Electronic Band Structure

5.1 Free Electron Model

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:

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

5.2 Density of States

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.

5.3 Bloch”s Theorem

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.

5.4 Nearly Free Electron Model

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 $2\lvert V_{\mathbf{G}}\rvert$. $\blacksquare$

5.5 Drude Model

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.

5.6 Sommerfeld Model

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.

5.7 Tight-Binding 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 $\varepsilon = \varepsilon_0$.

5.8 Effective Mass

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 $\omega_c = eB/m^*$.

:::caution Common Pitfall The effective mass is a tensor quantity . For crystals with cubic symmetry, it reduces to A scalar, but for anisotropic crystals (e.g., graphite, silicon), different effective masses apply Along different crystallographic directions. Always check the crystal symmetry before assuming $m^*$ is a scalar.

5.9 Band Structure Calculations

Modern band structure calculations are based on density functional theory (DFT), formulated by Hohenberg, Kohn, and Sham (1964—1965).

Hohenberg—Kohn theorems. (1) The ground-state energy of a many-electron system is a unique Functional of the electron density $n(\mathbf{r})$. (2) The correct ground-state density minimises This functional.

Kohn—Sham equations. The interacting system is mapped to a fictitious system of non-interacting Electrons in an effective potential:

\left[-\frac{\hbar^2}{2m}\nabla^2 + V_{\mathrm{eff}(\mathbf{r})\right]\psi_i(\mathbf{r}) = \varepsilon_i\psi_i(\mathbf{r})}

Where $V_{\mathrm{eff} = V_{\mathrm{ext} + V_H[n] + V_{\mathrm{xc}[n]}}}$. Here $V_{\mathrm{ext}}$ is the External (ionic) potential, $V_H$ is the Hartree (classical Coulomb) potential, and $V_{\mathrm{xc}}$ Is the exchange-correlation potential.

The electron density is $n(\mathbf{r}) = \sum_i \lvert\psi_i(\mathbf{r})\rvert^2$ (summing over occupied States). The Kohn—Sham equations are solved self-consistently.

Common approximations for $V_{\mathrm{xc}}$:

• Local density approximation (LDA): $V_{\mathrm{xc}(\mathbf{r}) = V_{\mathrm{xc}^{\mathrm{hom}(n(\mathbf{r}))}}}$ using the exchange-correlation energy of a homogeneous electron gas. Good for simple metals but tends to underestimate band gaps.
• Generalised gradient approximation (GGA): Includes the density gradient $\nabla n(\mathbf{r})$Improving accuracy for structural properties and band gaps.
• Hybrid functionals (e.g., HSE06): Mix a fraction of exact Hartree—Fock exchange with DFT exchange, giving improved band gaps at higher computational cost.

DFT accurately predicts structural properties (lattice constants, elastic constants, phonon Frequencies within a few percent) but is less reliable for band gaps (LDA underestimates By 30—50%) and strongly correlated systems (e.g., transition metal oxides).

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