Free electron model

This article is about the solid-state model for metals. For the model of a free electron gas, see Fermi gas.

In solid-state physics, the free electron model is a simple 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;
• 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 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.

Ideas and assumptions

The free model follows the Drude model in which four basic 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. In contrast with 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 $\tau$, 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 must 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 crystal lattice is not explicitly taken into account in the free electron model, but a quantum-mechanical justification is given by Bloch's Theorem: an unbound electron moves in a periodic potential as a free electron in vacuum, except for the electron mass m becoming an effective mass m* which may deviate considerably from m (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

Main article: 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 for Ohm's law, that is

$\sigma ={\frac {ne^{2}\tau }{m_{e}}},$

where $n$ is the electronic density (number of electrons/volume), $e$ is the electron electric charge, and $m_{e}$ is the electron mass.

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

Main article: Fermi 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 an electron gas we can define a Fermi energy as

$E_{F}={\frac {\hbar ^{2}}{2m_{e}}}\left(3\pi ^{2}n\right)^{\frac {2}{3}},$

where $\hbar$ is the reduced Planck constant. The Fermi energy defines the Fermi level, i.e. the maximal energy an electron in the metal can have at zero temperature. For metals the Fermi energy is in the order of units of electronvolts.

In three dimensions, the density of states of a gas of fermions is proportional to the square root of the kinetic energy of the particles.

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:

$g(E)={\frac {m_{e}}{\pi ^{2}\hbar ^{3}}}{\sqrt {2m_{e}E}}={\frac {3}{2}}{\frac {n}{E_{F}}}{\sqrt {\frac {E}{E_{F}}}},$

where ${\textstyle E\geq 0}$ is the energy of a given electron. This formula takes into account the spin degeneracy but does not consider a possible an 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.

Chemical potential

Additionally the Fermi energy is used to define chemical potential $\mu$. Sommerfeld expansion is a technique used to calculate the chemical potential for higher energies, that is

$\mu (T)=E_{F}\left[1-{\frac {\pi ^{2}}{12}}\left({\frac {T}{T_{F}}}\right)^{2}-{\frac {\pi ^{4}}{80}}\left({\frac {T}{T_{F}}}\right)^{4}+\cdots \right],$

where $T$ is the temperature and we define $T_{F}=E_{F}/k_{B}$ as the Fermi temperature ($k_{B}$ is Boltzmann constant). The perturbative approach is justified as the Fermi temperature is usually of about $10^{5}$ 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 ${\textstyle T=0}$) can also be calculated by integrating over the phase space of the system, we obtain

$u(0)={\frac {3}{5}}nE_{F},$

which does not depend of temperature. Compare with the energy per electron of an ideal gas: ${\frac {3}{2}}k_{B}T$, 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

$P=-\left({\frac {\partial U}{\partial V}}\right)_{T,\mu }={\frac {2}{3}}u(0),$

where ${\textstyle V}$ is the volume and ${\textstyle U(T)=u(T)V}$ 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

$B=-V\left({\frac {\partial P}{\partial V}}\right)_{T,\mu }={\frac {5}{3}}P={\frac {2}{3}}nE_{F}.$

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.

Additional predictions

Specific heat capacity

Further information: Electron heat capacity

One open problem in solid-state physics before the arrival of the free electron model was related to the low specific heat capacity of metals. Even when the Drude model was a good approximation for the Lorentz number of the Wiedemann-Franz law, the classical argument is based on the idea that the heat capacity of an ideal gas is

$c_{v}^{\text{Drude}}={\frac {3}{2}}nk_{B}$.

If this was the case, the heat capacity of a metal could be much higher due to this electronic contribution. Nevertheless, such a large heat capacity was never measured, rising suspicions about the argument. By using Sommerfeld expansion one can obtain corrections of the energy density at finite temperature and obtain the specific heat of an electron gas, given by:

$c_{v}=\left({\frac {\partial u}{\partial T}}\right)_{n}={\frac {\pi ^{2}}{2}}{\frac {T}{T_{F}}}nk_{B}$,

where the prefactor to $nk_{B}$is considerably smaller than the 3/2 found in $c_{v}^{\text{Drude}}$, about 100 times smaller at room temperature and much smaller at lower ${\textstyle T}$. The good estimation of the Lorentz number in the Drude model was a result of the classical mean velocity of electron being about 100 larger than the quantum version, compensating the large value of the classical specific heat. The free electron model calculation of the Lorentz factor is about twice the value of Drude's and its closer to the experimental value.

Mean free path

The mean free paths $\ell =v_{F}\tau$ (where $v_{F}={\sqrt {2E_{F}/m_{e}}}$ is the Fermi speed) in the free electron model are in the order of hundreds of ångströms, at least one order of magnitude larger than any possible classical calculation.

Inaccuracies and extensions

The free electron model presents several inadequacies that are contradicted by experimental observation. We list some inaccuracies below:

Temperature dependence

The free electron model presents several physical quantities that have the wrong temperature dependence, or no dependence at all like the electrical conductivity. The thermal conductivity and specific heat are well predicted for alkali metals at low temperatures, but fails to predict high temperature behaviour coming from ion motion and phonon scattering.

Hall effect

The Hall coefficient has a constant value $R_{H}=-(nec)^{-1}$ in the free electron model. This value is independent of temperature and the strength of the magnetic field. The Hall coefficient is actually dependent on the band structure and the difference with the model can be quite dramatic when studying elements like magnesium and aluminium that have a strong magnetic field dependence.

Directional

The conductivity of some metals can depend of the orientation of the sample with respect to the electric field. Sometimes even the electrical current is not parallel to the field. This possibility is not described because the model does not integrate the crystallinity of metals, i.e. the existence of a periodic lattice of ions.

The magnetoresistance

The free electron model predicts that the resistance in a wire perpendicular to a uniform magnetic field does not depend on the strength of the field. In almost all the cases it does.

Diversity in the conductivity

Not all metals are electrical conductors, some do not conduct electricity very well (insulators), some can conduct when impurities are added like semiconductors. Semimetals, with narrow conduction bands also exist. This diversity is not predicted by the model and can only by explained by analysing valence and conduction bands. Additionally, electrons are not the only charge carriers in a metal, electron vacancies or holes can be seen as quasiparticles carrying positive electric charge.

Other inadequacies are presented in the Wiedemann-Franz law at intermediate temperatures, the frequency-dependence of metals in the optical spectrum and the sign of the thermoelectric field.

More exact values for the electrical conductivity and Wiedemann-Franz law can be obtained by softening the relaxation-time approximation by appealing to Boltzmann transport equations or Kubo formula.

The spin is mostly neglected in the free electron model and its consequences can lead to emergent magnetic phenomena like Pauli paramagnetism and ferromagnetism.

An immediate continuation to the free electron model can be obtained by assuming the empty lattice approximation, which forms the basis of the band structure model known as the nearly free electron model.

Interestingly, adding repulsive interactions between electrons does not change very much the picture presented here. Lev Landau showed that a Fermi gas under repulsive interactions, can be seen as a gas of equivalent quasiparticles that slightly modify the properties of the metal. Landau's model is now known as the Fermi liquid theory. More exotic phenomena like superconductivity, where interactions can be attractive, require a more refined theory.

See also

• Bloch wave
• Tight binding
• Two-dimensional electron gas
• Bose–Einstein statistics
• Fermi surface
• White dwarf

References

General

• Kittel, Charles (1953). Introduction to Solid State Phsysics. University of Michigan: Wiley.
• Ashcroft, Neil; Mermin, N. David (1976). Solid State Physics. New York: Holt, Rinehart and Winston. ISBN 978-0-03-083993-1.
• Sommerfeld, Arnold; Bethe, Hans (1933). Elektronentheorie der Metalle. Berlin Heidelberg: Springer Verlag. ISBN 3642950027. {{cite book}}: ISBN / Date incompatibility (help)