Solid-State Chemistry/Part 3/Metallic Bonding & Free Electron Model

3.2 Metallic Bonding & Free Electron Model

Metals exhibit high electrical and thermal conductivity, luster, and ductility. The free electron model treats valence electrons as a gas of non-interacting particles, providing surprisingly accurate predictions for many metallic properties.

Free Electron Model

In the free electron model (Sommerfeld model), conduction electrons move freely within the metal with the dispersion relation:

$$E = \frac{\hbar^2 k^2}{2m}$$

where $\hbar$ is the reduced Planck constant, $k$ is the wave vector, and $m$ is the electron mass.

Density of States

The number of electronic states per unit energy per unit volume in 3D is:

$$g(E) = \frac{V}{2\pi^2}\left(\frac{2m}{\hbar^2}\right)^{3/2}\sqrt{E}$$

The density of states grows as $\sqrt{E}$, meaning there are more available states at higher energies.

Fermi Energy

At $T = 0$, all states up to the Fermi energy $E_F$ are filled. The Fermi energy depends only on the electron density $n$:

$$E_F = \frac{\hbar^2}{2m}\left(3\pi^2 n\right)^{2/3}$$

Typical Fermi energies for metals are 2--12 eV, corresponding to Fermi temperatures$T_F = E_F/k_B$ of order $10^4$--$10^5$ K.

Fermi-Dirac Distribution

At finite temperature, electrons obey Fermi-Dirac statistics:

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

At $T = 0$, this is a step function: $f = 1$ for $E < E_F$ and$f = 0$ for $E > E_F$. At finite $T$, the step is smeared over an energy range $\sim k_BT$ around $E_F$.

Drude Model of Conductivity

The Drude model treats electrons as classical particles scattered with a mean free time $\tau$. The electrical conductivity is:

$$\sigma = \frac{ne^2\tau}{m}$$

The resistivity $\rho = 1/\sigma$ increases with temperature due to increased scattering (shorter $\tau$).

Electronic Heat Capacity

Only electrons near $E_F$ (within $\sim k_BT$) can be thermally excited. The electronic contribution to heat capacity is:

$$C_{el} = \gamma T = \frac{\pi^2}{2}\frac{Nk_B^2 T}{E_F}$$

This is much smaller than the classical prediction $\frac{3}{2}Nk_B$ because only a fraction $\sim k_BT/E_F$ of electrons participate in thermal excitation.

Video Lectures

17. Metals

Python: Fermi-Dirac Distribution & DOS

Visualization of the Fermi-Dirac distribution at different temperatures and the free-electron density of states with occupied state regions.

Fermi-Dirac Distribution & Density of States

Python

Plots f(E) at various temperatures and occupied DOS for copper

script.py82 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
# ─────────────────────────────────────────────────
# Fermi-Dirac Distribution & Density of States
# ─────────────────────────────────────────────────

kB = 8.617e-5  # eV/K

# Fermi energy for copper
Ef = 7.0  # eV (approximate for Cu)

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 6))
fig.patch.set_facecolor('#0a0a1a')
for ax in (ax1, ax2):
    ax.set_facecolor('#0a0a1a')
    for spine in ax.spines.values():
        spine.set_edgecolor('#334155')

# --- Left: Fermi-Dirac distribution at different temperatures ---
E = np.linspace(0, 12, 1000)
temperatures = [0.01, 300, 1000, 3000, 5000]
colors = ['#22d3ee', '#a78bfa', '#f59e0b', '#ef4444', '#22c55e']

print("=" * 55)
print("Fermi-Dirac Distribution & Density of States")
print("=" * 55)
print(f"\nFermi energy Ef = {Ef:.1f} eV (Cu)")

for T, color in zip(temperatures, colors):
    if T < 1:
        f_E = np.where(E < Ef, 1.0, 0.0)
        label = 'T = 0 K'
    else:
        f_E = 1.0 / (np.exp((E - Ef) / (kB * T)) + 1.0)
        label = f'T = {T} K'
    ax1.plot(E, f_E, color=color, linewidth=2, label=label)

ax1.axvline(x=Ef, color='white', linestyle='--', alpha=0.5, linewidth=1)
ax1.text(Ef + 0.1, 0.5, f'$E_F$ = {Ef} eV', color='white', fontsize=10)
ax1.set_xlabel('Energy (eV)', color='#94a3b8', fontsize=12)
ax1.set_ylabel('f(E)', color='#94a3b8', fontsize=12)
ax1.set_title('Fermi-Dirac Distribution', color='white', fontsize=14)
ax1.legend(fontsize=10, facecolor='#0a0a1a', edgecolor='#334155', labelcolor='white')
ax1.tick_params(colors='#94a3b8')
ax1.grid(True, alpha=0.2)
ax1.set_ylim(-0.05, 1.05)

# --- Right: Density of states and occupied states ---
# g(E) proportional to sqrt(E) for free electron model
# Using arbitrary units for visualization
E_pos = np.linspace(0, 12, 1000)
g_E = np.sqrt(E_pos)  # proportional to sqrt(E)
g_E = g_E / np.max(g_E)  # normalize for display

# Occupied states at T = 300 K and T = 3000 K
f_300 = 1.0 / (np.exp((E_pos - Ef) / (kB * 300)) + 1.0)
f_3000 = 1.0 / (np.exp((E_pos - Ef) / (kB * 3000)) + 1.0)

ax2.plot(E_pos, g_E, color='#94a3b8', linewidth=2, label='g(E) (DOS)', linestyle='--')
ax2.fill_between(E_pos, 0, g_E * f_300, alpha=0.4, color='#22d3ee', label='Occupied (300 K)')
ax2.fill_between(E_pos, 0, g_E * f_3000, alpha=0.3, color='#ef4444', label='Occupied (3000 K)')
ax2.axvline(x=Ef, color='white', linestyle='--', alpha=0.5, linewidth=1)
ax2.set_xlabel('Energy (eV)', color='#94a3b8', fontsize=12)
ax2.set_ylabel('g(E) (arb. units)', color='#94a3b8', fontsize=12)
ax2.set_title('Density of States & Occupation', color='white', fontsize=14)
ax2.legend(fontsize=10, facecolor='#0a0a1a', edgecolor='#334155', labelcolor='white')
ax2.tick_params(colors='#94a3b8')
ax2.grid(True, alpha=0.2)

print("\nFermi-Dirac distribution values at E = Ef:")
for T in [300, 1000, 3000]:
    print(f"  T = {T} K: f(Ef) = 0.500 (always 1/2 at Ef)")
print("\nThermal broadening ~ kT:")
for T in [300, 1000, 3000]:
    print(f"  T = {T} K: kT = {kB*T*1000:.1f} meV")

plt.tight_layout()
plt.savefig('fermi_dirac.png', dpi=150, bbox_inches='tight', facecolor='#0a0a1a')
print("\nPlot saved as 'fermi_dirac.png'")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran: Fermi Energy & Heat Capacity

Computes the Fermi energy, Fermi velocity, Fermi temperature, and electronic heat capacity coefficient for common metals using the free electron model.

Fermi Energy & Electronic Properties of Metals

Fortran

Free electron model calculations for Cu, Al, Ag, Au, Na, K

fermi_energy.f9066 lines

program fermi_energy_metals
  implicit none
  integer, parameter :: dp = selected_real_kind(15)
  real(dp), parameter :: pi = 3.14159265358979_dp
  real(dp), parameter :: hbar = 1.0546d-34    ! J*s
  real(dp), parameter :: me = 9.109d-31       ! kg
  real(dp), parameter :: eV = 1.602d-19       ! J
  real(dp), parameter :: kB_J = 1.381d-23     ! J/K
  real(dp), parameter :: kB_eV = 8.617d-5     ! eV/K
  real(dp), parameter :: Na = 6.022d23        ! Avogadro

integer, parameter :: nmet = 6
  character(len=4) :: names(nmet)
  real(dp) :: rho(nmet), A_m(nmet), valence(nmet)
  real(dp) :: n_e, Ef, vf, Tf, gamma_e
  integer :: i

! Metal properties: density (kg/m^3), atomic mass (g/mol), valence electrons
  names =    (/ 'Cu  ', 'Al  ', 'Ag  ', 'Au  ', 'Na  ', 'K   ' /)
  rho =      (/ 8960.0_dp, 2700.0_dp, 10490.0_dp, 19300.0_dp, 971.0_dp, 862.0_dp /)
  A_m =      (/ 63.55_dp, 26.98_dp, 107.87_dp, 196.97_dp, 22.99_dp, 39.10_dp /)
  valence =  (/ 1.0_dp, 3.0_dp, 1.0_dp, 1.0_dp, 1.0_dp, 1.0_dp /)

write(*,'(A)') '=========================================================='
  write(*,'(A)') ' Free Electron Model: Fermi Energy & Electronic Properties'
  write(*,'(A)') '=========================================================='
  write(*,'(A)') ''
  write(*,'(A)') 'Metal  n_e(m-3)     Ef(eV)   vF(m/s)    TF(K)     gamma'
  write(*,'(A)') '                                                   (mJ/mol K^2)'
  write(*,'(A)') '-----  --------     ------   ------     -----     -----'

do i = 1, nmet
    ! Free electron density
    n_e = rho(i) * Na * 1000.0_dp * valence(i) / (A_m(i) * 1.0d-3)
    ! n_e in m^-3: rho in kg/m^3, Na in mol^-1, A_m in kg/mol
    n_e = rho(i) * Na * valence(i) / (A_m(i) * 1.0d-3)

! Fermi energy
    Ef = (hbar**2 / (2.0_dp * me)) * (3.0_dp * pi**2 * n_e)**(2.0_dp/3.0_dp)

! Fermi velocity
    vf = sqrt(2.0_dp * Ef / me)

! Fermi temperature
    Tf = Ef / kB_J

! Electronic heat capacity coefficient (Sommerfeld)
    ! gamma = pi^2 * n_e * kB^2 / (2 * Ef)  per unit volume
    ! Per mole: gamma = pi^2 * Na * kB^2 / (2 * Ef)
    gamma_e = pi**2 * Na * kB_J**2 / (2.0_dp * Ef)
    ! Convert to mJ/mol/K^2
    gamma_e = gamma_e * 1000.0_dp

write(*,'(A4,2X,ES11.3,2X,F6.2,3X,ES9.3,2X,ES9.3,2X,F6.3)') &
      names(i), n_e, Ef/eV, vf, Tf, gamma_e
  end do

write(*,'(A)') ''
  write(*,'(A)') 'Notes:'
  write(*,'(A)') '  Ef = (hbar^2/2m)(3*pi^2*n)^(2/3)'
  write(*,'(A)') '  vF = sqrt(2*Ef/m)'
  write(*,'(A)') '  TF = Ef/kB'
  write(*,'(A)') '  gamma = pi^2*Na*kB^2/(2*Ef)  [Sommerfeld coefficient]'
  write(*,'(A)') ''
  write(*,'(A)') 'Done.'
end program fermi_energy_metals

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

← 3.1 Semiconductors & Doping Part 4: Crystallography →