In solid-state physics, the free electron model is a quantum mechanical model for the behaviour of charge carriers in a metallic solid. It was developed in 1927, principally by Arnold Sommerfeld, who combined the classical Drude model with quantum mechanical Fermi–Dirac statistics and hence it is also known as the Drude–Sommerfeld model.
Given its simplicity, it is surprisingly successful in explaining many experimental phenomena, especially the Wiedemann–Franz law which relates electrical conductivity and thermal conductivity; the temperature dependence of the electron heat capacity; the shape of the electronic density of states; the range of binding energy values; electrical conductivities; the Seebeck coefficient of the thermoelectric effect; thermal electron emission and field electron emission from bulk metals.
The free electron model solved many of the inconsistencies related to the Drude model and gave insight into several other properties of metals. The free electron model considers that metals are composed of a quantum electron gas where ions play almost no role. The model can be very predictive when applied to alkali and noble metals.
History
Issues with the Drude model In 1900, Paul Drude developed the Drude model of electrical conduction, which was refined by Hendrik Lorentz. Drude model was successful in predicting Wiedemann–Franz law. However physicists like Albert Einstein and quickly pointed out various problems.
By 1911, it was the consensus that the Drude model provided an incorrect prediction of the heat capacity. The same year, in his dissertation Niels Bohr pointed out that Drude model could not predict the diamagnetism of metals. In the same way, Karl Baedeker pointed out that all galvanomagnetic predictions were wrong.
In 1924, the fourth Solvay Conference was organized about the problems with the Drude model. About this time, theories of degeneracy were being discussed. Einstein working with Satyendra Nath Bose, tried to apply the recently developed Bose–Einstein statistics to electrons in a metal but he showed it was not compatible. In 1925, Wolfgang Pauli proposed the exclusion principle for electrons. Enrico Fermi and Paul Dirac came independently with Fermi–Dirac statistics in 1926. Specially Fermi was motivated by the failures of the Drude model.
Sommerfeld's work Arnold Sommerfeld became involved in the theory of metals during a meeting in University of Hamburg in the spring 1927. During the meeting, Pauli shared his work with Sommerfeld who quickly realized that the classical Maxwell–Boltzmann statistics of the Drude model could be the source of the issue. When Sommerfeld returned to the Ludwig-Maximilians-Universität München, he applied Fermi–Dirac statistics and found that the new prediction of Wiedemann–Franz law was even more accurate than the result of the Drude model.
In the summer of 1927, Sommerfeld lectured about his result and showed that the specific heat problem was solved by using the right statistics. That year, he published his results in Die Naturwissenschaften and presented his conclusions during the Como Conference. After the conference, many physicists, including Yakov Frenkel, Fermi, Georg Joos and Walter Schottky, continued to discuss the model with Sommerfeld by mail.
Promoting the theory During winter 1927/1928 he gave a lecture to his students, including Rudolf Peierls, Carl Eckart, William V. Houston and Linus Pauling. Peierls remembers that "Sommerfeld... was going around with a little book by Karl Baedeker, which was then the reference book about the properties of metals and definitions of the various coefficients, and seeing how far things could be made to agree with the theory and how far they couldn't." Eckart remembers that later Sommerfeld got them involved for a year in working on the Drude–Lorentz model using Fermi–Dirac statistics. Sommerfeld wrote a two-part treatise on the topic in Zeitschrift für Physik in 1928. At least, 14 other articles were published that year on the topic in that journal, including equations for field electron emission developed by Lothar Wolfgang Nordheim, student of Sommerfeld and Ralph H. Fowler.
The same year, Bohr wrote to William Hume-Rothery that Sommerfeld work "meant a decisive step as regards to the adequate quantum theoretical treatment of the metallic problem." Einstein indicated that Sommerfeld's model was under his impression the "correct recovery, in principle, of what is true in the original electron theory of metals." Some were not convinced, including Lawrence Bragg.
Sommerfeld nevertheless promoted his work in a trip to Japan in 1928 and to the United States in 1931. John Bardeen, who later developed the transistor, attended Sommerfeld's lecture on the free electron model in Wisconsin. Sommerfeld also wrote a review of the model in 1931 with for Reviews of Modern Physics. And included a chapter in 1933 with Bethe.
Succeeding theory A major breakthrough with respect to the Drude and free electron models came in 1928, with Felix Bloch dissertation, under the supervision of Werner Heisenberg. Bloch's theorem introduced the electron wave behavior and made use of the lattice periodicity. Contrary to Sommerfeld's semi-classical model, Bloch's theory was the first time that the electron was treated quantum mechanically using the recently developed quantum mechanics by Heisenberg and Erwin Schrödinger. Bethe, years later, wrote that he believed in Sommerfeld until the work of Felix Bloch appeared.
Ideas and assumptions
In the free electron model four main assumptions are taken into account: Free electron approximation: The interaction between the ions and the valence electrons is mostly neglected, except in boundary conditions. The ions only keep the charge neutrality in the metal. Unlike in the Drude model, the ions are not necessarily the source of collisions. Independent electron approximation: The interactions between electrons are ignored. The electrostatic fields in metals are weak because of the screening effect. Relaxation-time approximation: There is some unknown scattering mechanism such that the electron probability of collision is inversely proportional to the relaxation time , which represents the average time between collisions. The collisions do not depend on the electronic configuration. Pauli exclusion principle: Each quantum state of the system can only be occupied by a single electron. This restriction of available electron states is taken into account by Fermi–Dirac statistics (see also Fermi gas). Main predictions of the free-electron model are derived by the Sommerfeld expansion of the Fermi–Dirac occupancy for energies around the Fermi level.
The name of the model comes from the first two assumptions, as each electron can be treated as free particle with a respective quadratic relation between energy and momentum.
The crystal lattice is not explicitly taken into account in the free electron model, but a quantum-mechanical justification was given a year later (1928) by Bloch's theorem: an unbound electron moves in a periodic potential as a free electron in vacuum, except for the electron mass me becoming an effective mass m* which may deviate considerably from me (one can even use negative effective mass to describe conduction by electron holes). Effective masses can be derived from band structure computations that were not originally taken into account in the free electron model.
From the Drude model
Many physical properties follow directly from the Drude model, as some equations do not depend on the statistical distribution of the particles. Taking the classical velocity distribution of an ideal gas or the velocity distribution of a Fermi gas only changes the results related to the speed of the electrons.
Mainly, the free electron model and the Drude model predict the same DC electrical conductivity σ for Ohm's law, that is with
where is the current density, is the external electric field, is the electronic density (number of electrons/volume), is the mean free time and is the electron electric charge.
Other quantities that remain the same under the free electron model as under Drude's are the AC susceptibility, the plasma frequency, the magnetoresistance, and the Hall coefficient related to the Hall effect.
Properties of an electron gas
Many properties of the free electron model follow directly from equations related to the Fermi gas, as the independent electron approximation leads to an ensemble of non-interacting electrons. For a three-dimensional electron gas we can define the Fermi energy as
where is the reduced Planck constant. The Fermi energy defines the energy of the highest energy electron at zero temperature. For metals the Fermi energy is in the order of units of electronvolts above the free electron band minimum energy.
Density of states The 3D density of states (number of energy states, per energy per volume) of a non-interacting electron gas is given by:
where is the energy of a given electron. This formula takes into account the spin degeneracy but does not consider a possible energy shift due to the bottom of the conduction band. For 2D the density of states is constant and for 1D is inversely proportional to the square root of the electron energy.
Fermi level The chemical potential of electrons in a solid is also known as the Fermi level and, like the related Fermi energy, often denoted . The Sommerfeld expansion can be used to calculate the Fermi level () at higher temperatures as:
where is the temperature and we define as the Fermi temperature ( is Boltzmann constant). The perturbative approach is justified as the Fermi temperature is usually of about 105 K for a metal, hence at room temperature or lower the Fermi energy and the chemical potential are practically equivalent.
Compressibility of metals and degeneracy pressure The total energy per unit volume (at ) can also be calculated by integrating over the phase space of the system, we obtain
which does not depend on temperature. Compare with the energy per electron of an ideal gas: , which is null at zero temperature. For an ideal gas to have the same energy as the electron gas, the temperatures would need to be of the order of the Fermi temperature. Thermodynamically, this energy of the electron gas corresponds to a zero-temperature pressure given by
where is the volume and is the total energy, the derivative performed at temperature and chemical potential constant. This pressure is called the electron degeneracy pressure and does not come from repulsion or motion of the electrons but from the restriction that no more than two electrons (due to the two values of spin) can occupy the same energy level. This pressure defines the compressibility or bulk modulus of the metal
This expression gives the right order of magnitude for the bulk modulus for alkali metals and noble metals, which show that this pressure is as important as other effects inside the metal. For other metals the crystalline structure has to be taken into account.
Magnetic response According to the Bohr–Van Leeuwen theorem, a classical system at thermodynamic equilibrium cannot have a magnetic response. The magnetic properties of matter in terms of a microscopic theory are purely quantum mechanical. For an electron gas, the total magnetic response is paramagnetic and its magnetic susceptibility given by