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In Chapter 1, we explained how the Drude expression for conductivity is useful for estimating  the average time electrons can propagate before suffering momentum loss. We should note however, that the Drude model is a classical theory, and therefore is not suitable for describing the quantum mechanical nature of electrons. Some further limitations stem from the following assumptions which are made; electrons scatter only through collisions with ion cores and in-between collisions do not feel the potential of the crystal lattice, that is, they are free electrons which behave like an ideal gas. The latter posed a huge discrepancy between experiment and theory, because the electrons were assumed to have a thermal velocity v ~ T 1/2 , which implied a mean free path of a few Angstroms, the distance between atoms in the lattice49. A number of other experimental signatures proved this value to be far too small. For example, the resistivity was found to depend strongly on the density of impurities within a sample, even though the spacing between impurities was far larger than the distance between atoms. The mean free path inferred from this empirical experiment is actually an order of magnitude larger than the spacing between atoms. It was then quite a mystery how electrons could avoid atoms for such large distances.

A vast improvement on the transport theory was made when Sommerfeld used a quantum mechanical treatment, combining the classical Drude model with quantum mechanical Fermi-Dirac statistics. Similar to Drude, Sommerfeld considered electrons as free particles. He then proceeded by solving the Schrödinger equation for free particles in a box of volume V, which described the free electrons in a crystal sample of volume V.

−ℎ_2𝑚2∇2

𝑒 𝜓(𝑟) = 𝐸𝜓(𝑟) (12) By choosing appropriate boundary conditions (Born-Von Karman), he found the following solution 𝜓𝑟 (𝑟) =_√𝑉1 𝑒𝑖𝑘.𝑟 (13)

This means the electrons behave as plane waves carrying a momentum k, that is quantised and takes discrete values which satisfy the Born-Von Karman boundary conditions. This treatment allows us to calculate the density of states (DOS) which in turn can accurately predict many different thermodynamic properties49. In addition, the Fermi velocity, vF, was defined (an order of magnitude larger than the thermal velocities assumed in the Drude model) and gave much more reasonable

values of the mean free path 𝑙. However, the Sommerfeld model still could not explain the microscopic origin of insulating and metallic like behaviour in different crystals. Not to mention there was still no physical insight in to how electrons propagate over such large distance without bumping in to the ion cores (in the Sommerfeld model, the potential of the ion cores were ignored). To understand these things, we have to consider the effects of the periodic potential of the lattice.

Bloch’s Theoreom

Now we turn on the effects of the crystal lattice, which is described by a periodic potential

𝑈(𝑟) = 𝑈(𝑟 + 𝑅) (14) Where 𝑅 corresponds to one crystal lattice vector. We then proceed by solving the Schrödinger equation which now includes the positive periodic potential

−ℎ_2𝑚2∇2

𝑒 𝜓(𝑟) + 𝑈(𝑟)𝜓(𝑟) = 𝐸𝜓(𝑟) (15) Remarkably, Bloch found the following solution

𝜓𝑘(𝑟) = 𝑒𝑖𝑘.𝑟𝑢𝑘(𝑟) (16)

Where 𝑢𝑘(𝑟) = 𝑢𝑘(𝑟 + 𝑅) is a function which has the periodicity of the lattice (for a detailed

derivation see Ref. 34 Chapter 17). The wave function is simply a plane wave modulated by some periodic function. In other words, the electrons behave as “nearly free” particles, propagating as plane waves without scattering. The term “nearly free” refers to the fact that electrons travelling in the crystal move slower than in free space because they now feel the lattice potential. This is quantified by ascribing an “effective” mass (m*) to the electrons.Regardless of their speed, Bloch’s proof shows that electrons can move without scattering (at T = 0 K and in ideal, defect free crystals) even in the presence of the positively charged atomic centres. The physical reason stems from the fact that the electron wave functions exhibit translational symmetry around the crystal lattice. This means that the wave function of an electron in some state k is the same when you move one lattice vector in real space, differing only by a phase factor i.e

𝜓𝑘 (𝑟 + 𝑅) = 𝑒𝑖𝑘∙𝑅𝜓𝑘(𝑟) (17)

Where r is the position and R is the distance corresponding to one lattice vector. Figure 11 illustrates the electron wavefunction in a periodic potential. This means the electron wavefunction is essentially delocalised around the whole crystal lattice, which is why it can move such large distance and explains the relatively large conductivity of metals.

Figure 11| Bloch electrons in crystals. a, The electron wavefunction 𝜓𝑘(𝑥) (top panel) in a one-

dimensional periodic potential V(x) (bottom panel) is plotted. Note the wavefunction shares the same periodicity as the periodic potential. b, The energy dispersion 𝜀𝑘(𝑘) for Bloch states49. Here,

four energy bands are present which are periodic over the reciprocal lattice vector G = 2/a. Illustration is taken from ref. 49.

Finally, we note one more essential property of Bloch electrons, that is, Bloch states differing by one reciprocal lattice vector are identical.

𝜓𝑘+𝐺 (𝑟) = 𝜓𝑘(𝑟) (18)

Where G is one reciprocal lattice vector. Accordingly, the energies of both Bloch states in equation (18) are identical. This means that the electronic dispersion is periodic with the reciprocal lattice (Fig. 11 b) and therefore the full energy spectrum is described by electron states within the first Brillouin zone only.

To calculate those energies, we then proceed by solving the Schrödinger equation, which however is recast in the form of a Fourier expansion of the periodic structure, and is known instead as the central equation. The resulting solution gives a set of bands separated by gaps which describes all the possible allowed electronic states of the Bloch wave functions (Fig. 11b). The solution found in (17) finally answers our question raised at the beginning of Chapter 3. Because of the wave nature of electrons, they can travel for large distances in a periodic potential without scattering, so long as there are no impurities or thermal vibrations in the lattice. Electrons behave like this provided their wave function retains its translational symmetry with respect to the crystal lattice.