Electronic Band Structure & Bloch Electron Dynamics

1. Electrons in a Periodic Crystal Potential

An electron in a perfect crystalline solid experiences a potential energy $V(\vec{r})$ that possesses the full discrete translational periodicity of the Bravais lattice:

The single-electron Schrödinger equation is:

2. Formal Statement & Proof of Bloch's Theorem

Because the Hamiltonian commutes with all lattice translation operators $\hat{T}_{\vec{R}}$ ($[\hat{H}, \hat{T}_{\vec{R}}] = 0$), simultaneous eigenstates of $\hat{H}$ and $\hat{T}_{\vec{R}}$ can be constructed. Bloch's Theorem states that the eigenstates of an electron in a periodic potential can be chosen in the form of a plane wave modulated by a periodic function $u_{n,\vec{k}}(\vec{r})$ having the periodicity of the lattice:

where $u_{n,\vec{k}}(\vec{r} + \vec{R}) = u_{n,\vec{k}}(\vec{r})$ for all lattice vectors $\vec{R}$. Equivalently, translating by any lattice vector $\vec{R}$ merely shifts the phase of the wavefunction:

The vector $\vec{k}$ is the crystal wavevector, and the integer $n = 1, 2, 3, \dots$ is the band index.

3. Crystal Momentum vs. Physical Momentum

The quantity $\hbar \vec{k}$ is known as crystal momentum. It is not the eigenvalue of the physical momentum operator $-i\hbar\nabla$ (since momentum is not conserved in the presence of the lattice potential). Instead, crystal momentum is conserved in electron-photon, electron-phonon, and electron-electron scattering events modulo an arbitrary reciprocal lattice vector $\vec{G}$:

1. The Kronig-Penney 1D Solvable Model

R. de L. Kronig and W. G. Penney (1931) introduced an idealized one-dimensional periodic array of rectangular potential barriers of height $V_0$, barrier width $b$, and well width $a$ (lattice constant $d = a + b$):

In Region I ($0 < x < a$), $\psi_1(x) = A e^{i K x} + B e^{-i K x}$, where $K = \sqrt{2mE/\hbar^2}$.

In Region II ($-b < x < 0$), for $E < V_0$, $\psi_2(x) = C e^{Q x} + D e^{-Q x}$, where $Q = \sqrt{2m(V_0 - E)/\hbar^2}$.

2. The Dirac Delta-Function Barrier Limit

Taking the limit where the barriers become infinitely thin and tall ($b \to 0, V_0 \to \infty$) while their barrier area $P = \lim \frac{m V_0 b a}{\hbar^2}$ remains constant, matching wavefunctions and their derivatives across the boundary using Bloch's theorem yields the famous Kronig-Penney dispersion relation:

where $P$ is the dimensionless barrier strength (representing the strength of the periodic crystal potential).

3. Origin of Energy Bands and Band Gaps

Because the right-hand side is $\cos(k a)$, real solutions for crystal wavevector $k$ exist if and only if the left-hand side satisfies:

• Allowed Energy Bands: Values of $E = \frac{\hbar^2 K^2}{2m}$ where the condition is satisfied represent allowed energy bands.
• Forbidden Band Gaps ($E_g$): Energy ranges where the magnitude of the left-hand side exceeds unity ($> +1$ or $< -1$). No traveling Bloch waves can propagate through the crystal at these energies; electron waves undergo destructive interference and total Bragg reflection.
• Limits:
  • $P \to 0$ (Free Electrons): $\cos(Ka) = \cos(ka) \implies K = k \implies E = \frac{\hbar^2 k^2}{2m}$ (continuous parabolic spectrum).
  • $P \to \infty$ (Isolated Atoms): $\sin(Ka) = 0 \implies Ka = n\pi \implies E_n = \frac{n^2 \pi^2 \hbar^2}{2m a^2}$ (discrete bound atomic levels).

1. Representation of Energy Bands: Zone Schemes

Because energy eigenvalues satisfy $E_n(k + G) = E_n(k)$, the band structure can be represented in three equivalent representations:

1. Extended Zone Scheme: Band $n$ is plotted in the $n$-th Brillouin zone: band 1 in $[-\pi/a, +\pi/a]$, band 2 in $[-2\pi/a, -\pi/a] \cup [+\pi/a, +2\pi/a]$, etc. Most closely resembles the free-electron parabola $E = \hbar^2 k^2/2m$.
2. Reduced Zone Scheme: All energy bands are mapped back into the First Brillouin Zone ($-\pi/a \le k \le +\pi/a$) by translating by appropriate reciprocal lattice vectors $G = 2\pi n / a$. Standard representation used in modern solid-state physics.
3. Periodic (Repeated) Zone Scheme: The First Brillouin Zone dispersion is repeated periodically across all $k$-space with period $2\pi/a$. Convenient for visualizing semiclassical electron trajectories.

2. The Nearly Free Electron (NFE) Approximation

When the periodic crystal potential $V(x)$ is weak compared to electron kinetic energies, it can be treated as a perturbation. Expanding the potential in a Fourier series over reciprocal lattice vectors $G$:

Away from the Brillouin zone boundaries, standard non-degenerate perturbation theory shifts energies insignificantly. However, near the zone boundary $k = \pm G/2$, the two unperturbed plane-wave states $|k\rangle$ and $|k - G\rangle$ are degenerate ($E_0(k) \approx E_0(k - G)$).

Applying degenerate perturbation theory with trial state $\psi = c_1 |k\rangle + c_2 |k - G\rangle$ yields the $2 \times 2$ secular equation:

At the exact zone boundary $k = G/2$, $E_0(k) = E_0(k - G) = \frac{\hbar^2 (G/2)^2}{2m} = E_0$:

A band gap of magnitude $\Delta E_g = 2 |V_G|$ opens up at every Brillouin zone boundary. The standing wave solutions $\psi_+ \sim \cos(Gx/2)$ and $\psi_- \sim \sin(Gx/2)$ pile up electronic charge density either at the ion cores (lowering energy by $-|V_G|$) or midway between the ion cores (raising energy by $+|V_G|$).

1. Physical Foundation of Tight-Binding

In contrast to the nearly free electron model, the Tight-Binding Approximation assumes that electrons are tightly bound to individual atomic cores, spending most of their time in localized atomic orbitals $\phi(\vec{r} - \vec{R}_n)$. When atoms are brought together into a crystal lattice, the overlap between adjacent atomic wavefunctions broadens discrete atomic energy levels into continuous energy bands.

2. Formulation of the Bloch Sum

To satisfy Bloch's theorem, the trial crystal wavefunction is formed as a coherent linear combination of atomic orbitals (LCAO) summed over all $N$ lattice sites $\vec{R}_n$:

3. Derivation of 1D and 3D Tight-Binding Dispersion

Evaluating the expectation value of the crystal Hamiltonian $\hat{H} = \hat{H}_{\text{atom}} + \Delta U(\vec{r})$:

Retaining only on-site and nearest-neighbor matrix elements:

• On-site atomic energy: $\varepsilon_0 = \int \phi^*(\vec{r}) \hat{H} \phi(\vec{r}) d^3r \approx - E_0 - \alpha$.
• Nearest-neighbor transfer (hopping) integral: $t = - \int \phi^*(\vec{r}) \hat{H} \phi(\vec{r} - \vec{a}) d^3r > 0$.

For a one-dimensional chain with nearest-neighbor distance $a$:

The total bandwidth of the tight-binding band is $W = E_{\text{max}} - E_{\text{min}} = (\varepsilon_0 + 2t) - (\varepsilon_0 - 2t) = 4t$.

For a 3D Simple Cubic lattice with nearest-neighbor spacing $a$:

1. Semiclassical Equations of Motion for Bloch Electrons

An electron in an energy band $E_n(\vec{k})$ is described by a localized wave packet centered at position $\vec{r}$ and mean wavevector $\vec{k}$. Its dynamical evolution under external electromagnetic fields $\vec{E}$ and $\vec{B}$ is governed by the semiclassical equations of motion:

1. Group Velocity:
   $$\vec{v}(\vec{k}) = \frac{1}{\hbar} \nabla_{\vec{k}} E(\vec{k})$$
2. Rate of Change of Crystal Momentum:
   $$\hbar \frac{d\vec{k}}{dt} = \vec{F}_{\text{ext}} = - e [\vec{E} + \vec{v}(\vec{k}) \times \vec{B}]$$

2. Derivation of the Effective Mass Tensor $m^*$

Differentiating the group velocity with respect to time:

Substituting $\hbar \frac{dk_j}{dt} = F_j$:

Comparing with Newton's second law $\vec{a} = (m^*)^{-1} \vec{F}$ defines the Effective Mass Tensor $(m^*)_{ij}$:

In an isotropic 1D band: $m^* = \hbar^2 / \left(\frac{d^2E}{dk^2}\right)$.

• Near the bottom of a band ($\frac{d^2E}{dk^2} > 0$): $m^* > 0$. The electron accelerates in the direction of the applied electric force like a free electron.
• Near the top of a band ($\frac{d^2E}{dk^2} < 0$): $m^* < 0$. The electron accelerates in the direction opposite to the external force because of Bragg reflections from the lattice potential.

3. The Concept of Positive Holes

In a nearly filled band, the absence of an electron from a state with wavevector $\vec{k}_e$, energy $E_e$, and charge $-e$ is described as a quasiparticle called a hole:

• Wavevector: $\vec{k}_h = - \vec{k}_e$
• Energy: $E_h(\vec{k}_h) = - E_e(\vec{k}_e)$
• Velocity: $\vec{v}_h = \vec{v}_e$
• Charge: $q_h = + e$ (positive charge)
• Effective Mass: $m_h^* = - m_e^* > 0$ (positive effective mass at the band top)

Treating valence band transport in terms of positive holes simplifies the quantum statistical mechanics of semiconductors and metals.