Past exam of the quantum condensed matter course of the University of Cambridge / 2008 / Section B / B2

The tight-binding approximation starts with localized atomic orbitals. Their overlap allows hopping, broadening each atomic energy into a band. For example, a single orbital on a simple cubic lattice with nearest-neighbor hopping gives $E(\mathbf{k})=E_0-2t[\cos (k_{x}a)+\cos (k_{y}a)+\cos (k_{z}a)]$. Small overlap means a narrow band; orbital symmetry and hopping signs determine its extrema and dispersion.

The nearly-free electron model starts instead with free-electron parabolas, folded into the Brillouin zone, and treats the periodic crystal potential as a perturbation. Near a Bragg plane, states differing by a reciprocal vector become degenerate and mix, opening a gap approximately $2|V_{\mathbf{G}}|$. Away from such degeneracies, the energy remains close to the free-electron dispersion. These are complementary limits of the same Bloch-band framework, not different counts of electronic states. Both can give metals or insulators depending on filling and band overlap, but their natural basis functions and useful expansion parameters differ.

Thomas-Fermi screening treats the carrier density locally as that of a uniform electron gas whose chemical potential is shifted by the slowly varying electrostatic potential. Linearizing gives $\delta n=(\partial n/\partial \mu )e\phi$ for electrons. The resulting screened Poisson equation has inverse screening lengthFor a degenerate gas, $\partial n/\partial \mu =g_{\mathrm{vol}}(E_F)$ including both spins, giving Thomas-Fermi screening length $k_s^{-1}$. A nondegenerate semiconductor instead has $\partial n/\partial \mu =n/(k_{B}T)$, giving the Debye length $\sqrt{\epsilon k_{B}T/(n{e}^2)}$. Electrons and holes contribute additively to the charge susceptibility when both are present; the lattice permittivity replaces the vacuum value.

The approximation requires a potential varying slowly compared with microscopic electronic length scales and a small perturbation for the linearized result. It captures smooth electrostatic screening, but not the full nonlocal response or Friedel oscillations. Low carrier density or carrier freeze-out can make a semiconductor's screening much weaker than a metal's.

The Hubbard model combines band motion and local repulsion:Hopping favors itinerant states, while positive $U$ penalizes simultaneous opposite-spin occupation on one site. In a uniform mean-field treatment, a spin imbalance reduces the interaction cost but raises the band-filling energy. The Stoner criterion for instability of the unpolarized metal is $U{g}_{\uparrow }(E_F)>1$ when $g_{\uparrow }$ is the density of states per site, per energy, for one spin. Using a total two-spin density of states changes the displayed numerical convention.

The ordered mean-field state has exchange-split spin bands and a spontaneous magnetization, with the kinetic and interaction costs balancing at equilibrium. This is a model of itinerant ferromagnetism, not a guarantee that every repulsive Hubbard system becomes ferromagnetic. Filling, lattice structure and competing order matter: near half filling on a bipartite lattice, antiferromagnetism and a Mott insulator can be favored instead. The approximation also omits important spin fluctuations and strong-correlation effects.

In a two-dimensional electron system under a perpendicular magnetic field, orbital energies form Landau levels with spacing $\hslash {\omega }_c$. Each resolved spin level has $eBA/h$ states in area $A$. At integer filling, the integer quantum Hall effect has Hall conductance ${\sigma }_{xy}=\nu e^2/h$ in magnitude and vanishing longitudinal conductance in the ideal plateau regime; inversion of the conductivity tensor gives Hall resistance $h/(\nu e^2)$.

A Hall-bar experiment cools the sample, passes a small current and records longitudinal and transverse voltages while sweeping the field. The Hall resistance develops plateaus while the longitudinal resistance develops minima. Disorder broadens the Landau levels and localizes many bulk states, allowing the chemical potential to move through a range of carrier density without changing the number of conducting edge channels. Extended states near a Landau-level center allow transitions between plateaus. Spin splitting determines which integer sequence is resolved. The robustness comes from the quantum state and its edge transport, rather than the Hall bar having a specially calibrated geometric width.