Past exam of the quantum condensed matter course of the University of Cambridge / 2005 / Section B / B4

Exchange interaction arises from Coulomb interaction together with the antisymmetry required for identical electrons. Different spin states require different spatial symmetry and therefore different pair correlations and Coulomb energy; exchange is not the classical magnetic dipole force between spins. For electrons in distinct atomic orbitals, positive direct exchange favors parallel spins, underlying Hund's rules and the formation of local moments.

In a solid with localized moments, a common convention is $H=-J{\sum }_{⟨ij⟩}{\mathbf{S}}_i\cdot {\mathbf{S}}_j$. Then $J>0$ favors ferromagnetism and $J<0$ favors antiferromagnetism. Superexchange through a ligand and other virtual hopping processes can produce either effective sign depending on orbital occupancy and geometry; half-filled orbitals often give antiferromagnetic coupling. For itinerant electrons, the Stoner criterion describes a competition between exchange energy gain and the kinetic cost of splitting the spin populations.

Experiments distinguish these descriptions through both static and dynamic response. Magnetic neutron diffraction locates magnetic Bragg peaks and therefore the ordering wavevector; an antiferromagnet can have such peaks despite zero bulk moment. Inelastic neutron scattering measures spin-wave dispersion, from which an exchange scale can be fitted. A Curie-Weiss susceptibility intercept gives complementary information about dominant coupling, but its sign alone does not establish the actual ordered structure.

A Peierls transition is a lattice distortion coupled to a gap in an initially partly filled quasi-one-dimensional electronic band. A modulation at $2k_F$ couples the opposite Fermi points. At half filling with one orbital per atom, it doubles the cell and alternates strong and weak bonds; the dimerized tight-binding chain shows explicitly how the gap opens at the Fermi energy. In the ideal weak-coupling zero-temperature model, the Peierls energy gain has a negative term proportional to $-{\delta }^2\ln (t/|\delta |)/t$, which can outweigh a positive quadratic elastic cost.

Observation should connect electronic and structural evidence: cooling through the transition can reduce conductivity as the gap develops, while diffraction satellites identify the ordering wavevector. A softened phonon near $2k_F$ tests the precursor electron-lattice coupling. In inelastic neutron scattering of TTF-TCNQ, measurements from $60$ to $300\mathrm{K}$ found an increasing Kohn anomaly near $2k_F$ as the $54\mathrm{K}$ transition was approached. Those particular measurements were above the transition, so they probe the precursor dynamics rather than a static low-temperature diffraction pattern.^[ref]

A real finite-temperature transition also depends on interchain coupling and fluctuations. The ideal one-dimensional energy argument by itself does not prove finite-temperature long-range order.

A p-n junction joins p-type and n-type semiconductors. Initial carrier diffusion exposes ionized acceptors and donors, creating a depletion region and an internal electric field. At equilibrium the Fermi level is uniform and drift balances diffusion, so the built-in potential produces no net terminal current. For nondegenerate abrupt doping, its magnitude is $V_{\mathrm{bi}}=(k_{B}T/e)\ln (N_A{N}_D/n_i^2)$.

Forward bias lowers the barrier and injects minority carriers across the junction. In the ideal low-injection diffusion model, the Shockley diode equation is $I=I_s[e^{eV/(k_{B}T)}-1]$. Reverse bias broadens the depletion region and leaves a small leakage current until breakdown mechanisms become important.

The asymmetric current-voltage curve enables rectification. Illumination produces electron-hole pairs that the junction field can separate, giving a photodiode or solar cell. Forward injection and radiative recombination in a suitable direct-gap material give a light-emitting diode. Measuring dark and illuminated current-voltage curves distinguishes electrical injection from photogenerated current; an ideal junction need not have zero reverse leakage or identical behavior in every material.

In a heavy fermion material, correlated low-energy quasiparticles have a large electronic density of states and effective mass. A Kondo lattice with hybridization between localized f-electron states and conduction states is an important setting. The large mass is a low-energy dynamical property, not a change in the electron's bare mass or the ionic mass.

Low-temperature calorimetry measures $C/T\simeq \gamma +\beta T^2$ in a regime where an electronic linear term and a Debye lattice term are appropriate. The large intercept $\gamma =({\pi }^2{k}_B^2/3)D(E_F)$ measures the enhanced quasiparticle density of states, with volume or molar conventions kept consistent. Apply a known heat pulse to a thermally characterized sample platform and measure the temperature rise, then vary the base temperature to separate contributions.

In low-temperature calorimetry and resistivity of CeAl3, measurements below $0.2\mathrm{K}$ gave $\gamma =1.62\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-2}$ and $\rho ={\rho }_0+A{T}^2$ with $A=35\mu \Omega \mathrm{cm}{\mathrm{K}}^{-2}$. The large thermal coefficient and enhanced quadratic transport coefficient are complementary evidence for a strongly renormalized Fermi liquid.^[ref] Enhanced spin susceptibility provides another probe. Where coherent quantum oscillations can be resolved, their thermal damping determines a cyclotron effective mass, giving a band-sensitive comparison with the thermodynamic mass inferred from heat capacity.