of their end point pure compounds, as one would expect from a random distribution of ions on the alloying sublattices (ideal solution). (Note that ions alloyed on a sublattice are second nearest neighbors and hence interact much more weakly than atoms alloyed on the entire lattice, for instance, in (Si, Ge) alloys.) However, small curvatures are observable, and are of great practical importance for bandgap tunability; the curvatures are successfully predicted by treating the alloys as regular solutions.
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Applications
As noted above, many compound semiconductors have electronic structures with valence and conduction band edges at k¼ 0 (the Brillouin zone center, often denoted conventionally by G), so that their lowest energy optical transitions are direct, a necessary condition for semiconductor quantum well lasers. The quantum wells themselves enhance the overlap between excited electron and hole wave functions, increasing oscillator strengths, lowering lasing thresholds, and increasing laser intensities. The transparency window of silica-based optical fibers is fixed by nature in the near infrared, and because of bandgap engineering it is possible to design semiconductor lasers that emit light at any desired wavelength in this window. The actual alloys used are quaternary (In, Ga) (P, As) alloys, and it is semiconductor multiquantum well lasers based on these alloys that have revolutionized civilization in the twenty-first century. The lifetimes of these lasers are fixed mainly by the quality of interfaces (semiconductor and semiconductor-metal). Other compound semiconductors are utilized in many other applications as far infrared detectors, and light emitting diodes are now manufactured over a wide range of colors up to the ultraviolet ((InGa)N).
Further Reading Benedict, L.X., Wethkamp, T., Wilmers, K., et al., 1999. Dielectric function of wurtzite GaN and AlN thin films. Solid State Communication 112, 129. Bonapasta, A.A., Giannozzi, P., 2000. Effects of strain and local charge on the formation of deep defects in III−V ternary alloys. Physical Review Letters 84, 3923. Brust, D., Phillips, J.C., Bassani, F., 1962. Critical points and ultraviolet reflectivity of semiconductors. Physical Review Letters 9, 94. Cohen, M.L., Chelikowsky, J.R., 1988. Electronic Structure and Optical Properties of Semiconductors. Heidelberg: Springer. Kroemer, H., 2001. Nobel lecture quasielectric fields and band offsets: Teaching electrons new tricks. Reviews Modern Physics 73, 783. Mahadevan, P., Zunger, A., 2002. Room-temperature ferromagnetism in Mn-doped semiconducting CdGeP2. Physiological Review Letters 88, 047205. Maxisch, T., Binggeli, N., Baldereschi, A., 2003. Intermetallic bonds and midgap interface states at epitaxial Al/GaAs(0 0 1) junctions. Physical Reviews B 67, 125315. Nagahama, S., Yanamoto, T., Sano, M., Mukai, T., 2002. Blue−violet nitride lasers. Physica Status Solidi A 194, 423. Pauling, L., 1960. Nature of the Chemical Bond. Ithaca, NY: Cornell University Press. Phillips, J.C., Lucovsky, G., 2009. Bonds and Bands in Semiconductors. New York, NY: Momentum Press. Wei, S.H., Zhang, S.B., Zunger, A., 1999. Band structure and stability of zinc-blende-based semiconductor polytypes. Physiological Reviews B 59, R2478.
1 2 3 4 5 6 7 8 9 10 11 12 13 Further Reading
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Chemical Bonding in Octet Compounds Pseudopotential Form Factors Ionicity Charge Densities Energy Bands of Octet Compounds Elastic Constants and Phonons Fundamental Optical Spectra Electronic States of Defects Surfaces and Interfaces PbS or ANB10 N Compounds Ternary Octet Compounds Electronic Structures of Alloys Applications
1 2 3 4 4 5 6 7 9 9 9 10 10 10
Chemical Bonding in Octet Compounds
Because of their tetrahedrally coordinated structures, the electronic states of octet compounds and alloys can be described approximately as superpositions of Pauling's sp3 hybrid atomic orbitals (Figure 1). In a diatomic structure there are two kinds of such hybrids, those pointing toward and away from the nearest neighbors, corresponding to bonding (valence) and antibonding (conduction) band states. When the two atoms in the unit cell are different and have different electronegativities, the covalent bonds found in elemental semiconductors (Si, Ge) become partially ionic. For many decades, the question of how to define the ionic and covalent fractions of chemical bonds was open and considered by many to be unanswerable, but this problem was solved for the octet compounds by Phillips' dielectric theory. It turns out that not only the electronic states but also almost all the physical properties of octet semiconductors are smooth functions (usually linear) of ionicity defined dielectrically, thus microscopically justifying much of Pauling's heuristic discussion of chemical bonding in molecules and solids, and explaining its many great successes. In semiconductors, there are many energy gaps between the valence and conduction band states, the most important one for transport properties being the smallest gap. However, all the gaps follow chemical trends that are best described in terms of the average gap Eg defined dielectrically εð0Þ ¼ 1 þ Aðℏωp =Eg Þ2
½1
where A is a fixed number of order unity that corrects for band dispersion, and ωp is the plasma frequency defined in terms of the valence electron density n0 by ω2p ¼ 4πn0 e2 =m
½2
Of course, in most cases, the average gap Eg is much larger than the smallest gap Emin ¼ E0. For example, in (Si, Ge) the gaps in electron volts are Eg ¼ (12.5,11.5) and Eg ¼ (1.1,0.7). All the gaps follow chemical trends as functions of ionicity similar to those followed by the average gap, which is decomposed into its covalent or homopolar (Eh) and ionic or charge transfer (C) parts according to the Cartesian relation E2g ¼ E2h þ C2
½3
The covalent gap Eh is a function only of the lattice constant, and is easily interpolated for compounds from the values of C, Si, Ge, and (gray) Sn. The ionic gap C is then obtained either from the known values of ε(0) in the octet compounds, or from the analytic systematics of Eh and C obtained from other compounds. In horizontal series such as Ge, GaAs, and ZnSe, composed of elements
☆
Change History: February 2015. J.C. Phillips added Keywords, PACS indices removed, and one new reference added.
Reference Module in Materials Science and Materials Engineering
doi:10.1016/B978-0-12-803581-8.01065-1
1
2
Electronic States of Semiconductor Compounds and Alloys
Figure 1 (a) Directed bonding and (b) antibonding orbitals in ANB8–N semiconductors. The A atoms are represented by smaller spheres and are more electropositive, while the B atoms are represented by larger spheres and are more electronegative. The bonding orbitals have lower energy both because they are directed toward their nearest neighbors (covalency) and because they are centered predominantly on the more electronegative ion (ionic effect). These apparently qualitative effects are represented quantitatively (accuracy B1%) by the energy gaps Eh and C. Reproduced from Phillips, J.C., 1973. Bonds and Bands in Semiconductors. New York: Academic Press, with permission from Elsevier.
from the same period, where the lattice constant changes are very small, it is found that C is proportional to N – 4. Thus, C behaves as a dielectrically defined electronegativity difference.
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Pseudopotential Form Factors
The actual wave functions in semiconductors are not described accurately by the hybrid atomic orbitals shown in Figure 1, but they are described very accurately outside the atomic cores by superpositions of several hundred plane waves. The surprising aspect of these wave functions is that they can be derived from Cohen–Chelikowsky pseudopotential form factors Vp(q) that depend on only a few parameters/atoms (if they are derived empirically from observed optical spectra) or that can be calculated from atomic potentials very precisely with no adjustable parameters; the values obtained empirically or from first principles are in excellent
Electronic States of Semiconductor Compounds and Alloys
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agreement. That is why one says that pseudopotentials are ‘transparent, reliable and transferable (TRT).’ (Aside: this is a rare, virtually unprecedented, occurrence in quantum theory.) Without pseudopotentials, the knowledge and understanding of electronic states in solids would still be in a primitive, disorganized, and erratically inaccurate state. Comparing the screened pseudopotentials for Al and Si shown in Figure 2, normalized to the free-electron Fermi energy EF, it is to be noted that the form factors Vp(q) of these adjacent elements in the periodic table as functions of x¼ q/kF appear at first to be quite similar, as one would expect from the fact that their atomic cores are isoelectronic. A closer inspection shows that the nodal value q0 is smaller for Si than for Al, and the repulsive overshoot (due to exclusion of valence electrons from the atomic cores) is slightly larger. Most striking, however, are the positions of the reciprocal lattice vectors labeled Gi. In Al, with its close-packed 12fold coordinated structure, the first two G's lie in the repulsive region beyond q0, and covalent bonding is not possible. However, for Si q0 lies well inside kF and is large in magnitude and negative, favoring the formation of covalent bonds.
Figure 2 The electronic structures of solids arise from a complex interplay of electronic interactions at atoms and interatomic interactions between multiply scattered waves. Two such radically different materials as Al, a nearly free-electron metal, and Si, the semiconductor that is the basis of microelectronics, appear to have quite similar TRT pseudopotential form factors. The most striking differences arise because the reciprocal lattice vectors Gi are different for close-packed Al and tetrahedrally coordinated Si, leading to different values for V(Gi), marked by the solid circles. However, a close examination of the form factors reveals other differences as well. Note that the number of valence electrons alone does not account for the differences in physical properties: Pb, like Si, is tetravalent, but it has the same close-packed crystal structure as Al. A close examination of the TRT pseudopotential calculations shows that the G that is largely responsible for this difference is G2 ¼(200), because V(G2) reverses the sign between Si and Pb. Reproduced from Phillips, J.C., 1973. Bonds and Bands in Semiconductors. New York: Academic Press, with permission from Elsevier.
Thus far the atomic orbital and pseudopotential descriptions are quite similar. Now, however, a miracle happens. The covalent and ionic parts of the average energy gap are described quite accurately by Eh =2 ¼ Vs ðG2 Þ þ ½Vs ðG1 Þ2 =DT
½4
C=2 ¼ 2Vs ðG1 ÞVa ðG1 Þ=DT
½5
where DT is a free-electron energy difference that depends only on the lattice constant, and Vs,a(C) ¼ VA(G)7VB(G). These energy gaps are central to understanding chemical trends in the electronic states of ANB8–N compound semiconductors and their alloys. They have been derived from tabulated atomic pseudopotential form factors without elaborate calculation. Their transferability has been tested elegantly for the series Mg2X (X ¼ Si, Ge, Sn), with a triatomic cubic fluorite structure and many new reciprocal lattice vectors, with the results shown in Figure 3. TRT successes of this kind are unique to pseudopotentials.
3
Ionicity
Given the covalent or homopolar (Eh) and ionic or charge transfer (C) energy gaps, one defines the bond ionicity as fi ¼ C2 =ðE2h þ C2 Þ. Almost all bulk properties of ANB8–N compound semiconductors and their alloys are linear functions of fi.
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Electronic States of Semiconductor Compounds and Alloys
Figure 3 To test the assumption that pseudopotential form factors are transferable, self-consistent electronic structures were calculated for the X elements shown, and for the compound semiconductors Mg2X. Values of the form factors derived from elemental band structures are represented by filled circles, while values for the compounds are marked by the empty circles. The overall consistency is spectacular, and validates the TRT description. Reproduced with permission from Au-Yang, M.Y., Cohen, M.L., 1969. Electronic structure and optical properties of Mg2Si, Mg2Ge, and Mg2Sn. Physical Review 178, 1358–1364; © American Physical Society.
(Strictly speaking, this statement has been tested only for 16 compounds. These are: Si, Ge, AlSb, GaP, GaAs, GaSb, InP, InAs, InSb, ZnS, ZnSe, ZnTe, CdS, CdSe, CdTe, and CuCl. High-quality samples are gradually becoming available for AlN and GaN, and these should soon join the list.) The largest database against which the bond ionicity has been tested is 70 octet compounds that are either tetrahedrally or octahedrally coordinated. Success in predicting this fundamental covalent/ionic structural dichotomy had eluded all other attempts, but the parameter-free dielectric approach is completely successful, as shown in Figure 4.
4
Charge Densities
The pseudoatom charge densities calculated using several hundred plane waves are in excellent agreement with X-ray results, after subtraction of the chemically uninteresting core background. A surprising feature of the charge densities is the appearance of a bond charge between the atoms. In elemental cases (diamond, Si, Ge, gray Sn), this charge is, of course, situated halfway between two atoms, but for compounds it moves closer to the anion. The displacement is linear in the ionicity fi. Alternatively, one can integrate the excess charge density above the background. As shown in Figure 5, this excess charge extrapolates to zero as fi approaches the critical ionicity derived in Figure 4.
5
Energy Bands of Octet Compounds
The energy bands of compound semiconductors have been calculated with great precision over a wide energy range (detailed comparisons with experiment cover an energy range of 10–20 eV) using the pseudopotential form factors illustrated in the next paragraph. The same form atomic factor occurs in many compounds, and the variations from one compound to another for the same atom are so small as to be scarcely perceptible. Two kinds of form factors have been used: semi-empirical and self-consistent
Electronic States of Semiconductor Compounds and Alloys
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Figure 4 Values of Eh and C for B70 octet compounds, including both tetrahedral semiconductors and octahedral insulators. The line corresponding to fi ¼ C 2 =ðEh2 þ C 2 Þ ¼ 0:785 (5) separates the two classes exactly. Reproduced from Phillips, J.C., 1973. Bonds and Bands in Semiconductors. New York: Academic Press, with permission from Elsevier.
first-principle, including in some cases corrections for many-electron correlation and exchange. To illustrate the differences between elemental band structures and compound band structures, one can compare (Figure 6) the energy bands of Ge and GaAs, which are quite similar because the main change in the crystal potential comes from the ionic potential C produced by the difference between Ga and As. The most important difference is that the smallest energy gap Emin ¼ E0 is indirect in the elemental semiconductors Si and Ge, that is, the valence and conduction band edges lie at different points in the crystal momentum space k, but it is direct in most compound semiconductors, where both edges fall at k¼ 0. Almost all optical applications, including the most important one, multiquantum well lasers, require a direct optical gap. The gap at k¼ 0 is denoted conventionally by, that is associated with states near the (1 1 1) Brillouin zone face by E1, and that is associated with states near the (2 0 0) Brillouin zone face by E2.
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Elastic Constants and Phonons
The pseudopotential theory yields phonon spectra ωi(q) for semiconductors by a convolution of the pseudopotential form factors in the conventional dynamical matrix theory, but this must be done on a case-by-case basis. More interesting are the general trends in the normalized shear modulus, shown in Figure 7; it is nearly linear in the ionicity fi, except for collapsing in the Cu halides as the ionicity approaches the critical value 0.785 (Figure 4) for transition to the rock salt structure.
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Electronic States of Semiconductor Compounds and Alloys
Figure 5 The bond charge as a function of dielectric ionicity for two isoelectronic series of octet semiconductors (third and fourth periods) clearly extrapolates to zero for the critical ionicity derived in Figure 4. Reproduced with permission from Walter, J.P., Cohen, M.L., 1971. Physical Review 4B, 1877; © American Physical Society.
Figure 6 The energy bands of Ge and GaAs over an energy range of B20 eV are quite similar; all the important differences in their physical properties stem from scarcely perceptible shifts in energies of their lowest conduction bands at the symmetry points Γ, X, and L in crystal momentum space k. These shifts, B0.3 eV, are accurately predicted by TRT pseudopotential band structures, as these are calculated with an accuracy B0.03 eV over the entire range of 20 eV. Accuracies of this kind are generally not achieved in quantum calculations even for polyvalent atomic spectra. Reproduced with permission from Chelikowsky, J. R., Cohen, M.L., 1973. Physical Review Letters 31, 1582; © American Physical Society.
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Fundamental Optical Spectra
At first sight, the optical spectra of semiconductors seem to be uninformative, because the energy bandwidth is large compared to the spacing of structures generated by crystal symmetry. However, modern TRT pseudopotential calculations yield very accurate
Electronic States of Semiconductor Compounds and Alloys
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Figure 7 Increasing ionicity in octet compound semiconductors reduces the bond charge (Figure 5) and leads to the archetypical tetrahedraloctahedral phase transition (Figure 4). It also dramatically reduces the resistance of the open tetrahedral structure to shear. As shown here, this reduction appears to extrapolate to zero at fi ¼1, but in the Cu halides, there is a dramatic collapse as fi approaches the critical value 0.785. This means that although there is a large volume change at the transition, so that it is technically first order, quantum mechanically it appears to be continuous, that is, second order, presumably because the bond charge (Figure 5) extrapolates to zero for the critical ionicity derived in Figure 4. Reproduced from Phillips, J.C., 1973. Bonds and Bands in Semiconductors. New York: Academic Press, with permission from Elsevier.
energy bands, and oscillator strengths are routinely calculated including corrections for the Coulombic final state (excitonic) interactions. The resulting optical spectra agree so well with experiment (see Figure 8 for GaAs) that often the discrepancies result merely from experimental problems with sample surfaces. Moreover, lock-in modulation methods uncover the fine structure in the reflectivity spectra (see Figure 9 for GaAs) with an improvement in resolution of at least a factor of 10, as well as doubling the number of identified symmetry transitions. The static dielectric constant ε1(0) is connected to the absorptive dielectric function ε2(ω) through the Kramers–Kronig relation as an integral over ε2(ω). Thus if the average energy gap is E2R ¼ E2h þ C2 , then all the symmetry-derived peaks in ε2(ω) should follow similar trends, using the same values of the dielectric electronegativity difference C. This is indeed the case; there are small downward shifts in peak energies relative to one-electron calculations, but most of these are already incorporated into the definition of C from ε1(0), which itself includes similar corrections.
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Electronic States of Defects
The most common defects in compound semiconductors are vacancies and antisite defects (i.e., A ions on B sites). The energies of formation and the electronic states associated with these defects depend on the position of the Fermi energy (the doping level) and strain. These energies can be estimated very accurately using pseudopotentials. The antisite defect energies are at least 10 times smaller than the vacancy energies, and are comparable to thermal energies at growth temperatures. Thus antisite defects are the most important defects in most compound semiconductors. An example is strain at quantum well interfaces. Macroscopically, one would expect that such interfaces would be most stable when strain-free. In practice, a small amount of interfacial misfit is found
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Electronic States of Semiconductor Compounds and Alloys
Figure 8 TRT pseudopotential calculations of the reflectance spectrum of GaAs (including oscillator strength redistribution to lower energies caused by interactions between excited electrons and holes) are in excellent agreement with experiment. Reproduced with permission from Rohlfing, M., Louie, S.G., 1998. Electron–hole excitations in semiconductors and insulators. Physical Review Letters 81, 2312; © American Physical Society.
Figure 9 The derivative of the reflectance spectrum shown in Figure 8 has also been calculated by one-electron TRT pseudopotentials; this derivative is less sensitive to many-electron interactions than the original spectrum, and the calculated results are in spectacularly good agreement with experiment. Reproduced from Zucca, R.L.R., Walter, J.P., Shen, Y.R., Cohen, M.L., 1970. Solid State Communications 8(8), 627–632, with permission from Elsevier.
to be optimal. This misfit may suppress diffusion of antisite defects; it is larger, for example, in InP-based cases (large antisite size difference) than in GaAs-based cases (small antisite size difference). Empirically, in manufacturing devices, the presence of these defects is reduced by the use of proprietary (confidential) methods. In other words, here science and technology are separated by important business considerations.
Electronic States of Semiconductor Compounds and Alloys
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Surfaces and Interfaces
Another technologically important topic is the nature of the electronic states at surfaces and interfaces. The chemistry of these states is different in compound semiconductors than in elemental semiconductors. In the most important case – Schottky barriers between metals and compound semiconductors such as GaAs – the interface states for a midgap Fermi energy are taken mainly from the metal and the conduction band of the semiconductor.
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PbS or ANB10N Compounds
The two extra electrons outside the sp3 octets are an extra s2 lone pair associated with the cation: other examples of these compounds are GeTe, SnSe, SnTe, PbSe, and PbTe, as well as the elemental materials As, Sb, and Bi. The crystal structure is usually rock salt (sixfold coordinated). The weakly bound s2 electrons produce strong absorption in the infrared region (B0.2 eV). The spin–orbit splittings of heavy elements such as Pb and Te are larger than the energy gaps, and produce unusual band structures calculated again very accurately by pseudopotential methods, in excellent agreement with optical data. The band structures of these compounds can be quite complex, as illustrated for the example of SnTe in Figure 10. Again TRT pseudopotential derivative reflectivity spectra are in excellent agreement with experiment, as shown for the example of PbSe in Figure 11.
Figure 10 The structure of the energy bands of SnTe near the energy gap is complex. The complexity is caused by a combination of factors, including inversion of energy levels relative to octet compounds by the presence of lone pair electrons, and very large spin–orbit splittings. Reproduced from Tung, Y.W., Cohen, M.L., 1969. Physics Letters A 29(5), 236–237, with permission from Elsevier.
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Ternary Octet Compounds
Further refinements and additional tunability (sometimes called band-structure engineering) are possible in two ways: through alloying (next paragraph) and through supercell formation (ternary compounds), the most important example being the octet chalcopyrite compounds AIIBIVCV2 (CuBIIICVI 2 ) which are pseudo III–V (II–VI) compounds with tetragonally distorted structures. Phase diagrams and accordingly sample preparation are complex in the chalcopyrites, in order for the cations to order on their respective sublattices in a unique polytype. Thus, the TRT pseudopotential theory has played a very important part in the development of these materials.
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Electronic States of Semiconductor Compounds and Alloys
Figure 11 The modulated (derivative) reflectivity spectra of ANB10N compounds calculated by one-electron TRT pseudopotentials are also in spectacularly good agreement with experiment. The example shown here is PbSe. Reproduced with permission from Kohn, S.E., Yu, P.Y., Petroff, Y., et al., 1973. Physical Review B 8, 1477–1488; © American Physical Society.
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Electronic Structures of Alloys
The most important optical materials in practice (see following section) are semiconductor III–V alloys. The energy bands and optical transitions of these alloys are given approximately by linear interpolation between the corresponding structures of their end point pure compounds, as one would expect from a random distribution of ions on the alloying sublattices (ideal solution). (Note that ions alloyed on a sublattice are second nearest neighbors and hence interact much more weakly than atoms alloyed on the entire lattice, for instance, in (Si, Ge) alloys.) However, small curvatures are observable, and are of great practical importance for bandgap tunability; the curvatures are successfully predicted by treating the alloys as regular solutions.
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Applications
As noted above, many compound semiconductors have electronic structures with valence and conduction band edges at k¼ 0 (the Brillouin zone center, often denoted conventionally by G), so that their lowest energy optical transitions are direct, a necessary condition for semiconductor quantum well lasers. The quantum wells themselves enhance the overlap between excited electron and hole wave functions, increasing oscillator strengths, lowering lasing thresholds, and increasing laser intensities. The transparency window of silica-based optical fibers is fixed by nature in the near infrared, and because of bandgap engineering it is possible to design semiconductor lasers that emit light at any desired wavelength in this window. The actual alloys used are quaternary (In, Ga) (P, As) alloys, and it is semiconductor multiquantum well lasers based on these alloys that have revolutionized civilization in the twenty-first century. The lifetimes of these lasers are fixed mainly by the quality of interfaces (semiconductor and semiconductor-metal). Other compound semiconductors are utilized in many other applications as far infrared detectors, and light emitting diodes are now manufactured over a wide range of colors up to the ultraviolet ((InGa)N).
Further Reading Benedict, L.X., Wethkamp, T., Wilmers, K., et al., 1999. Dielectric function of wurtzite GaN and AlN thin films. Solid State Communication 112, 129. Bonapasta, A.A., Giannozzi, P., 2000. Effects of strain and local charge on the formation of deep defects in III−V ternary alloys. Physical Review Letters 84, 3923. Brust, D., Phillips, J.C., Bassani, F., 1962. Critical points and ultraviolet reflectivity of semiconductors. Physical Review Letters 9, 94. Cohen, M.L., Chelikowsky, J.R., 1988. Electronic Structure and Optical Properties of Semiconductors. Heidelberg: Springer. Kroemer, H., 2001. Nobel lecture quasielectric fields and band offsets: Teaching electrons new tricks. Reviews Modern Physics 73, 783. Mahadevan, P., Zunger, A., 2002. Room-temperature ferromagnetism in Mn-doped semiconducting CdGeP2. Physiological Review Letters 88, 047205. Maxisch, T., Binggeli, N., Baldereschi, A., 2003. Intermetallic bonds and midgap interface states at epitaxial Al/GaAs(0 0 1) junctions. Physical Reviews B 67, 125315. Nagahama, S., Yanamoto, T., Sano, M., Mukai, T., 2002. Blue−violet nitride lasers. Physica Status Solidi A 194, 423. Pauling, L., 1960. Nature of the Chemical Bond. Ithaca, NY: Cornell University Press. Phillips, J.C., Lucovsky, G., 2009. Bonds and Bands in Semiconductors. New York, NY: Momentum Press. Wei, S.H., Zhang, S.B., Zunger, A., 1999. Band structure and stability of zinc-blende-based semiconductor polytypes. Physiological Reviews B 59, R2478.