Electronic Band Structure | Physics

In the simplest model, conduction electrons move freely in a box of volume $V$ (the “jellium” model). The allowed wave vectors are:

$\mathbf{k} = \frac{2\pi}{L}(n_x, n_y, n_z), \quad n_i \in \mathbb{Z}$

The energy spectrum:

$\varepsilon(\mathbf{k}) = \frac{\hbar^2 k^2}{2m_e}$

The Fermi wave vector is determined by the electron density $n = N/V$:

$k_F = (3\pi^2 n)^{1/3}$

The Fermi energy:

$\varepsilon_F = \frac{\hbar^2 k_F^2}{2m_e}$

For a 3D free electron gas:

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

Derivation. The number of states with $\lvert\mathbf{k}\rvert \leq k$ is:

$N(k) = 2 \cdot \frac{V}{(2\pi)^3} \cdot \frac{4\pi k^3}{3}$

Where the factor of 2 accounts for spin. Differentiating: $g(k)\,dk = dN/dk\,dk = (Vk^2/\pi^2)\,dk$. Converting to energy: $g(\varepsilon) = g(k)\lvert dk/d\varepsilon\rvert = (Vk^2/\pi^2)(m_e/\hbar^2 k)$. $\blacksquare$

At the Fermi energy: $g(\varepsilon_F) = \frac{3N}{2\varepsilon_F}$.

The Fermi surface is the surface in $\mathbf{k}$-space defined by $\varepsilon(\mathbf{k}) = \varepsilon_F$. For the free electron gas, this is a sphere of radius $k_F$. The shape of the Fermi surface Strongly influences transport properties (conductivity, Hall effect, cyclotron resonance).

In real metals, the periodic potential distorts the Fermi surface from a sphere. At the Brillouin Zone boundaries, band gaps open and the Fermi surface can develop “necks” (connecting to adjacent Zones) or become multiply connected. The topology of the Fermi surface determines whether a material Is a metal or insulator: a material is metallic if the Fermi surface crosses any Brillouin zone Boundary.

The number of electrons per atom determines the filling: 1 electron/atom (e.g., Na, Cu) gives a Nearly spherical Fermi surface well within the first BZ. 2 electrons/atom (e.g., Mg) nearly fills The first BZ and the Fermi surface contacts the zone boundary. 3—4 electrons/atom (e.g., Al, Pb) Produce complex multiply-connected Fermi surfaces.

Theorem 5.1 (Bloch, 1928). The eigenstates of the one-electron Hamiltonian in a periodic Potential $V(\mathbf{r} + \mathbf{R}) = V(\mathbf{r})$ can be written as:

$\psi_{n\mathbf{k}}(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}} u_{n\mathbf{k}}(\mathbf{r})$

Where $u_{n\mathbf{k}}(\mathbf{r})$ has the periodicity of the lattice: $u_{n\mathbf{k}}(\mathbf{r} + \mathbf{R}) = u_{n\mathbf{k}}(\mathbf{r})$.

Proof. The translation operators $\hat{T}_{\mathbf{R}}$ commute with the Hamiltonian $\hat{H} = -\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r})$ since $V$ is periodic. Therefore, the Eigenstates of $\hat{H}$ can be chosen as simultaneous eigenstates of all $\hat{T}_{\mathbf{R}}$:

$\hat{T}_{\mathbf{R}}\psi(\mathbf{r}) = \psi(\mathbf{r} + \mathbf{R}) = c_{\mathbf{R}}\psi(\mathbf{r})$

From the composition rule $\hat{T}_{\mathbf{R}_1}\hat{T}_{\mathbf{R}_2} = \hat{T}_{\mathbf{R}_1 + \mathbf{R}_2}$:

$c_{\mathbf{R}_1 + \mathbf{R}_2} = c_{\mathbf{R}_1} c_{\mathbf{R}_2}$

The only solution of this functional equation is $c_{\mathbf{R}} = e^{i\mathbf{k}\cdot\mathbf{R}}$. Therefore $\psi(\mathbf{r} + \mathbf{R}) = e^{i\mathbf{k}\cdot\mathbf{R}}\psi(\mathbf{r})$Which is Satisfied by $\psi(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}}u_{\mathbf{k}}(\mathbf{r})$ with $u_{\mathbf{k}}$ periodic. $\blacksquare$

Consequences:

• $\mathbf{k}$ is defined only up to a reciprocal lattice vector: $\mathbf{k}$ and $\mathbf{k} + \mathbf{G}$ are equivalent.
• The energy spectrum consists of bands $\varepsilon_n(\mathbf{k})$Each labelled by a band index $n$.
• Band gaps appear between allowed energy bands.

Starting from the free electron model, a weak periodic potential $V(\mathbf{r}) = \sum_{\mathbf{G}} V_{\mathbf{G}} e^{i\mathbf{G}\cdot\mathbf{r}}$ Opens gaps at the Brillouin zone boundaries where $\lvert\mathbf{k}\rvert = \lvert\mathbf{k} + \mathbf{G}\rvert$ (Bragg Condition).

At the zone boundary $\mathbf{k} = \mathbf{G}/2$The gap is:

$\Delta\varepsilon = 2\lvert V_{\mathbf{G}}\rvert$

Derivation. Near the zone boundary, the free electron states at $\mathbf{k}$ and $\mathbf{k} - \mathbf{G}$ Are degenerate: $\varepsilon_{\mathbf{k}}^0 = \varepsilon_{\mathbf{k} - \mathbf{G}}^0$. Degenerate Perturbation theory gives:

$\det\begin{pmatrix} \varepsilon_{\mathbf{k}}^0 - E & V_{\mathbf{G}} \\ V_{\mathbf{G}}^* & \varepsilon_{\mathbf{k} - \mathbf{G}}^0 - E \end{pmatrix} = 0$

At $\mathbf{k} = \mathbf{G}/2$: $E = \varepsilon_{\mathbf{G}/2}^0 \pm \lvert V_{\mathbf{G}}\rvert$ So the gap is $2\lvert V_{\mathbf{G}}\rvert$. $\blacksquare$

The Drude model (1900) treats conduction electrons as a classical ideal gas scattering off Static ions with a mean free time $\tau$ (relaxation time).

Equation of motion. Under an electric field $\mathbf{E}$:

$m_e\frac{d\mathbf{v}}{dt} = -e\mathbf{E} - \frac{m_e\mathbf{v}}{\tau}$

The second term represents a frictional drag with characteristic time $\tau$.

DC conductivity. In steady state ($d\mathbf{v}/dt = 0$): $\mathbf{v}_d = -\frac{e\tau}{m_e}\mathbf{E}$. The current density: $\mathbf{J} = -ne\mathbf{v}_d = \frac{ne^2\tau}{m_e}\mathbf{E}$.

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

AC conductivity. For $\mathbf{E}(t) = \mathbf{E}_0\,e^{-i\omega t}$The Drude model gives:

$\sigma(\omega) = \frac{ne^2\tau/m_e}{1 - i\omega\tau} = \frac{\sigma_0}{1 - i\omega\tau}$

The real part $\mathrm{Re}[\sigma(\omega)] = \frac{\sigma_0}{1 + \omega^2\tau^2}$ describes absorption, Peaking at $\omega = 0$ (the Drude peak). This explains the metallic reflectivity in the infrared.

Hall effect. With $\mathbf{B} = B\hat{z}$ applied, the steady-state equation becomes:

$-e\mathbf{E} - \frac{m_e\mathbf{v}}{\tau} - e\mathbf{v} \times \mathbf{B} = 0$

For current $\mathbf{J} = J_x\hat{x}$A transverse field $E_y$ develops:

$R_H = \frac{E_y}{J_x B} = -\frac{1}{ne}$

This provides a direct measurement of the carrier density $n$.

Successes: Ohm”s law ($\mathbf{J} = \sigma\mathbf{E}$), Wiedemann—Franz law ($\kappa/\sigma T = \frac{\pi^2 k_B^2}{3e^2}$), Hall effect.

Failures: Predicts $\chi \propto T^{-1}$ (Curie law) for magnetic susceptibility, but real Metals have nearly temperature-independent Pauli paramagnetism. Predicts $C_V = \frac{3}{2}nk_B$ But experiments give $C_V \ll \frac{3}{2}nk_B$ at room temperature.

The Sommerfeld model (1928) corrects the Drude model by treating electrons as a Fermi gas Obeying Fermi—Dirac …/4-statistics-and-probability/2_statistics:

$f(\varepsilon) = \frac{1}{e^{(\varepsilon - \mu)/k_B T} + 1}$

At $T = 0$The chemical potential equals the Fermi energy: $\mu(0) = \varepsilon_F$. At finite $T$:

$\mu(T) = \varepsilon_F\left[1 - \frac{\pi^2}{12}\left(\frac{k_B T}{\varepsilon_F}\right)^2 + \cdots\right]$

Since $\varepsilon_F/k_B \sim 10^4$ K for metals, the correction at room temperature is negligible: The chemical potential is essentially constant.

Electronic specific heat. By the Sommerfeld expansion:

$C_e = \frac{\pi^2}{3}k_B^2\,g(\varepsilon_F)\,T = \gamma T$

Where $\gamma = \frac{\pi^2}{2}\frac{Nk_B^2}{\varepsilon_F}$. At room temperature, only electrons within $\sim k_B T$ of $\varepsilon_F$ can be thermally excited, which is a tiny fraction $\sim T/T_F \sim 1/100$ of the total. This explains why $C_e \ll \frac{3}{2}Nk_B$.

Pauli paramagnetism. The spin susceptibility of a degenerate electron gas:

$\chi_P = \mu_0\mu_B^2\,g(\varepsilon_F) = \frac{3\mu_0\mu_B^2 N}{2\varepsilon_F}$

This is independent of $T$ (up to corrections of order $(T/T_F)^2$), in contrast to the Curie law $\chi \propto 1/T$ of the Drude model.

The tight-binding model starts from isolated atomic orbitals and treats the overlap between Neighbours as a perturbation. For a 1D chain with lattice constant $a$ and a single $s$-orbital Of energy $\varepsilon_0$:

$\psi_k(r) = \frac{1}{\sqrt{N}}\sum_n e^{ikna}\,\phi(r - na)$

Where $\phi(r - na)$ is the atomic orbital centred at site $n$.

Dispersion relation (nearest-neighbour approximation):

$\varepsilon(k) = \varepsilon_0 - 2t\cos(ka)$

Where $t = -\int \phi^*(r - na)\,\hat{H}\,\phi(r - (n+1)a)\,dr$ is the hopping integral ($t > 0$ for typical $s$-orbitals).

Key features:

• Band width: $W = 4t$.
• Minimum at $k = 0$: $\varepsilon_{\min} = \varepsilon_0 - 2t$.
• Maximum at $k = \pm\pi/a$: $\varepsilon_{\max} = \varepsilon_0 + 2t$.
• Effective mass at band bottom: $m^* = \hbar^2/(2ta^2)$.

Extension to 3D: For a simple cubic lattice with nearest-neighbour hopping:

$\varepsilon(\mathbf{k}) = \varepsilon_0 - 2t(\cos k_x a + \cos k_y a + \cos k_z a)$

The band width is $W = 12t$ and the density of states develops a van Hove singularity at $\varepsilon = \varepsilon_0$.

Near a band extremum at $\mathbf{k}_0$The energy can be expanded:

$\varepsilon(\mathbf{k}) = \varepsilon_0 + \frac{\hbar^2}{2}\sum_{ij}(m^{-1})_{ij}(k_i - k_{0,i})(k_j - k_{0,j})$

The effective mass tensor $(m^{-1})_{ij} = \frac{1}{\hbar^2}\frac{\partial^2 \varepsilon}{\partial k_i \partial k_j}$ Determines the response to external fields. For isotropic bands, $m^* = \hbar^2/(d^2\varepsilon/dk^2)$.

A large effective mass means a flat band (small group velocity). A small effective mass means a Steep band (high mobility).

The effective mass can be negative near a band maximum (holes). Cyclotron resonance experiments Measure $m^*$ directly: the resonance frequency is $\omega_c = eB/m^*$.

flowchart TD
    A[5_Electronic Band Structure] --> B[Key Concepts]
    A --> C[Core Principles]
    A --> D[Practical Applications]
    B --> E[Fundamental definitions]
    C --> F[Design patterns]
    D --> G[Real-world usage]

Band structure is like a highway system for electrons. In free space, electrons can have any energy, but a crystal lattice acts like a periodic toll booth that blocks certain energy ranges entirely. These forbidden zones are band gaps, and they are why some materials conduct while others insulate. The Brillouin zone is the set of unique momentum states an electron can occupy in the lattice, analogous to how a repeating wallpaper pattern has a single tile that encodes the whole design. Effective mass captures how strongly the lattice potential slows or accelerates an electron, allowing us to treat it as a free particle with modified inertia.

• Transport Properties: Uses the band structure and Fermi surface to explain electrical and thermal conductivity via the Drude and Boltzmann models.

• Superconductivity: Cooper pairing and the superconducting gap arise from the electronic structure at the Fermi surface.

• Topological Insulators and Semimetals: Extends band theory to include topological invariants that classify insulating and semimetallic phases.

• Calculus

• Linear Algebra

• Vector Calculus

Mistake 1: Assuming that the free electron model applies to all metals. The free electron model works well for simple metals (e.g., alkali metals) but fails for transition metals and other materials with complex band structures. Do not assume that the free electron model is universally applicable.

Mistake 2: Confusing the Fermi energy with the Fermi level. The Fermi energy $\varepsilon_F$ is the energy of the highest occupied state at $T = 0$, while the Fermi level $E_F$ is the chemical potential at finite temperature. In metals, they are approximately equal, but in semiconductors they can differ significantly. Do not confuse the two concepts.

Mistake 3: Forgetting that band gaps arise from the periodic potential. Band gaps open at Brillouin zone boundaries due to the periodic potential of the lattice. Without a periodic potential (free electron model), there are no band gaps. Do not assume that band gaps exist in all materials; they require a periodic potential.

Mistake 4: Assuming that DFT gives exact band gaps. DFT with LDA or GGA functionals underestimates band gaps by 30—50%. Hybrid functionals or GW calculations are needed for accurate band gaps. Do not assume that DFT band gaps are quantitatively accurate.

Mistake 5: Confusing the effective mass with the rest mass. The effective mass $m^*$ describes how an electron responds to external fields in a periodic potential. It can be larger or smaller than the rest mass $m_e$, and can even be negative. Do not assume that the effective mass equals the rest mass.