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:

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

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 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 ${\displaystyle \mu }$ of electrons in a solid is also known as the Fermi level and, like the related Fermi energy, often denoted ${\displaystyle E_{\rm {F}}}$. The Sommerfeld expansion can be used to calculate the Fermi level (${\displaystyle T>0}$) at higher temperatures as:

${\displaystyle E_{\rm {F}}(T)=E_{\rm {F}}(T=0)\left[1-{\frac {\pi ^{2}}{12}}\left({\frac {T}{T_{\rm {F}}}}\right)^{2}-{\frac {\pi ^{4}}{80}}\left({\frac {T}{T_{\rm {F}}}}\right)^{4}+\cdots \right],}$

where ${\displaystyle T}$ is the temperature and we define ${\textstyle T_{\rm {F}}=E_{\rm {F}}/k_{\rm {B}}}$ as the Fermi temperature (${\displaystyle k_{\rm {B}}}$ 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 ${\displaystyle E_{\rm {F}}(T=0)}$ and the chemical potential ${\displaystyle E_{\rm {F}}(T>0)}$ 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

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

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

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

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

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

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.

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

${\displaystyle \chi ={\frac {2}{3}}\mu _{0}\mu _{\mathrm {B} }^{2}g(E_{\mathrm {F} }),}$

where ${\textstyle \mu _{0}}$ is the vacuum permittivity and the ${\textstyle \mu _{\rm {B}}}$ is the Bohr magneton. This value results from the competition of two contributions: a diamagnetic contribution (known as Landau's diamagnetism) coming from the orbital motion of the electrons in the presence of a magnetic field, and a paramagnetic contribution (Pauli's paramagnetism). The latter contribution is three times larger in absolute value than the diamagnetic contribution and comes from the electron spin, an intrinsic quantum degree of freedom that can take two discrete values and it is associated to the electron magnetic moment.

Heat capacity

One open problem in solid-state physics before the arrival of quantum mechanics was to understand the heat capacity of metals. While most solids had a constant volumetric heat capacity given by Dulong–Petit law of about ${\displaystyle 3nk_{\rm {B}}}$at large temperatures, it did correctly predict its behavior at low temperatures. In the case of metals that are good conductors, it was expected that the electrons contributed also the heat capacity.

The classical calculation using Drude's model, based on an ideal gas, provides a volumetric heat capacity given by

${\displaystyle c_{V}^{\text{Drude}}={\frac {3}{2}}nk_{\rm {B}}}$.

If this was the case, the heat capacity of a metals should be 1.5 of that obtained by the Dulong–Petit law.

Nevertheless, such a large additional contribution to the heat capacity of metals was never measured, raising suspicions about the argument above. By using Sommerfeld's expansion one can obtain corrections of the energy density at finite temperature and obtain the volumetric heat capacity of an electron gas, given by:

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

where the prefactor to ${\displaystyle nk_{B}}$is considerably smaller than the 3/2 found in ${\textstyle c_{V}^{\text{Drude}}}$, about 100 times smaller at room temperature and much smaller at lower ${\textstyle T}$.

Evidently, the electronic contribution alone does not predict the Dulong–Petit law, i.e. the observation that the heat capacity of a metal is still constant at high temperatures. The free electron model can be improved in this sense by adding the contribution of the vibrations of the crystal lattice. Two famous quantum corrections include the Einstein solid model and the more refined Debye model. With the addition of the latter, the volumetric heat capacity of a metal at low temperatures can be more precisely written in the form,

${\displaystyle c_{V}\approx \gamma T+AT^{3}}$,

where ${\displaystyle \gamma }$ and ${\displaystyle A}$ are constants related to the material. The linear term comes from the electronic contribution while the cubic term comes from Debye model. At high temperature this expression is no longer correct, the electronic heat capacity can be neglected, and the total heat capacity of the metal tends to a constant given by the Dulong–petit law.

Mean free path

Notice that without the relaxation time approximation, there is no reason for the electrons to deflect their motion, as there are no interactions, thus the mean free path should be infinite. The Drude model considered the mean free path of electrons to be close to the distance between ions in the material, implying the earlier conclusion that the diffusive motion of the electrons was due to collisions with the ions. The mean free paths in the free electron model are instead given by ${\textstyle \lambda =v_{\rm {F}}\tau }$ (where ${\textstyle v_{\rm {F}}={\sqrt {2E_{\rm {F}}/m_{e}}}}$ is the Fermi speed) and are in the order of hundreds of ångströms, at least one order of magnitude larger than any possible classical calculation. The mean free path is then not a result of electron–ion collisions but instead is related to imperfections in the material, either due to defects and impurities in the metal, or due to thermal fluctuations.^[3]

Thermal conductivity and thermopower

While Drude's model predicts a similar value for the electric conductivity as the free electron model, the models predict slightly different thermal conductivities.

The thermal conductivity is given by ${\displaystyle \kappa =c_{V}\tau \langle v^{2}\rangle /3}$ for free particles, which is proportional to the heat capacity and the mean free path which depend on the model (${\displaystyle \langle v^{2}\rangle ^{1/2}}$ is the mean (square) speed of the electrons or the Fermi speed in the case of the free electron model). This implies that the ratio between thermal and electric conductivity is given by the Wiedemann–Franz law,

${\displaystyle {\frac {\kappa }{\sigma }}={\frac {m_{\rm {e}}c_{V}\langle v^{2}\rangle }{3ne^{2}}}=LT}$

where ${\displaystyle L}$ is the Lorenz number, given by

${\displaystyle L=\left\{{\begin{matrix}\displaystyle {\frac {3}{2}}\left({\frac {k_{\rm {B}}}{e}}\right)^{2}\;,&{\text{Drude}}\\\displaystyle {\frac {\pi ^{2}}{3}}\left({\frac {k_{\rm {B}}}{e}}\right)^{2}\;,&{\text{free electron model.}}\end{matrix}}\right.}$

The free electron model is closer to the measured value of ${\displaystyle L=2.44\times 10^{-18}}$ V²/K² while the Drude prediction is off by about half the value, which is not a large difference. The close prediction to the Lorenz number in the Drude model was a result of the classical kinetic energy of electron being about 100 smaller than the quantum version, compensating the large value of the classical heat capacity.

However, Drude's mode predicts the wrong order of magnitude for the Seebeck coefficient (thermopower), which relates the generation of a potential difference by applying a temperature gradient across a sample ${\displaystyle \nabla V=S\nabla T}$. This coefficient can be showed to be ${\displaystyle S=-{c_{\rm {V}}}/{|ne|}}$, which is just proportional to the heat capacity, so the Drude model predicts a constant that is hundred times larger than the value of the free electron model. While the latter get as coefficient that is linear in temperature and provides much more accurate values in the order of a few tens of µV/K at room temperature.