Electronic Band Theory of Solids

The electronic behavior of metallic solids was first addressed quantum mechanically by Arnold Sommerfeld (1928), who combined the classical Drude free-electron gas with Pauli's exclusion principle and Fermi-Dirac statistics.

Quantum Particle-in-a-Box State Counting

Consider $N$ non-interacting conduction electrons confined within a macroscopic cubic crystal of volume $V = L^3$. The time-independent Schrödinger equation for a free electron is:

Imposing periodic Born-von Kármán boundary conditions $\psi(x+L, y, z) = \psi(x, y, z)$ gives plane-wave eigenstates:

The parabolic dispersion relation is:

Fermi Sphere, Fermi Energy & Density of States

At absolute zero ($T = 0\text{ K}$), electrons occupy the lowest available energy levels up to the Fermi energy $E_F$. In reciprocal $\mathbf{k}$-space, occupied states form a sphere of radius $k_F$ (the Fermi wavevector). Each $\mathbf{k}$-state occupies a volume of $(2\pi/L)^3 = 8\pi^3/V$. Including spin degeneracy ($g_s = 2$):

where $n = N/V$ is the conduction electron number density. The Fermi energy is:

The three-dimensional Density of States (DOS) $g(E)$, defined such that $g(E)dE$ is the number of electron states per unit volume between $E$ and $E+dE$, is derived from $n(E) = \frac{1}{3\pi^2}\left(\frac{2mE}{\hbar^2}\right)^{3/2}$:

Fermi-Dirac Distribution Function

At non-zero temperature ($T > 0\text{ K}$), the thermal occupation probability of an orbital at energy $E$ is dictated by the Fermi-Dirac distribution:

where $\mu$ is the chemical potential (which satisfies $\mu(0) = E_F$ and $\mu(T) \approx E_F [1 - \frac{\pi^2}{12}(k_B T/E_F)^2]$). Only electrons within an energy window of approximately $\sim k_B T$ around $E_F$ participate in thermal conduction, electrical transport, and Pauli paramagnetism.

The Sommerfeld model fails to explain why some materials are insulators or semiconductors, why Hall coefficients are sometimes positive, and why electrons can travel thousands of atomic spacings without scattering. The resolution lies in the periodic crystal potential $V(\mathbf{r} + \mathbf{R}) = V(\mathbf{r})$, where $\mathbf{R}$ is any direct lattice translation vector.

Bloch's Theorem

Felix Bloch (1928) proved that the stationary wavefunctions of a single electron moving in a spatially periodic potential take the form of a plane wave modulated by a function having the periodicity of the Bravais lattice:

where the periodic Bloch function $u_{\mathbf{k}}(\mathbf{r})$ satisfies:

An equivalent formulation of Bloch's theorem is the phase-shift relation:

Physical Implications of Bloch's Theorem

1. Crystal Momentum $\hbar\mathbf{k}$: The wavevector $\mathbf{k}$ is not the true physical momentum (the electron experiences forces from the lattice), but a quantum number called crystal momentum.

2. Translation Operator Invariance: If $\mathbf{G}$ is any reciprocal lattice vector, $e^{i \mathbf{G} \cdot \mathbf{R}} = 1$. Consequently, wavevectors differing by $\mathbf{G}$ are physically identical:

3. The First Brillouin Zone (1BZ):

Because $E(\mathbf{k})$ is strictly periodic in reciprocal space, all unique energy eigenvalues are completely contained within the First Brillouin Zone, defined as the Wigner-Seitz primitive cell of the reciprocal lattice centered at $\mathbf{k} = \mathbf{0}$ ($\Gamma$ point). High-symmetry points inside the 1BZ (such as $\Gamma, X, L, K$ in FCC or $\Gamma, H, P, N$ in BCC) define the band dispersion curves.

To understand how energy band gaps arise at zone boundaries, consider a weak periodic perturbation potential acting on free electrons:

where $G = 2\pi/a$ is the shortest 1D reciprocal lattice vector.

Bragg Reflection & Standing Waves at Zone Boundaries

For electron wavevectors far from the zone boundary ($k \ll \pi/a$), the perturbation has negligible effect, and $E(k) \approx \hbar^2 k^2 / 2m$. However, as $k$ approaches the first Brillouin zone boundary:

the electron satisfies the Bragg condition for diffraction:

At this boundary, the forward-propagating plane wave $e^{i \pi x/a}$ and the backscattered wave $e^{-i \pi x/a}$ interfere coherently, forming two stationary standing waves:

Energy Splitting and the Band Gap $E_g$

The probability densities of these two standing waves are:

• $\rho_+(x) = |\psi_+(x)|^2 = 2 \cos^2(\pi x / a) = 1 + \cos(2\pi x / a)$
• $\rho_-(x) = |\psi_-(x)|^2 = 2 \sin^2(\pi x / a) = 1 - \cos(2\pi x / a)$

For attractive atomic potentials, the potential energy $V(x)$ has negative minima located at the atomic cores ($x = 0, \pm a, \dots$):

• $\psi_+$ piles up electron charge directly onto the positively charged atomic nuclei ($x = 0$), lowering its electrostatic energy:

• $\psi_-$ concentrates electron charge between the nuclei ($x = \pm a/2$), raising its electrostatic energy:

This creates an energy discontinuity—the fundamental band gap $E_g$—at the Brillouin zone boundary:

No traveling electron waves can propagate through the crystal with energies inside this forbidden interval.

Ralph Kronig and William Penney (1931) introduced an analytically solvable 1D model that demonstrates how periodic square potential wells naturally generate continuous energy bands and discrete forbidden band gaps.

Formulation of the 1D Kronig-Penney Potential

The periodic potential consists of a regular array of square barriers of height $V_0$ and width $b$, separated by wells of width $a$ (lattice period $d = a + b$):

For energies $0 < E < V_0$, the Schrödinger solutions in the two regions are:

• Region 1 ($0 < x < a$, $V = 0$):

• Region 2 ($-b < x < 0$, $V = V_0$):

Applying Bloch's theorem $\psi(x + d) = e^{i k d} \psi(x)$ along with wavefunction and derivative continuity boundary conditions at $x = 0$ and $x = a$ generates a $4 \times 4$ secular determinant for coefficients $A, B, C, D$.

The Dirac Delta Comb Limit

In the limiting case where the barrier width vanishes ($b \to 0$) while barrier height approaches infinity ($V_0 \to \infty$) such that the barrier strength $P = \lim \frac{m V_0 b a}{\hbar^2}$ remains finite, the secular determinant reduces to the celebrated Kronig-Penney dispersion relation:

where:

• $\alpha = \sqrt{2mE}/\hbar$
• $P$ is the dimensionless barrier strength parameter
• $k$ is the electron Bloch wavevector in the 1BZ ($-\pi/a \le k \le +\pi/a$).

Physical Interpretation of Kronig-Penney Solutions

Because the right-hand side is strictly bounded by $[-1, +1]$:

• Allowed Energy Bands: Values of $\alpha a$ (and therefore energy $E = \hbar^2 \alpha^2 / 2m$) for which the left-hand function lies between $-1$ and $+1$ yield real wavevectors $k$, forming continuous bands.
• Forbidden Band Gaps: Regions where $\left|P \frac{\sin(\alpha a)}{\alpha a} + \cos(\alpha a)\right| > 1$ require complex $k$, corresponding to decaying evanescent waves that cannot propagate through the lattice.
• Limits:
• As $P \to 0$ (free electrons): $\cos(\alpha a) = \cos(ka) \implies \alpha = k \implies E = \hbar^2 k^2 / 2m$ (single continuous parabolic band).
• As $P \to \infty$ (infinitely bound isolated atoms): $\sin(\alpha a) = 0 \implies \alpha a = n\pi \implies E_n = \frac{n^2 \pi^2 \hbar^2}{2m a^2}$ (discrete bound states).

While the nearly free electron model begins from free electrons slightly perturbed by a lattice, the tight-binding approximation (Linear Combination of Atomic Orbitals, LCAO) approaches the solid from the opposite limit: isolated atomic orbitals perturbed by neighboring atoms.

Mathematical Formulation

Let $\phi_m(\mathbf{r} - \mathbf{R}_n)$ be an atomic orbital centered at lattice site $\mathbf{R}_n$. The Bloch-adapted crystal wavefunction is:

The expectation value of the crystal Hamiltonian $\hat{H} = -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{crystal}}(\mathbf{r})$ is:

Assuming orthogonalized Wannier/atomic orbitals ($\langle \phi(\mathbf{r} - \mathbf{R}_n) | \phi(\mathbf{r} - \mathbf{R}_{n'}) \rangle = \delta_{n,n'}$):

Derivation for a 1D Monatomic Chain

Consider a 1D chain with lattice constant $a$. Retaining only on-site and nearest-neighbor interactions:

• On-site Coulomb integral ($\alpha$ or $\varepsilon_0$):

• Nearest-neighbor transfer integral (hopping parameter) ($-t$ or $-\beta$):

The energy dispersion reduces to:

where $-\pi/a \le k \le +\pi/a$.

Band Properties: Bandwidth and Effective Mass

1. Total Bandwidth $W$:

• Minimum energy: at $k = 0$, $E_{\min} = \varepsilon_0 - 2t$.
• Maximum energy: at $k = \pm \pi/a$, $E_{\max} = \varepsilon_0 + 2t$.

The bandwidth is directly proportional to the hopping integral $t$: stronger orbital overlap broadens the band.

2. Effective Mass $m^*$:

Expanding $E(k)$ near the band bottom ($k \approx 0$):

Comparing with $E(k) = E_{\min} + \frac{\hbar^2 k^2}{2m^*}$:

Stronger orbital overlap (larger $t$) results in lighter, more mobile charge carriers.

The fundamental distinction between electrical conductors, semiconductors, and insulators is determined by the degree of band filling and the magnitude of the band gap at the Fermi energy $E_F$.

Classification by Electronic Band Topology

1. Metals (Conductors):

• Possess a partially filled band at $T = 0\text{ K}$ (e.g., monovalent alkali metals $\text{Na}, \text{Cu}$ where the valence $s$-band is half-filled).
• Alternatively, exhibit band overlap where the top of a filled valence band lies higher in energy than the bottom of an empty conduction band (e.g., divalent alkaline earths $\text{Mg}, \text{Ca}$ with overlapping $s$- and $p$-bands).
• Because empty electronic states are infinitesimally close to occupied states ($g(E_F) > 0$), applying an infinitesimal electric field shifts electron momentum, generating high electrical conductivity ($\sigma \sim 10^7\text{ S/m}$) that decreases with temperature ($d\sigma/dT < 0$) due to phonon scattering.

2. Insulators:

• Possess a completely filled valence band separated from an entirely empty conduction band by a large forbidden band gap:

• Examples: Diamond ($E_g = 5.47\text{ eV}$), quartz $\text{SiO}_2$ ($E_g = 9.0\text{ eV}$), sapphire $\text{Al}_2\text{O}_3$ ($E_g = 8.8\text{ eV}$).
• Thermal excitation across $E_g$ at room temperature is negligible ($k_B T \approx 0.0259\text{ eV} \ll E_g$). Electrical conductivity is practically zero ($\sigma < 10^{-12}\text{ S/m}$).

3. Semiconductors:

• Possess the same electronic topology as insulators (completely filled valence band at $0\text{ K}$), but with a moderate band gap:

• Examples: $\text{Ge}$ ($0.66\text{ eV}$), $\text{Si}$ ($1.12\text{ eV}$), $\text{GaAs}$ ($1.42\text{ eV}$), $\text{GaN}$ ($3.44\text{ eV}$).
• Significant thermal promotion of electrons into the conduction band occurs at ambient temperatures, yielding measurable conductivity ($\sigma \sim 10^{-4} - 10^2\text{ S/m}$) that increases exponentially with temperature ($d\sigma/dT > 0$).

4. Semimetals:

• Possess a very small overlap between valence and conduction bands, or point touching at Dirac points with zero density of states at $E_F$ (e.g., bismuth $\text{Bi}$, graphite, and graphene).

In a pure, undoped intrinsic semiconductor, thermal excitation generates equal concentrations of conduction-band electrons ($n$) and valence-band holes ($p$):

Effective Density of States

Near the extrema of isotropic parabolic bands:

• Conduction band bottom ($E_c$): $E(k) = E_c + \frac{\hbar^2 k^2}{2m_e^*}$.
• Valence band top ($E_v$): $E(k) = E_v - \frac{\hbar^2 k^2}{2m_h^*}$.

The density of states in the conduction and valence bands are:

Integrating the Maxwell-Boltzmann approximation of the Fermi-Dirac distribution:

where $N_c$ and $N_v$ are the effective density of states:

The Law of Mass Action & Intrinsic Carrier Concentration

The product $np$ is strictly independent of the Fermi level $E_F$:

The carrier concentration exhibits an Arrhenius temperature dependence with effective activation energy equal to half the band gap ($E_g/2$).

The Intrinsic Fermi Level ($E_i$)

Equating $n = p$ and solving for the Fermi level:

At $T = 0\text{ K}$, the intrinsic Fermi level lies exactly at the center of the band gap ($E_g/2$). As temperature increases, if $m_h^ > m_e^$, $E_i$ shifts slightly upward toward the conduction band to maintain carrier equality.

The electrical conductivity of semiconductors and transition metal oxides can be tuned over ten orders of magnitude through the controlled introduction of chemical impurities (doping and controlled valency).

Extrinsic Semiconductors: Donors and Acceptors

1. $n$-type Semiconductors:

• Formed by substituting host atoms (e.g., tetravalent $\text{Si}$) with pentavalent donors (e.g., $\text{P}, \text{As}, \text{Sb}$).
• The fifth valence electron is loosely bound in a hydrogen-like orbit below the conduction band edge with binding energy:

Because $E_d \sim k_B T$, all donors are ionized at room temperature, making $n \approx N_d^+$.

2. $p$-type Semiconductors:

• Formed by trivalent acceptors (e.g., $\text{B}, \text{Al}, \text{Ga}$). Accepts an electron from the valence band, generating a hole with shallow binding energy $E_a \approx 10 - 50\text{ meV}$.

Temperature Regimes of Extrinsic Carriers

As temperature increases, carrier concentration passes through three regimes:

1. Freeze-out Regime ($T < 100\text{ K}$): Thermal energy is insufficient to ionize dopants; carriers remain trapped ($n \propto e^{-E_d/2k_B T}$).

2. Extrinsic (Saturation/Exhaustion) Regime ($100\text{ K} < T < 500\text{ K}$): All dopants are fully ionized ($n = N_d = \text{constant}$).

3. Intrinsic Regime ($T > 600\text{ K}$): Thermally generated intrinsic carriers swamp the dopant concentration ($n_i \gg N_d$), and $n(T)$ resumes exponential $e^{-E_g/2k_B T}$ growth.

Principle of Controlled Valency (Verwey Principle)

In transition metal oxides, stoichiometric monoxides (such as pure $\text{NiO}$) are insulating antiferromagnetic Mott insulators ($d^8$). Evert Verwey (1950) demonstrated that substituting monovalent $\text{Li}^+$ for divalent $\text{Ni}^{2+}$ forces an equivalent number of neighboring nickel cations to oxidize from $\text{Ni}^{2+}$ to $\text{Ni}^{3+}$ to maintain charge neutrality:

Conduction occurs via small-polaron hopping of electrons between adjacent $\text{Ni}^{2+}$ and $\text{Ni}^{3+}$ centers with activation energy $E_h$:

This transform pure insulating $\text{NiO}$ ($\sigma < 10^{-13}\text{ S/cm}$) into a $p$-type semiconductor with $\sigma > 1\text{ S/cm}$.