Patterson, Bailey - Solid State Physics Introduction to theory

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later (Sect. 11.6). They are important for determining how easily crystals deform and may also relate to crystal growth.

Finally, there are defects that occur over a whole volume. It is usually hard to grow a single crystal. In a single crystal, the lattice planes are all arranged as expected–in a perfectly regular manner. When we are presented with a chunk of material, it is usually in a polycrystal form. That is, many little crystals are stuck together in a somewhat random way. The boundary between crystals is also a twodimensional defect called a grain boundary. We have summarized these ideas in Table 11.1.

11.2Shallow and Deep Impurity Levels in Semiconductors (EE)

We start by considering a simple chemical model of shallow donor and acceptor defects. We will give a better definition later, but for now, by “shallow”, we will mean energy levels near the bottom of the conduction band for donor level and near the top of the valence band for acceptors.

Consider Si14 as the prototype semiconductor. In the usual one-electron shell notation, its electron structure is denoted by

1s2 2s2 2p6 3s2 3p2 .

There are four valence electrons in the 3s23p2 shell, which requires eight to be filled. We think of neighboring Si atoms sharing electrons to fill the shells. This sharing lowers energy and binds the electrons. We speak of covalent bonds. Schematically, in two dimensions, we picture this occurring as in Fig. 11.3. Each line represents a shared electron. By sharing, each Si in the outer shell has eight electrons. This is of course like the discussion we gave in Chap. 1 of the bonding of C to form diamond.

Fig. 11.3. Chemical model of covalent bond in Si

Now, suppose we have an atom, say As, which substitutionally replaces a Si. The sp shell of As has five electrons (4s24p3) and only four are needed to “fill the shell”. Thus, As acts as a donor with an additional loosely bound electron (with a large orbit encompassing many atoms), which can be easily excited into the conduction band at room temperature.

An acceptor like In (with three electrons in its outer sp shell (5s25p)) needs four electrons to complete its covalent bonds. Thus, In can accept an electron from the valence band, leaving behind a hole. The combined effects of effective mass and dielectric constant cause the carrier to be bound much less tightly than in an analogous hydrogen atom. The result is that donors introduce energy levels just below the conduction-band minimum and acceptors introduce levels just above the top of the valence band. We discuss this in more detail below.

In brief, it turns out that the ground-state donor energy level is given by (atomic units, see the appendix)

where m*/m is the effective mass ratio typically about 0.25 for Si and ε is the dielectric constant (about 11.7 in Si). En in (11.1) is measured from the bottom of the conduction band. Except for the use of the dielectric constant and the effective mass, this is the same result as obtained from the theory of the energy levels of hydrogen. A similar, remarkably simple result holds for acceptor states. These results arise from pioneering work by Kohn and Luttinger as discussed in [11.17], and we develop the basics below.

11.3Effective Mass Theory, Shallow Defects, and Superlattices (A)

11.3.1 Envelope Functions (A)

The basic model we will use here is called the envelope approximation.3 It will allow us to justify our treatments of effective mass theory and of shallow defects in semiconductors. With a few more comments, we will then be able to relate it to a simple approach to superlattices, which will be discussed in more detail in Chap. 12.

Let

3 Besides [11.17], see also Luttinger and Kohn [11.22] and Madelung [11.23].

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where V(r) is the periodic potential. Let H = H0 + U where U = VD(r) is the extra

defect potential. Now, H0ψn(k, r) = Enψn(k, r) and Hψ = Eψ. We expand the wave function in Bloch functions

where n is the band index. Also, since En(k) is a periodic function in k-space, we can expand it in a Fourier series with the sum restricted to lattice points

We define an operator En(−i ) by substituting −i for k:

En (−i )ψn (k, r) = ∑m FnmeRm ψn (k, r)

Substituting (11.3) into Hψ = Eψ, we have (using the fact that ψn is an eigenfunction of H0 with eigenvalue En(k))

11.3.2 First Approximation (A)

We neglect band-to-band interactions and hence, neglect the summation over n. Dropping n entirely from (11.9), we have

11.3.3 Second Approximation (A)

We assume a large extension in real space that means that only a small range of k values are important – say the ones near a parabolic (assumed for simplicity) minimum at k = 0 (Madelung op. cit. Chap. 9).

We assume, then,

But, ψ(0, r + Rm) = ψ(0, r), so it can be cancelled. Thus retracing our steps, we have

This simply means that a rapidly varying function has been replaced by a slowly varying function F(r) called the “envelope” function. This immediately leads to the concept of shallow donors. Consider the bottom of a parabolic conductor band near k = 0 and expand about k = 0;

594 11 Defects in Solids

And, if VD = e2/4πεr, our resulting equation is

Except for the use of ε and m*, these solutions are just hydrogenic wave functions and energies, and so our use of the hydrogenic solution (11.1) is justified.

Now let us discuss briefly electron and hole motion in a perfect crystal. If U = 0, we simply write

On the other hand, suppose U is still 0, but consider a valence band with a maximum at k = 0. We then can expand about that point with the following result:

Using the hole mass, which has the opposite sign for the electron mass (mh = −me), we can write

so the relevant Schrödinger equation becomes

Looking at (11.23) and (11.26), we see how discontinuities in band energies can result in effective changes in the potential for the carriers, and we see why the hole energies are inverted from the electron energies.

Now let us consider superlattices with a set of layers so there is both a lattice periodicity in each layer and a periodicity on a larger scale due to layers (see Sect. 12.6). The layers A and B could for example be laid down as ABABAB…

There are several more considerations, however, before we can apply these results to superlattices. First, we have to consider that if we are to move from a region of one band structure (layer) to another (layer), the effective mass changes since adjacent layers are different. With the possibility of change in effective mass, the Hamiltonian is often written as

rather than in the more conventional way. This allows the Hamiltonian to remain Hermitian, even with varying m*, and it leads to a probability current density of

from which we apply the requirement of continuity on ψ and ∂ψ/(m*∂z) rather than ψ and ∂ψ/∂z.

We have assumed the thickness of each layer is sufficient that the band structure of the material can be established in this thickness. Basically, we will need both layers to be several monolayers thick. Also, we assume in each layer that the electron wave function is an envelope function (different for different monolayers) times a Bloch function (see (11.13)). Finally, we assume that in each layer U = U0 (a constant appropriate to the layer) and the carrier motion perpendicular to the layers is free-electron-like so,

There are many, many complications to the above. We have assumed, e.g., that m*x,y = m*z which may not be so in all cases. The book by Bastard [11.1] can be consulted. See also, Mitin et al [11.25].

In semiconductors, shallow levels are often defined as being near a band edge and deep levels as being near the center of the forbidden energy gap. In more recent years, a different definition has been applied based on the nature of the causing agent. Shallow levels are now defined as defect levels produced by the long-range Coulomb potential of the defect and deep levels4 are defined as being produced by the central cell potential of the defect, which is short ranged. Since

4 See, e.g., Li and Patterson [11.20, 11.21] and references cited therein.

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the potential is short range, a modification of the Slater-Koster model, already discussed in Chap. 2, is a convenient starting point for discussing deep defects. Some reasons for the significance of shallow and deep defects are given in Table 11.2. Deep defects are commonly formed by substitutional, interstitial, and antisite atoms and by vacancies.

Table 11.2. Definition and significance of deep and shallow levels

Shallow levels are defect levels produced by the long-range Coulomb potential of defects. Deep Levels are defect levels produced by the central cell potential of defect.

11.4 Color Centers (B)

The study of color centers arose out of the curiosity as to what caused the yellow coloration of rock salt (NaCl) and other coloration in similar crystals. This yellow color was particularly noted in salt just removed from a mine. Becquerel found that NaCl could be colored by placing the crystal near a discharge tube. From a fundamental point of view, NaCl should have an infrared absorption due to vibrations of its ions and an ultraviolet absorption due to excitation of the electrons. A perfect NaCl crystal should not absorb visible light, and should be uncolored. The coloration of NaCl must be due, then, to defects in the crystal. The main absorption band in NaCl occurs at about 4650 Å (the “F”-band). This blue absorption is responsible for the yellow color that the NaCl crystal can have. A further clue to the nature of the absorption is provided by the fact that exposure of a colored crystal to white light can result in the bleaching out of the color. Further experiments show that during the bleaching, the crystal becomes photoconductive, which means that electrons have been promoted to the conduction band. It has also been found that NaCl could be colored by heating it in the presence of Na vapor. Some of the Na atoms become part of the NaCl crystal, resulting in a deficiency of Cl and, hence, Cl− vacancies. Since photoconductive experiments show that F-band defects can release electrons, and since Cl− vacancies can trap electrons, it seems very suggestive that the defects responsible for the F-band (called “F-centers”) are electrons trapped at Cl− vacancies. (Note: the “F” comes from the German farbe, meaning “color”.) This is the explanation accepted today. Of course, since some Cl− vacancies are always present in a NaCl crystal in thermodynamic equilibrium, any sort of radiation that causes electrons to be

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M-center models

Fig. 11.4. Models of the M-center: (a) Seitz, (b) Van Doorn and Haven. [Reprinted with permission from Rhyner CR and Cameron JR, Phys Rev 169(3), 710 (1968). Copyright 1968 by the American Physical Society.]

Fig. 11.5. Four proposed models for Z1-centers [Reprinted with permission from Paus H and Lüty F, Phys Rev Lett 20(2), 57 (1968). Copyright 1968 by the American Physical Society.]

11.5 Diffusion (MET, MS)

Point defects may diffuse through the lattice, while vacancies may provide a mechanism to facilitate diffusion. Diffusion and defects are intimately related, so we give a brief discussion of diffusion. If C is the concentration of the diffusing quantity, Fick’s Law says the flux of diffusing quantities is given by