Advanced Band Structure Methods & Many-Body Physics in Solids

1. The Many-Body Hamiltonian & Jellium Model

In real solids, electrons interact not only with the periodic ionic lattice potential $V_{\text{ion}}(\vec{r})$, but strongly with each other via the long-range Coulomb repulsion. In the uniform electron gas (jellium) model:

$$\hat{H} = \sum_{i=1}^N \frac{p_i^2}{2m} + \frac{1}{2}\sum_{i \ne j} \frac{e^2}{4\pi\epsilon_0 |\vec{r}_i - \vec{r}_j|} + \hat{H}_{\text{bg}} + \hat{H}_{\text{e-bg}}$$

The positive ion background $\hat{H}_{\text{bg}}$ and electron-background interaction $\hat{H}_{\text{e-bg}}$ exactly cancel the direct (Hartree) classical electrostatic divergence at $\vec{q} = 0$.

2. The Hartree-Fock Exchange Energy

In the Hartree-Fock approximation, the $N$-electron wavefunction is represented by a single Slater determinant of plane waves. The total energy per electron is:

$$E_{\text{HF}} / N = \langle \hat{T} \rangle + E_{\text{exchange}} = \frac{3}{5} E_F - \frac{3}{4} \frac{e^2 k_F}{4\pi^2 \epsilon_0}$$

Introducing the dimensionless Wigner-Seitz density parameter $r_s \equiv r_0 / a_0$ (where $\frac{4}{3}\pi r_0^3 = 1/n$):

$$\frac{E_{\text{HF}}}{N} = \left[ \frac{2.21}{r_s^2} - \frac{0.916}{r_s} \right] \text{ Rydbergs}$$

• At high densities ($r_s \ll 1$): Kinetic energy ($\propto 1/r_s^2$) dominates $\implies$ Free electron gas behavior.
• At low densities ($r_s \gg 1$): Exchange and correlation energies ($\propto -1/r_s$) dominate $\implies$ Wigner electron crystallization!

Failure of Hartree-Fock at the Fermi Surface: The single-particle Hartree-Fock energy dispersion is: $$\epsilon_{\text{HF}}(k) = \frac{\hbar^2 k^2}{2m} - \frac{e^2 k_F}{2\pi^2 \epsilon_0} \left[ 1 + \frac{k_F^2 - k^2}{2 k k_F} \ln\left| \frac{k + k_F}{k - k_F} \right| \right]$$ Differentiating to obtain the group velocity $v_g = \frac{1}{\hbar}\frac{d\epsilon}{dk}$, the logarithmic term causes the velocity to diverge logarithmically to infinity at the Fermi surface ($k = k_F$), forcing the density of states $g(E_F) \propto 1/v_g$ to drop unphysically to zero! This catastrophe occurs because Hartree-Fock completely neglects dynamic dielectric screening of the Coulomb interaction by other electrons.

1. Linear Dielectric Response Formalism

When an external test charge $\rho_{\text{ext}}(\vec{r}, t) = \rho_0 e^{i(\vec{q}\cdot\vec{r} - \omega t)}$ is introduced into the electron gas, it induces a screening charge density $\rho_{\text{ind}}(\vec{q}, \omega)$. In linear response theory:

$$\rho_{\text{ind}}(\vec{q}, \omega) = \chi(\vec{q}, \omega) \phi_{\text{total}}(\vec{q}, \omega)$$

where $\chi(\vec{q}, \omega)$ is the dynamic polarizability. The frequency- and wavevector-dependent dielectric function $\epsilon(\vec{q}, \omega)$ is defined by:

$$\phi_{\text{total}}(\vec{q}, \omega) = \frac{\phi_{\text{ext}}(\vec{q}, \omega)}{\epsilon(\vec{q}, \omega)} \implies \epsilon(\vec{q}, \omega) = 1 - \frac{e^2}{\epsilon_0 q^2} \chi(\vec{q}, \omega)$$

2. The Lindhard Dielectric Function

Jens Lindhard (1954) derived the exact quantum polarizability of a degenerate Fermi gas in the Random Phase Approximation (RPA):

$$\epsilon(\vec{q}, \omega) = 1 + \frac{e^2}{\epsilon_0 q^2} \sum_{\vec{k}} \frac{f(\vec{k}) - f(\vec{k}+\vec{q})}{\epsilon(\vec{k}+\vec{q}) - \epsilon(\vec{k}) - \hbar\omega - i\eta}$$

In the static limit ($\omega = 0$), the static Lindhard dielectric function is:

$$\epsilon(q, 0) = 1 + \frac{q_{\text{TF}}^2}{q^2} F\left( \frac{q}{2k_F} \right)$$

where $q_{\text{TF}} = \sqrt{\frac{3 e^2 n}{2\epsilon_0 E_F}}$ is the Thomas-Fermi screening wavevector, and $F(x)$ is the Lindhard function ($x \equiv q / 2k_F$):

$$F(x) = \frac{1}{2} + \frac{1 - x^2}{4x} \ln\left| \frac{1 + x}{1 - x} \right|$$

• Long-Wavelength Limit ($q \ll 2k_F$, $x \to 0$): $F(x) \to 1$, giving the classical Thomas-Fermi screening: $$\epsilon(q) \approx 1 + \frac{q_{\text{TF}}^2}{q^2} \implies V(r) = \frac{e}{4\pi\epsilon_0 r} \exp(-q_{\text{TF}} r)$$ The Coulomb potential is screened exponentially over the Thomas-Fermi length $\lambda_{\text{TF}} = 1/q_{\text{TF}} \sim 0.5\text{ \AA}$.
• Zone Edge Singularity ($q = 2k_F$, $x = 1$): The derivative $dF/dx$ diverges logarithmically to $-\infty$ due to the sharp step in the Fermi-Dirac distribution at $T = 0\text{ K}$.

1. Real-Space Friedel Oscillations

Because the static Lindhard dielectric function $\epsilon(q, 0)$ possesses a non-analytic logarithmic derivative at $q = 2k_F$, taking the inverse Fourier transform of the screened potential $\phi(r) = \int \frac{\phi_{\text{ext}}(q)}{\epsilon(q)} e^{i\vec{q}\cdot\vec{r}} \frac{d^3q}{(2\pi)^3}$ produces long-range spatial ripples rather than pure exponential decay:

$$\delta\rho(r) \propto \frac{\cos(2k_F r)}{r^3} \quad (r \gg 1/k_F)$$ $$\phi(r) \propto \frac{\cos(2k_F r)}{r^3}$$

These quantum interference ripples are Friedel oscillations. They arise from the sharp edge of the Fermi sphere: electrons cannot screen perturbations on spatial scales shorter than the Fermi wavelength $\lambda_F = 2\pi/k_F$. Friedel oscillations govern the oscillatory sign of the RKKY magnetic exchange interaction and mediate long-range vacancy-impurity interactions in alloys.

2. Kohn Anomalies & The Friedel Sum Rule

• Kohn Anomaly in Phonon Dispersion: In 1959, Walter Kohn showed that because ions are screened by conduction electrons, the singularity in $\epsilon(q, 0)$ at $q = 2k_F$ is directly imprinted onto the phonon dispersion curve $\omega(\vec{q})$. At phonon wavevectors spanning the Fermi surface ($|\vec{q}| = 2k_F$), the phonon frequency exhibits a sharp dip or kink: $$\left. \frac{d\omega}{dq} \right|_{q = 2k_F} \to -\infty$$ In low-dimensional conductors (1D and 2D), this anomaly triggers the Peierls structural transition and charge density waves (CDWs).
• The Friedel Sum Rule: When an impurity ion with excess nuclear charge $Z e$ is embedded in a metal, it scatters conduction electrons. The phase shifts $\delta_l(E_F)$ of the partial waves at the Fermi energy must completely screen the impurity charge: $$Z = \frac{2}{\pi} \sum_{l=0}^\infty (2l + 1) \delta_l(E_F)$$ This exact relation connects microscopic electron scattering phase shifts to macroscopic residual electrical resistivity $\Delta\rho$.

1. The Harrison Construction & Fermi Surface Topology

The Fermi surface is the constant-energy locus $E(\vec{k}) = E_F$ separating occupied from unoccupied states in reciprocal space at $T = 0\text{ K}$.

In the nearly free electron approximation, the Harrison construction determines the Fermi surface geometry:

1. Draw the free electron Fermi sphere of radius $k_F = (3\pi^2 n)^{1/3}$ centered at every reciprocal lattice point $\vec{G}$.
2. Points enclosed by at least one sphere belong to the first Brillouin zone; points enclosed by at least two spheres belong to the second zone, and so on.
3. Applying periodic potential gaps rounds off the sharp intersections, decomposing the Fermi surface into:
   • Closed Electron Pockets: Enclosing lower unoccupied states.
   • Closed Hole Pockets: Enclosing unoccupied states.
   • Open Orbits: Non-periodic trajectories that traverse the entire Brillouin zone without closing, producing unsaturated transverse magnetoresistance ($\rho_{xx} \propto B^2$).

2. The de Haas-van Alphen (dHvA) Effect & Onsager Relation

When a strong magnetic field $B\hat{z}$ is applied to a clean metal at cryogenic temperatures ($\omega_c \tau \gg 1, k_B T \ll \hbar\omega_c$), the continuous electronic density of states condenses into discrete Landau tubes in reciprocal space with cross-sectional area:

$$S_n = (n + \gamma) \frac{2\pi e B}{\hbar}$$

As $B$ increases, the Landau tubes expand radially. Each time a tube crosses the extremal cross-section of the Fermi surface $S_e$, the free energy and magnetic moment undergo a sharp oscillation.

According to the Onsager Relation (1952), the magnetic susceptibility oscillates periodically in $1/B$:

$$\Delta\left( \frac{1}{B} \right) = \frac{2\pi e}{\hbar S_e}$$

By measuring the oscillation periods $\Delta(1/B)$ as a function of the magnetic field orientation, the complete 3D topological shape and extremal cross-sectional areas $S_e$ of the Fermi surface can be reconstructed with sub-percent precision.

1. The Tight-Binding Method

In the tight-binding (LCAO) approximation, the crystal wavefunction is constructed from linear combinations of isolated atomic orbitals $\phi_m(\vec{r})$:

$$\psi_{\vec{k}}(\vec{r}) = \frac{1}{\sqrt{N}} \sum_j e^{i \vec{k} \cdot \vec{R}_j} \phi(\vec{r} - \vec{R}_j)$$

Retaining only on-site energy $\epsilon_0$ and nearest-neighbor hopping matrix element $t = -\int \phi^*(\vec{r}) \Delta V(\vec{r}) \phi(\vec{r} - \vec{\delta}) d^3r$, the band dispersion on a 2D square lattice of constant $a$ is:

$$E(\vec{k}) = \epsilon_0 - 2t [\cos(k_x a) + \cos(k_y a)]$$

At the band saddle points ($\vec{k} = (\pi/a, 0)$ and $(0, \pi/a)$), the group velocity vanishes, producing a logarithmic divergence in the 2D density of states known as a van Hove singularity:

$$g(E) \propto \ln\left| \frac{16t}{E - \epsilon_0} \right|$$

2. Pseudopotentials & Density Functional Theory (DFT)

• Orthogonalized Plane Waves (OPW) & Pseudopotentials: Conduction wavefunctions must oscillate rapidly near atomic nuclei to remain orthogonal to tightly bound core states, requiring thousands of plane waves. Phillips and Kleinman proved that this core orthogonality acts as an effective repulsive potential that nearly cancels the deep Coulomb attraction: $$V_{\text{pseudo}}(\vec{r}) = V_{\text{true}}(\vec{r}) + \sum_c (E - E_c) |\phi_c\rangle\langle\phi_c|$$ The resulting smooth, shallow pseudopotential reproduces valence eigenvalues accurately with a small plane-wave basis.
• Density Functional Theory (Hohenberg-Kohn & Kohn-Sham, 1964-1965): DFT establishes that the ground-state properties of an interacting many-electron system are uniquely determined by the 3D ground-state electron density $n(\vec{r})$, reducing a $3N$-dimensional problem to 3 dimensions! The Kohn-Sham self-consistent single-particle equations are: $$\left[ -\frac{\hbar^2}{2m} \nabla^2 + V_{\text{ext}}(\vec{r}) + V_{\text{Hartree}}[n](\vec{r}) + V_{\text{xc}}[n](\vec{r}) \right] \psi_i(\vec{r}) = \epsilon_i \psi_i(\vec{r})$$ $$n(\vec{r}) = \sum_{i=1}^N |\psi_i(\vec{r})|^2$$ The exchange-correlation potential $V_{\text{xc}}$ is approximated via the Local Density Approximation (LDA) or Generalized Gradient Approximation (GGA). DFT forms the backbone of modern computational materials physics.

1. Failure of Independent-Electron Band Theory: Mott Insulators

According to conventional Bloch band theory, any crystal with an odd number of valence electrons per unit cell must have a half-filled energy band and should therefore be a metal.

However, materials such as $\text{NiO}$, $\text{CoO}$, and undoped cuprates ($\text{La}_2\text{CuO}_4$) have partially filled $3d$ shells, yet are transparent electrical insulators with optical bandgaps exceeding $4\text{ eV}$!

In 1937, Nevill Mott explained that single-particle band theory completely breaks down when the on-site Coulomb repulsion $U$ between two electrons on the same atom exceeds the kinetic hopping bandwidth $W = 2 z t$.

2. The Single-Band Hubbard Model

The minimal Hamiltonian capturing this competition between kinetic itinerancy and Coulomb localization is the Hubbard Model:

$$\hat{H} = -t \sum_{\langle ij \rangle, \sigma} \left( c_{i\sigma}^\dagger c_{j\sigma} + \text{h.c.} \right) + U \sum_i n_{i\uparrow} n_{i\downarrow}$$

• Hopping parameter $t$: Promotes electron delocalization into a metallic band of bandwidth $W = 2 z t$.
• On-site Coulomb repulsion $U$ ($U = \int |\phi(r_1)|^2 \frac{e^2}{r_{12}} |\phi(r_2)|^2$): Energy penalty incurred whenever two electrons with opposite spins occupy the same lattice site.

At half-filling (one electron per atom, $\langle n_i \rangle = 1$):

• Weak Coupling ($U / W \ll 1$): Itinerant metallic Fermi liquid.
• Strong Coupling ($U / W \gg 1$): Double occupancy is strictly suppressed. To conduct electricity, an electron must hop to an already occupied site, costing energy $U$. The single band splits into two sub-bands:
  • Lower Hubbard Band (LHB): Completely filled ($E \approx 0$).
  • Upper Hubbard Band (UHB): Completely empty ($E \approx U$).
  The gap separating them is the Mott-Hubbard gap $E_{\text{gap}} \approx U - W$. The material is a Mott Insulator!
• At second-order perturbation theory in $t/U$, virtual hopping between neighboring singly-occupied sites yields an effective antiferromagnetic Heisenberg exchange interaction: $$J = \frac{4 t^2}{U}$$ explaining why virtually all parent Mott insulators are robust antiferromagnets.

1. Green's Functions & The GW Approximation

Standard DFT notoriously underestimates electronic bandgaps by $30-50\%$ (e.g. predicting $0.5\text{ eV}$ for Si instead of $1.12\text{ eV}$, and predicting zero gap for Mott insulators) because Kohn-Sham eigenvalues are Lagrange multipliers of a non-interacting auxiliary system, not true quasiparticle excitation energies.

To accurately compute true quasiparticle spectra, many-body perturbation theory employs the single-particle Green's function $G(\vec{r}, \vec{r}', \omega)$. The quasiparticle equation is:

$$\left[ -\frac{\hbar^2}{2m} \nabla^2 + V_{\text{ion}}(\vec{r}) + V_{\text{Hartree}}(\vec{r}) \right] \psi_n(\vec{r}) + \int \Sigma(\vec{r}, \vec{r}', \epsilon_n) \psi_n(\vec{r}') d^3r' = \epsilon_n \psi_n(\vec{r})$$

where $\Sigma$ is the non-local, energy-dependent self-energy operator. In Lars Hedin's GW approximation, the self-energy is expanded to first order in the dynamically screened Coulomb interaction $W = \epsilon^{-1} v$:

$$\Sigma = i G W$$

The $GW$ method accurately reproduces experimental bandgaps across semiconductors and insulators to within $0.1\text{ eV}$.

2. Semiconductor Superlattices & Minibands

In 1970, Leo Esaki and Raphael Tsu proposed fabricating artificial 1D periodic structures by alternating ultra-thin epitaxial layers of two different semiconductors (e.g. $\text{GaAs}$ and $\text{Al}_x\text{Ga}_{1-x}\text{As}$) with period $d \sim 5-20\text{ nm} \gg a_{\text{lattice}}$.

This introduces an artificial periodic potential that folds the host Brillouin zone into a mini-Brillouin zone of width $\pi/d$.

• The conduction band splits into narrow minibands of width $\Delta \sim 10-50\text{ meV}$ separated by minigaps.
• Under an applied electric field $\mathcal{E}$, electrons accelerate to the mini-zone boundary and undergo Bragg reflection, executing periodic real-space Bloch oscillations at frequency $\omega_B = e\mathcal{E}d / \hbar$ without scattering.
• This produces negative differential resistance ($dI/dV < 0$), providing the operational foundation for resonant tunneling diodes (RTDs) and quantum cascade lasers (QCLs).