Consider a simple atom: sodium. Sodium has atomic number Z= 11 and in the ground state its electrons fill the orbitals as

Thus, sodium has 10 electrons which fill the two innermost n = 1 and 2 shells, and has a single electron further out in the n = 3 shell. (See Fig 1.)

Figure 1: The sodium atom and its electron levels in the ground state. [Source: (left figure) Wikimedia Commons.]

What happens as we bring a second sodium atom near to the first? The electrons of one atom will feel those of the other, leading to a shuffling of the energy levels. As the atoms are brought closer, this warping of energy levels becomes greater. (See Fig. 2)

Figure 2: As two atoms are brought within a distance r of each other, previously identical energy levels (here the 3s level is shown) will diverge into two separate levels, one becoming slightly higher and one becoming slightly lower in energy. [Source: (left figure) adapted from Wikimedia Commons, (right figure) Fig. 12.16, Modern Physics, Serway, Moses & Moyer, 2005.]

This process continues as we bring together more and more atoms until we form a continuous band of very closely spaced energy levels. (See Fig. 3.) These bands appear for all energy levels, though the electron orbitals close to the atom (holding the valence electrons) will not spread out as much as the orbitals further from the atom. In some cases, expanded energy bands can even overlap or split. For example, In the case of sodium, the 3s and 3p levels merge to form a single band (see Fig 4), whereas in the case of diamond, the states of the 2p orbital split into a lower and upper band (see Fig. 5).

Figure 3: As we add more atoms, – (a) 2 atoms, (b) 6 atoms, © many atoms – the energy splitting forms a continuous band. [Source: Fig. 12.16, Modern Physics, Serway, Moses & Moyer, 2005.]

Figure 4: The band structure of sodium showing the overlap of the conduction and valence bands that forms when the 3s and 3p orbitals merge. [Source: Fig. 3.7, Experiments in Modern Physics, Melissinos, 1966.]

Figure 5: The band structure of diamond showing the splitting of the 2p orbital and the formation of well-separated (several eV) valence and conduction bands. [Source: Fig. 3.6, Experiments in Modern Physics, Melissinos, 1966.]

We can define a few important terms. In the ground state, electrons will fill the bands from lowest to highest energy up to the Fermi level. We call the highest band with electrons in it in the ground state the valence band and we call the lowest band with open states the conduction band. Note that in sodium, the valence and conduction bands are the same (that is, the band is only partially filled), whereas in diamond we have a full valence band and an empty conduction band in the ground state.

We may categorize materials according to their band structure. The three main categories are as follows:

• Insulators: In an insulator, a full valence band is separated from an empty conduction band by a large band gap, typically several eV or more.
• Conductors: In a conductor, the valence and conduction bands overlap.
• Semiconductors: In a semiconductor, the bands are separate in the ground state (like an insulator), but the band gap energy between them is smaller, typically 0.1-1.0 eV. Because of the smaller gap, some electrons can be coaxed into the conduction band by adding small amounts of energy (e.g. by heating the material). When a significant number of electrons have jumped the gap, the semiconductor behaves more like a conductor than an insulator.

These three types of materials are illustrated in Fig. 6.

Figure 6: The three classifications of materials based on band structure. [Source: Fig. 9.1, Introduction to Solid State Physics, 3rd Ed., Kittel, 1966.]

Electrons in the valence band are far more influenced by their host nucleus than by neighbors and therefore remain localized to their host. Compare this to electrons in the conduction band which are far enough from the nuclei that they can move about; they are free electrons. Free electrons are the ones which participate in electrical conduction, or the movement of charge around the crystal when an electric field is applied. When inside a partially-filled conduction band, these electron can move up or down within the band via very small changes in energy, whereas there is a large energy barrier (the band gap energy) required to move an electron out of the valence band.

Figure 7: The free versus localized nature of electrons in a sodium lattice. [Source: Adapted from  Bonding in Metals and Semiconductors ]

When an electric field is applied, electrons feel a force

They scatter with an average time $\tau$ between collisions and develop a drift velocity

Therefore, we have a net current $I$  and can define a current density,

where $n$  is the number of free electrons per unit volume, and $A$  is the cross sectional area of the material. (See Fig. 8.)

Figure 8: A schematic of an electron drift path through a material.

If we rearrange this, we find the fundamental form of Ohm's law,

where

is the conductivity, or

is the resistivity.

So far we have not mentioned what causes the electrons to scatter. While we might initially guess that it is the positive ions which form the crystal lattice, this is actually not the case. It can be shown quantum mechanically that a perfect lattice will present a uniform background through which the electrons will pass, with no scattering at all.

Instead, the electrons will scatter off anything which breaks the periodicity. The two main suspects are therefore defects in the crystal and lattice vibrations.

Understanding (and quantifying) how the atoms of a crystal lattice vibrate was one of the most important problems of the early 20th century. This was motived – at least in part – by those who wanted to understand the curious experimentally-measured temperature dependence of the specific heat in different metals. At high temperatures, metals all approach a universal value for the molar specific heat,

which is called the Dulong-Petit law. At low temperatures, all materials show a $T^3$  temperature dependence and metals show an additional linear-T dependence. We now know that the linear dependence is an effect related to the conduction electrons in metals, but the $T^3$  behavior at low temperatures and the constant behavior at high temperatures can only be explained by understanding exactly how energy can be stored in the vibrating atoms of the lattice.

Albert Einstein proposed that the atoms of a lattice all vibrate with the same frequency, but vibrate independently in individual quantum harmonic oscillator wells. (See Fig. 9.)

Figure 9: The energy levels of a quantum harmonic oscillator. [Source: Hyperphysics]

This model was only partly successful. It explained the Dulong-Petit law at high temperature – Eq. (8) above – but could not explain the $T^3$ dependence of the specific heat at low temperatures.

Peter Debye instead proposed that instead of independent oscillators, the atoms were instead connected by springs so that the crystal was a coupled oscillator. In such a case, we do not have $3N$ identical oscillators, but instead have $3N$ different normal modes where the frequencies are not equal and are not equally common. In fact, Debye postulated that the number of modes of a given frequency increased as $\omega^2$  – there are more high frequency (or short wavelength) modes than low frequency (or long wavelength) modes. Debye also said that this increase in the number of modes can't go on for even – remember that there are only $3N$  modes total – so he instituted a cut-off frequency, what we now call the Debye frequency, $\omega_D$ . This frequency is sometimes expressed as the Debye temperature $\theta_D$  given by

Examples of the density of states (i.e., the number of modes per unit volume) are plotted in Fig. 10. On the left is the density of states predicted by the Debye model, while on the right is a schematic of a more realistic density of states. While the two plots do not initially look similar, note that they agree in both the low frequency and high frequency range (where the density of states increase as $\omega^2$ in the first case, and tends to zero in the second).

Figure 10: (a) Debye model density of states. (b) A realistic density of states. [Source: Fig. 5-14, Introduction to Solid State Physics, 3rd Ed., Kittel, 1966]

So far, we have thought of lattice vibrations as waves, but we can instead consider them as particles (since this is a quantum mechanical system).

To see this, recall that the energy contained in a particular mode is given by the quantum harmonic oscillator energy,

where $\omega$  is the frequency of that mode and $n$ describes the quantum energy state. A vibrational mode can only gain or lose energy in discrete amounts, and we call these quanta of vibrational energy phonons. A mode in the $n$th energy state is occupied by n phonons, each with energy $E_p = \hbar\omega$ .

At very low temperatures, some metals undergo a transition from normal conductor to a superconductor. A superconductor is characterized by the following two important features:

• ZERO resistivity: A superconductor's resistivity drops to zero and the electrons moving through the material no longer dissipate energy through collisions. For example, if one can establish a current in a superconductor, that current will continue (without diminishing) so long as the material remains in the superconducting state.
• The Meissner effect: A superconductor will rearrange the electrons on its surface to create a magnetic field which cancels out any external field which would otherwise penetrate the material. This expulsion of magnetic field lines from the material means that outside the material you can observe very interesting magnetic field patterns. (See Fig. 11)

Figure 11: A magnet levitating above a superconducting material due to the fields created by the Meissner effect. [Source: Wikimedia Commons]

Typical superconducting temperatures are extremely low: $T_C \le 10 \;\mathrm{K}$.