In Part VI we developed a model of electrical conductivity in which the valence electrons of a metal form a loosely bound “sea of electrons.” The typical speed of an electron is the rms speed:

where $k_{\text{B}}$ is Boltzmann’s constant. Hence at room temperature, where $v_{\text{rms}}\approx 1\times {10}^5\text{ m/s,}$ the de Broglie wavelength of a typical conduction electron is

There is a range of wavelengths because the electrons have a range of speeds, but this is a typical value.

You’ve now seen many times that wave effects are significant only when the sizes of physical structures are comparable to or smaller than the wavelength. Because the de Broglie wavelength of conduction electrons is only a few nm, quantum effects are insignificant in electronic devices whose features are larger than about 100 nm. The electrons in macroscopic devices can be treated as classical particles, which is how we analyzed electric current in Chapter 27.

However, devices smaller than about 100 nm do exhibit quantum effects. Some semiconductor devices, such as the semiconductor lasers used in fiber-optic communications, now incorporate features only a few nm in size. Quantum effects play an important role in these devices.

FIGURE 40.16a shows a semiconductor diode laser through which a current travels from left to right. In the center is a very thin layer of the semiconductor gallium arsenide (GaAs). It is surrounded on either side by layers of gallium aluminum arsenide (GaAlAs), and these in turn are embedded within the larger structure of the diode. The electrons within the central GaAs layer begin to emit laser light when the current through the diode exceeds a threshold current. The laser beam diverges because of diffraction through the “slit” of the GaAs layer, with the wider axis of the laser beam corresponding to the narrower portion of the lasing region.

FIGURE 40.16 A semiconductor diode laser with a single quantum well.

You can learn in a solid-state physics or materials engineering course that the electric potential energy of an electron is slightly lower in GaAs than in GaAlAs. This makes the GaAs layer a potential well for electrons, with higher-potential-energy GaAlAs “walls” on either side. As a result, the electrons become trapped within the thin GaAs layer. Such a device is called a quantum-well laser.

As an example, FIGURE 40.16b shows a quantum-well device with a 1.0-nm-thick GaAs layer in which the electron’s potential energy is 0.300 eV lower than in the surrounding GaAlAs layers. A numerical solution of the Schrödinger equation finds that this potential well has only a single quantum state, $n=1$ with $E_1=0.125\text{eV}$. Every electron trapped in this quantum well has the same energy—a very nonclassical result! The fact that the electron energies are so well defined, in contrast to the range of electron energies in bulk material, is what makes this a useful device. You can also see from the probability density ${\left|\psi \right|}^2$ that the electrons are more likely to be found in the center of the layer than at the edges. This concentration of electrons makes it easier for the device to begin laser action.