Sommerfeld Expansion : Why and How?

As proposed by Drude and Lorentz, the free electron theory of metals considered the electrons as a gas of free classical particles. By “Free”, we mean that the contribution to the energy is only due to the kinetic energy of the particles and there is no potential energy. Thus the energy of an electron will be given by $$ E = \frac{p^2}{2m} $$ where \(p\) and \(m\) are the momentum and mass of the particle respectively. The term “classical” is incorporated mathematically by the assumption of Maxwell-Boltzmann Distribution of the velocities of the particle. As a consequence of this assumption, the calculation showed the value of the specific heat, due to the electrons, is large by a factor of about 100 – clearly, a fail of the classical theory. Moreover, classical theory (assumption of Maxwell-Boltzmann Distribution) also predicted that the specific heat should be a constant, contrary to the fact that it is linearly dependent on temperature. This calls for the need of incorporating quantum ideas.

As Quantum Mechanics developed, scientists realise that electrons are a particular kind of particles known as Fermions (Named after Fermi), and obey a principle called “Pauli’s Exclusion Principle”, which means, no two electrons can occupy identical state. This leads to a particular distribution of energy levels in a fermionic system, known as Fermi-Dirac Distribution, and is given by

$$f(E) = \frac{1}{\exp\left(\frac{E-E_F}{k_BT}\right) + 1}$$

Quantum Techniques employ a different method to calculate energy of number of particles of a system. To count the number of particles, in particular, to obtain the number density, the methodology used is to first calculate the density of states, and then multiply it with the occupation probability of the states.

The density of states (denoted by \( g(E)\)) is the number of energy levels per unit volume of the material per unit energy. Using these the number density \(n = N/V\) of the electrons and their energy density \( \epsilon = E/V \) is given by

$$n = \int_0^{\infty} g(E)f(E)dE $$

$$\epsilon = \int_0^{\infty} E g(E) f(E) dE $$

To calculate the specific heat, one needs to obtain the temperature derivative of the energy density. Although heuristic arguments resolve the classical anomalies, the exact pre-factors remain elusive. Sommerfeld expansions allows for the calculation of the pre-factors, which shall be explained now (Ref: Ashcroft and Mermin).

Note the the above integrals are of the form


At finite temperatures (\(T>0\)), the integral is tedious, and it is Sommerfeld, who did the math and computed the specific heat upto fourth order in temperature. In this post we are going to do the math, and see the relevant corrections.

To compute the energy density we will use \(H(E) = Eg(E)\), and for number density we will use \( H(E) = g(E) \). Note the following properties of \(H(E) \):

  1. \(H(E)\) vanishes as \(E\rightarrow 0\)
  2. \(H(E)\) diverges no rapidly than some power of \(E\) as \( E\rightarrow\infty \). What it means is that there exists a positive real number which is greater than \(p\), where \(p\) is the power of the energy in the expression of \(H(E)\). For example, for the case of energy density, \(H(E)\sim E^{3/2}\). This implies \(p=3/2\). It is obvious that there exist a real number greater than 3/2. It is crucial to note that the divergence of the function \(H(E)\) should be limited by a polynomial function of \(E\).

We define a function \(K(E)\), such that $$K(E) = \int_{-\infty}^{E}H(E’)dE’$$ This enables us to write $$H(E) = \frac{dK}{dE}$$.

Now, by using the technique of integration by parts we can write

$$\int_{-\infty}^{\infty} H(E)f(E)dE = \int_{-\infty}^{\infty}K(E)\left( – \frac{\partial f}{\partial E} \right) dE $$


Ph.D. student, Laboratory of Advanced Nanostructures for Photonics and Electronics, Indian Institute of Science, Bengaluru

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