15.5
Band Structure and Optical Properties
To the extent that electronic band structure is a good description of the properties of materials (and usually it is), one can attribute many of the optical properties of materials to this band structure. First one needs to know a few simple facts about light shown here in this table:
Color ~ω Infrared < 1.65 eV Red ∼ 1.8 eV Orange ∼ 2.05 eV Yellow ∼ 2.15 eV Green ∼ 2.3 eV Blue ∼ 2.7 eV Violet ∼ 3.1 eV Ultraviolet > 3.2 eV R 2,01 O Y G B V 1,65 2,11 2,17 2,50 2,75 3,27
15.5.1
Optical Properties of Insulators and Semiconductors
With this table in mind we see that if an insulator (or wide-bandgap semiconductor) has a bandgap of greater than 3.2 eV, then it appears transparent. The reason for this is that a single photon of visible light cannot excite an electron in the valence band into the conduction band. Since the valence band is completely filled, the minimum energy excitation is of the band gap energy — so the photon creates no excitations at all. As a result, the visible optical photons do not scatter from this material at all and they simply pass right through the material6. Materials such as quartz,
diamond, aluminum-oxide, and so forth are insulators of this type.
Semiconductors with somewhat smaller band gaps will absorb photons with energies above the band gap (exciting electrons from the valence to the conduction band), but will be transparent to photons below this band gap. For example, cadmium-sulfide (CdS) is a semiconductor with band gap of roughly 2.6 eV, so that violet and blue light are absorbed but red and green light are transmitted. As a result this material looks reddish. (See Fig. 15.8).
15.5.2
Direct and Indirect Transitions
While the band gap determines the minimum energy excitation that can be made in an insulator (or semiconductor), this is not the complete story in determining whether or not a photon can be absorbed by a material. It turns out to matter quite a bit at which values of k the maximum of the valence band and the minimum of the conduction band lies. If the value of k for the valence band maximum is the same as the value of k for the conduction band minimum, then we say that it is a direct band gap. If the values of k differ, then we say that it is an indirect band gap. For example, the system shown on the left of Fig. 15.2 is a direct band gap, where both the valence band maximum and the conduction band minimum are at the zone boundary. In comparison, if the band shapes were as in the right of Fig. 15.2, but the band gap were large enough such that it would be an insulator (just imagine the bands separated by more), this would be an indirect band gap since the valence band maximum is at the zone boundary, but the conduction band minimum is at k = 0.
6Very weak scattering processes can occur where, say, two photons together can excite an electron, or a photon
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Figure 15.8: Orange crystals of CdS. This particular crystal is the naturally occurring mineral called “Greenockite” which is CdS with trace amounts of impurity which can change its color somewhat.
One can also have both indirect and direct band gaps in the same material, as shown in Fig. 15.9. In this figure, the minimum energy excitation is the indirect transition — meaning an excitation of an electron across an indirect band gap, or equivalently a transition of nonzero crystal momentum7 where the electron is excited from the top of the valence band to the bottom of the
lower conduction band at a very different k. While this may be the lowest energy excitation that can occur, it is very hard for this type of excitation to result from exposure of the system to light — the reason for this is energy-momentum conservation. If a photon is absorbed, the system absorbs both the energy and the momentum of the photon. But given an energy E in the eV range, the momentum of the photon |k| = E/c is extremely small, because c is so large. Thus the system cannot conserve momentum while exciting an electron across an indirect band gap. Nonetheless, typically if a system like this is exposed to photons with energy greater than the indirect band gap a small number of electrons will manage to get excited — usually by some complicated process including absorbtion of a photon exciting an electron with simultaneous emission of a phonon8 to
arrange the conservation of energy and momentum. In comparison, if a system has a direct band gap, and is exposed to photons of energy matching this direct band gap, then it strongly absorbs these photons while exciting electrons from the valence band to the conduction band.
15.5.3
Optical Properties of Metals
The optical properties of metals, however, are a bit more complicated. Since these materials are very conductive, photons (which are electromagnetic) excite the electrons9, which then re-emit
light. This re-emission (or reflection) of light is why metals look shiny. Noble metals (gold, silver,
7By “nonzero” we mean, substantially nonzero – like a fraction of the Brillouin zone.
8Another way to satisfy the conservation of momentum is via a “disorder assisted” process. You recall that the
reason we conserve crystal momentum is because the system is perfectly periodic. If the system has some disorder, and is therefore not perfectly periodic, then crystal momentum is not perfectly conserved. Thus the greater the disorder level, the less crystal momentum needs to be conserved and the easier it is to make a transition across an indirect band gap.
9Note the contrast with insulators — when an electron is excited above the band gap, since the conductivity is
15.5. BAND STRUCTURE AND OPTICAL PROPERTIES 173
0
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Figure 15.9: Direct and Indirect transitions. While the indirect transition is lower energy, it is hard for a photon to excite an electron across an indirect band gap because photons carry very little momentum (since the speed of light, c, is large).
platinum) look particularly shiny because their surfaces do not form insulating oxides when exposed to air, which many metals (such as sodium) do within seconds.
Even amongst metals (ignoring possible oxide surfaces), colors vary. For example, Silver looks brighter than gold and copper, which look yellow or orange-ish. This again is a result of the band structure of these materials. Both of these materials have valence one meaning that a band should be half-filled. However, the total energy width of the conduction band is greater for silver than it is for gold or copper (In tight-binding language t is larger for silver, see chapter 10). This means that higher energy electronic transitions within the band are much more possible for silver than they are for gold and copper. For copper and gold, photons with blue and violet colors are not well absorbed and re-emitted, leaving these material looking a bit more yellow and orange. For silver on the other hand, all visible colors are re-emitted well, resulting in a more perfect (or “white”) mirror. While this discussion of the optical properties of metals is highly over- simplified10, it captures the correct essence — that the details of the band structure determine
which color photons are easily absorbed and/or reflected, and this in turn determines the apparent color of the material.
15.5.4
Optical Effects of Impurities
It turns out that small levels of impurities put into periodic crystals (particularly into semicon- ductors and insulators) can have dramatic effects on many of their optical (as well as electrical!) properties. For example, one nitrogen impurity per million carbon atoms in a diamond crystal gives the crystal a yellow-ish color. One boron atom per million carbon atoms give the diamond a blue-ish color11. We will discuss the physics that causes this in section 16.2.1 below.
10Really there are many bands overlapping in these materials and the full story addresses inter and intra-band
transitions.
11Natural blue diamonds are extremely highly prized and are very expensive. Possibly the world’s most famous
diamond, the Hope Diamond, is of this type (it is also supposed to be cursed, but that is another story). With modern crystal growth techniques, in fact it is possible to produce man-made diamonds of “quality” better than those that are mined. Impurities can be placed in as desired to give the diamond any color you like. Due to the
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