Energy Bands

Section 4.4 Energy Bands

Quantized energies are found for particles confined in any system, not just the infinite or finite square well potentials. Schrödinger's Equation can be solved to determine the allowed energies for particles in any potential energy function. (In fact, determining allowed energies for electrons in atoms is one fundamental aspect of physical chemistry.) Each element (and molecule) has its own pattern of energy levels. As we'll see later this chapter, this has implications for the light that is absorbed and emitted by different elements and molecules.

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But in addition to the case of a single electron trapped in a single potential well, there are many systems composed of multiple electrons trapped in a repeating pattern of potential wells. Consider a chunk of aluminum in a crystalline pattern, for example. The nuclei of the aluminum atoms produce a potential energy function that has repeating energy wells that mobile electrons experience.

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An interesting thing happens to the quantized energy levels when there is more than one side-by-side potential well. For simplicity, we will consider the case of side-by-side, 1D, finite square-well potentials. Figure 4.12) shows a representation of what happens to the energy levels when more and more square-well potentials are added to the system. Each energy level in the single finite-well potential (Figure 4.12a) splits into two nearby levels for the double-well potential (Figure 4.12b). If there are four side-by-side potential wells (Figure 4.12c), there are four energy levels clustered around each value. As the number of side-by-side wells increases, the number of energy levels in each “band” increases accordingly.

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Figure 4.12. Allowed energy levels for (a) infinite square well potential; (b) double, side-by-side potential wells, separated by a thin, finite barriers; and (c) four side-by-side potential wells.

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In a real solid (such as aluminum), there are many, many, many side-by-side potential wells, so many that it becomes difficult to distinguish the individual energy levels within each band (Figure 4.13). They are still there — in the figure, it may look like a continuous band of allowed energies, but there are still a finite number of allowed energy levels (although a very large number of finite values) in each energy band. So, even though the energies appear continuous within each band, they are still quantized.

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Figure 4.13. Valence and conduction bands for a typical solid composed of many atoms. Each of these bands contains a large (but finite) number of discrete energy levels.

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In Figure 4.13, we show only two bands of allowed energies, although there are typically more. The valence band is the highest energy band that is typically filled with electrons.¹ The conduction band always has available energy levels that are unoccupied. In Chapter 7, we'll discuss how the band structure discussed here is important in understanding electrical conductivity for solids, and how the quantum properties of these bands have been used to develop some of the most important building blocks of modern electronics.

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Figure 4.14. (a) Absorption of a single photon, causing an electron in the material to jump up to a higher (unoccupied) energy state. (b) Emission of a photon by an electron that drops from a higher to lower energy state.

As we will see in Chapter 6, the Pauli Exclusion Principle argues that two electrons cannot occupy the same state, so there is a finite number of electrons that can have energies in any band in a solid.