29 Jun 2025
Free-Electron Theory of Metals
Drude-Lorentz transport, Sommerfeld statistics, Fermi energy, Fermi level, Fermi velocity, density of states, and thermal conductivity.
The free-electron model replaces the detailed ionic potential inside a metal by a constant and treats collisions through a mean relaxation time. The classical Drude-Lorentz model explains Ohm’s law; Sommerfeld’s quantum statistics supplies the correct energy scale and thermal transport.
Drude-Lorentz electrical conductivity
Let $n$ be the conduction-electron number density, $m$ the electron mass, and $-e$ its charge. Between collisions an electric field accelerates an electron, while collisions relax its average velocity in time $\tau$:
\[m\frac{d\mathbf v_d}{dt}=-e\mathbf E-\frac{m\mathbf v_d}{\tau}.\]In a static steady state, $d\mathbf v_d/dt=0$, so
\[\mathbf v_d=-\frac{e\tau}{m}\mathbf E.\]The conventional current density is opposite to electron motion:
\[\mathbf j=-ne\mathbf v_d =\frac{ne^2\tau}{m}\mathbf E.\]Thus
\[\boxed{\sigma=\frac{ne^2\tau}{m}}, \qquad \boxed{\rho=\frac{1}{\sigma}=\frac{m}{ne^2\tau}}.\]$\sigma$ has SI unit $\mathrm{S\,m^{-1}}$ and $\rho$ has unit $\Omega\,\mathrm m$. The mean free path is $\ell=\bar v\tau$. The model attributes increasing resistivity to decreasing $\tau$, not to a change of electron charge or density.
For a harmonic field $\mathbf E=\operatorname{Re}[\mathbf E_0e^{-i\omega t}]$, take $\mathbf v_d=\operatorname{Re}[\mathbf v_0e^{-i\omega t}]$. The equation of motion gives
\[(-i\omega m+m/\tau)\mathbf v_0=-e\mathbf E_0,\]and therefore
\[\boxed{\sigma(\omega)=\frac{ne^2\tau/m}{1-i\omega\tau}}.\]The static result is recovered when $\omega\tau\ll1$; at higher frequency the current lags the field.
Classical thermal conductivity
Electrons moving between regions of different temperature transport energy. The kinetic-theory form is
\[\boxed{\kappa=\frac13c_e\bar v^{\,2}\tau},\]where $c_e$ is electron heat capacity per unit volume. Its unit is $\mathrm{W\,m^{-1}K^{-1}}$. Classical equipartition uses $c_e=3nk_B/2$ and $\bar v^{\,2}=3k_BT/m$, predicting an electronic heat capacity far larger than observed. This failure is removed by Fermi-Dirac statistics.
Sommerfeld theory and allowed states
Put the electrons in a cube of volume $V=L^3$ and impose periodic boundary conditions,
\[\psi(x+L,y,z)=\psi(x,y,z).\]For $\psi\propto e^{i\mathbf k\cdot\mathbf r}$ this gives
\[k_x=\frac{2\pi n_x}{L},\quad k_y=\frac{2\pi n_y}{L},\quad k_z=\frac{2\pi n_z}{L}.\]One $\mathbf k$ state occupies volume $(2\pi/L)^3$ in $k$ space and accepts two electrons of opposite spin. At $T=0$, states fill a sphere of radius $k_F$. Hence
\[N=2\frac{V}{(2\pi)^3}\frac{4\pi k_F^3}{3} =\frac{Vk_F^3}{3\pi^2}.\]With $n=N/V$,
\[\boxed{k_F=(3\pi^2n)^{1/3}}.\]For a free electron $E=\hbar^2k^2/(2m)$, so the highest occupied zero-temperature energy is
\[\boxed{E_F=\frac{\hbar^2}{2m}(3\pi^2n)^{2/3}}.\]The corresponding Fermi speed and temperature are
\[\boxed{v_F=\frac{\hbar k_F}{m}=\sqrt{\frac{2E_F}{m}}}, \qquad \boxed{T_F=\frac{E_F}{k_B}}.\]$E_F$ is measured in joules or electronvolts, $v_F$ in $\mathrm{m\,s^{-1}}$, and $T_F$ in kelvin.
Density of states
The number of spin states below wave number $k$ is
\[N(k)=\frac{Vk^3}{3\pi^2}.\]Using $k=(2mE/\hbar^2)^{1/2}$,
\[N(E)=\frac{V}{3\pi^2}\left(\frac{2mE}{\hbar^2}\right)^{3/2}.\]Differentiation gives the three-dimensional density of states
\[\boxed{ g(E)=\frac{dN}{dE} =\frac{V}{2\pi^2}\left(\frac{2m}{\hbar^2}\right)^{3/2}\sqrt E }.\]It has unit $\mathrm{J^{-1}}$. At the Fermi energy,
\[\boxed{g(E_F)=\frac{3N}{2E_F}}.\]
Fermi level
At temperature $T$, the occupation probability is
\[\boxed{f(E)=\frac{1}{e^{(E-\mu)/k_BT}+1}},\]where the chemical potential $\mu(T)$ is the Fermi level. Since $f(\mu)=1/2$, the Fermi level marks the midpoint of the occupation step. At $T=0$, $\mu=E_F$. For $k_BT\ll E_F$ at fixed density,
\[\mu(T)\simeq E_F\left[1-\frac{\pi^2}{12} \left(\frac{k_BT}{E_F}\right)^2\right].\]Thus ordinary temperatures alter only electrons close to $E_F$; the deeply occupied states cannot change because nearby final states are already filled.
Sommerfeld heat and thermal conductivity
The low-temperature Sommerfeld result for electronic heat capacity is
\[\boxed{C_{V,e}=\frac{\pi^2}{2}Nk_B\frac{T}{T_F}}\]or, per unit volume,
\[c_e=\frac{\pi^2}{2}nk_B^2\frac{T}{E_F}.\]Only the fraction $T/T_F$ of electrons near the Fermi surface is thermally active. In the transport formula the active electrons move with approximately $v_F$, so
\[\begin{aligned} \kappa &=\frac13c_ev_F^2\tau\\ &=\frac13\left(\frac{\pi^2}{2}nk_B^2\frac{T}{E_F}\right) \left(\frac{2E_F}{m}\right)\tau\\ &=\boxed{\frac{\pi^2nk_B^2T\tau}{3m}}. \end{aligned}\]Dividing by $\sigma T=(ne^2\tau/m)T$ eliminates $n,m,$ and $\tau$:
\[\boxed{ \frac{\kappa}{\sigma T}=L_0 =\frac{\pi^2}{3}\left(\frac{k_B}{e}\right)^2 }.\]This is the Wiedemann-Franz law with Lorenz number $L_0\simeq2.44\times10^{-8}\ \mathrm{W\,\Omega\,K^{-2}}$. Its simple form assumes the same relaxation time governs charge and heat currents near the Fermi surface.
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