H 1 I 11 I I I I H I l M 1 I I I 1 I I

5 10 lfl 15 20 25

Figure 2.7. Dependence of the observed mass spectra lines from Na* nanopartides on the cube root W/3 of the number of atoms N in the duster. The lines are labeled with the index n of their electronic and structural magic numbers obtained from Martin et al. (1990).

approximately equally spaced, with the spacing between the structural magic numbers about 2.6 times that between the electronic ones. This result provides evidence that small clusters tend to satisfy electronic criteria and large structures tend to be structurally determined.

2.1.4. Tetrahedrally Bonded Semiconductor Structures

The type II1-V and type II-VI binary semiconducting compounds, such as GaAs and ZnS, respectively, crystallize with one atom situated on a FCC sublattice at the positions 000, 55O, and 0and the other atom on a second FCC sublattice displaced from the first by the amount along the body diagonal, as shown in Fig. 2.8b. This is called the zinc blende or ZnS structure. It is clear from the figure that each Zn atom (white sphere) is centered in a tetrahedron of S atoms (black spheres), and likewise each S has four similarly situated Zn nearest neighbors. The small half-sized, dashed-line cube delineates one such tetrahedron. The same structure would result if the Zn and S atoms were interchanged.

The elements Si and Ge crystallize in this same structure by having the Si (or Ge) atoms occupying all the sites on the two sublattices, so there are eight identical atoms in the unit cell. This atom arrangement, sketched in Fig. 2.8a, is called the diamond structure. Both Si and Ge have a valence of 4, so from bonding considerations, it is appropriate for each to be bound to four other atoms in the shape of a regular tetrahedron.

In Appendix B we see that Table B.l lists the lattice constants a for various compounds with the zinc blende structure, and Table B.2 provides the crystal radii of their monatomic lattices in which the atoms are uncharged, as well as the ionic radii for ionic compounds in which the atoms are charged. We see from Table B.2 that the negative anions are considerably larger than the positive cations, in accordance with the sketch of the unit cell presented in Fig. 2.9, and this size differential is greater for the ni-V compounds than for the II-VI compounds. However, these size changes for the negative and positive ions tend to balance each other so that the III-V compounds have the same range of lattice constants as the II-VI compounds, with Si and Ge also in this range. Table B.4 gives the molecular masses, and Table B.5 gives the densities of these semiconductors. The three tables B.l, B.4, and B.5 show a regular progression in the values as one goes from left to right in a particular row,

Figure 2.8. Unit cell of the diamond structure (a), which contains only one type of atom, and corresponding unit cell of the zinc blende (sphalerite) structure (b), which contains two atom types. The rods represent the tetrahedral bonds between nearest-neighbor atoms. The small dashed line cube in (b) delineates a tetrahedron. (From G. Burns, Solid State Physics, Academic Press, Boston, 1985, p.148.)

Figure 2.8. Unit cell of the diamond structure (a), which contains only one type of atom, and corresponding unit cell of the zinc blende (sphalerite) structure (b), which contains two atom types. The rods represent the tetrahedral bonds between nearest-neighbor atoms. The small dashed line cube in (b) delineates a tetrahedron. (From G. Burns, Solid State Physics, Academic Press, Boston, 1985, p.148.)

and as one goes from top to bottom in a particular column. This occurs because of the systematic increase in the size of the atoms in each group with increasing atomic number, as indicated in Table B.2.

There are two simple models for representing these AC binary compound structures. For an ionic model the lattice A"~C"+ consists of a FCC arrangement

Figure 2.9. Packing of larger S atoms and smaller Zn atoms in the zinc blende (ZnS) structure. Each atom is centered in a tetrahedron of the other atom type. (From R. W. G. Wyckoff, Crystal Structures, Vol. 1, Wiley, New York, 1963, p. 109.)

of the large anions A"~ with the small cations C"+ located in the tetrahedral sites of the anion FCC lattice. If the anions touch each other, their radii have the value aQ = a/2\/2, where a is the lattice parameter, and the radius aT of die tetrahedral site aT = 0.2247a0 is given by Eq. (2.2). This is the case for the very small Al3+ cation in the AlSb structure. In all other cases the cations in Table B.2 are too large to fit in the tetrahedral site so they push the larger anions further apart, and the latter no longer touch each other, in accordance with Fig. 2.9. In a covalent model for the structure consisting of neutral atoms A and C the atom sizes are comparable, as the data in Table B.2 indicate, and the structure resembles that of Si or Ge. To compare these two models, we note that the distance between atom A at lattice position 000 and its nearest neighbor C at position is equal to | -/3a, and in Table B.3 we compare this crystallographically evaluated distance with the sums of radii of ions A*-, C+ from die ionic model, and with the sums of radii of neutral atoms A and C of the covalent model using die data of Table B.2. We see from the results on Table B.3 that neither model fits the data in all cases, but the neutral atom covalent model is closer to agreement. For comparison purposes we also list corresponding data for several alkali halides and alkaline-earth chalcogenides that crystallize in the cubic rock salt or NaQ structure, and we see that all of these compounds fit the ionic model very well. In these compounds each atom type forms a FCC lattice, with the atoms of one FCC lattice located at octahedral sites of the other lattice. The octahedral site has the radius a^, = 0.41411 a0 given by Eq. (2.1), which is larger than the tetrahedral one of Eq. (2.2).

Since the alkali halide and alkaline-earth chalcogenide compounds fit the ionic model so well, it is significant that neither model fits the structures of the semiconductor compounds. The extent to which the semiconductor crystals exhibit ionic or covalent bonding is not clear from crystallographic data. If the wavefunction describing the bonding is written in the form where the coefficients of the covalent and ionic wavefunction components are normalized then is the fractional covalency and afon is the fractional ionicity of the bond. A chapter (Poole and Farach 2001) in a book by Karl Boer (2001) tabulates the effective charges e* associated with various II-VI and Ifl-V semiconducting compounds, and this effective charge is related to the fractional covalency by the expression

where N = 2 for II-VI and N = 3 for III-V compounds. The fractional charges all lie in the range from 0.43 to 0.49 for the compounds under consideration. Using the e* tabulations in the Boer book and Eq. (2.7), we obtain the fractional covalencies of <4V ~ 0.81 for all the II-VI compounds, and a2mw ~ 0.68 for all the HI V compounds listed in Tables B.l, B.4, B.5, and so on. These values are consistent with the better fit of the covalent model to the crystallographic data for these compounds.

We conclude this section with some observations that will be of use in later chapters. Table B.l shows that the typical compound GaAs has the lattice constant a = 0.565 nm, so the volume of its unit cell is 0.180nm3, corresponding to about 22 of each atom type per cubic nanometer. The distances between atomic layers in the 100, 110, and 111 directions are, respectively, 0.565 nm, 0.400 nm, and 0.326 nm for GaAs. The various III-V semiconducting compounds under discussion form mixed crystals over broad concentration ranges, as do the group of II-VI compounds. In a mixed crystal of the type InxGaj_xAs it is ordinarily safe to assume that Vegard's law is valid, whereby the lattice constant a scales linearly with the concentration parameter x. As a result, we have the expressions a(x) = a(GaAs) + [a(InAs) - a(GaAs)J*

where 0 < x < 1. In the corresponding expression for the mixed semiconductor AljGaj^As the term +0.00 be replaces the term +0.04 lx, so the fraction of lattice mismatch 2|aA1As - Ag^I/Omas + «GaAs) = 0.0018 - 0.18% for this system is quite minimal compared to that pla^ - floaAsl/fainAs + flGaAs) = 0 070 = 7.0%] of the In^Ga^As system, as calculated from Eq. (2.8) [see also Eq. (10-3).] Table B.l gives the lattice constants a for various III-V and II-VI semiconductors with the zinc blende structure.

2.1.5. Lattice Vibrations

We have discussed atoms in a crystal as residing at particular lattice sites, but in reality they undergo continuous fluctuations in the neighborhood of their regular positions in the lattice. These fluctuations arise from the heat or thermal energy in the lattice, and become more pronounced at higher temperatures. Since the atoms are bound together by chemical bonds, the movement of one atom about its site causes the neighboring atoms to respond to this motion. The chemical bonds act like springs that stretch and compress repeatedly during oscillatory motion. The result is that many atoms vibrate in unison, and this collective motion spreads throughout the crystal. Every type of lattice has its own characteristic modes or frequencies of vibration called normal modes, and the overall collective vibrational motion of the lattice is a combination or superposition of many, many normal modes. For a diatomic lattice such as GaAs, there are low-frequency modes called acoustic modes, in which the heavy and light atoms tend to vibrate in phase or in unison with each other, and high-frequency modes called optical modes, in which they tend to vibrate out of phase.

A simple model for analyzing these vibratory modes is a linear chain of alternating atoms with a large mass M and a small mass m joined to each other by springs (~) as follows:

When one of the springs stretches or compresses by an amount Ax, a force is exerted on the adjacent masses with the magnitude C Ax, where C is the spring constant. As the various springs stretch and compress in step with each other, longitudinal modes of vibration take place in which the motion of each atom is along the string direction. Each such normal mode has a particular frequency co and a wavevector k = 2n/X, where 2 is the wavelength, and the energy E, associated with the mode is given by E = %<x>. There are also transverse normal modes in which the atoms vibrate back and forth in directions perpendicular to the line of atoms. Figure 2.10 shows the dependence of co on k for the low-frequency acoustic and the high-frequency optical longitudinal modes. We see that the acoustic branch continually increases in frequency co with increasing wavenumber k, and the optical branch continuously decreases in frequency. The two branches have respective limiting frequencies given by (2C/M)i/2 and (2C/m)1'2, with an energy gap between them at the edge of the

Figure 2.10. Dependence of the longitudinal normal-mode vibrational frequency to on the wavenumber k = 2n/X for a linear diatomic chain of atoms with alternating masses m < M . having an equilibrium spacing a, and connected by bonds with spring constant C. (From C. P. Poole, Jr., The Physics Handbook, Wiley, New York, 1998, p. 53.)

Figure 2.10. Dependence of the longitudinal normal-mode vibrational frequency to on the wavenumber k = 2n/X for a linear diatomic chain of atoms with alternating masses m < M . having an equilibrium spacing a, and connected by bonds with spring constant C. (From C. P. Poole, Jr., The Physics Handbook, Wiley, New York, 1998, p. 53.)

Brillouin zone k^ — n/a, where a is the distance between atoms m and M at equilibrium. The Brillouin zone is a unit cell in wavenumber or reciprocal space, as will be explained later in this chapter. The optical branch vibrational frequencies are in the infrared region of the spectrum, generally with frequencies in the range from 1012 to 3 x 1014Hz, and the acoustic branch frequencies are much lower. In three dimensions the situation is more complicated, and there are longitudinal acoustic (LA), transverse acoustic (TA), longitudinal optical (LO), and transverse optical (TO) modes.

The atoms in molecules also undergo vibratory motion, and a molecule containing N atoms has 3N — 6 normal modes of vibration. Particular molecular groups such as hydroxyl —OH, amino —NH2 and nitro —N02 have characteristic normal modes that can be used to detect their presence in molecules and solids.

The atomic vibrations that we have been discussing correspond to standing-wave types. This vibrational motion can also produce traveling waves in which localized regions of vibratory atomic motion travel through the lattice. Examples of such traveling waves are sound moving through the air, or seismic waves that start at the epicenter of an earthquake, and travel thousands of miles to reach a seismograph detector that records the earthquake event many minutes later. Localized traveling waves of atomic vibrations in solids, called phonons, are quantized with the energy %<d = hv, where v = a>/2n is the frequency of vibration of the wave. Phonons play an important role in the physics of the solid state.

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