Past exam of the advanced Quantum Condensed Matter Physics course of the University of Cambridge / 2013 / Paper 1 / 3

An organic semiconductor has conjugated molecular units with strong intramolecular bonding and comparatively weak intermolecular electronic overlap. Optical excitation commonly promotes a $\pi$ electron from an occupied molecular orbital toward an unoccupied one. Weak dielectric screening and narrow bands favour bound Frenkel excitons, sometimes mixed with intermolecular charge-transfer states, rather than immediate production of free carriers. The optical excitation energy is below the unbound electron–hole threshold by the exciton binding energy, with relaxation conventions specified when comparing measured gaps.

The electric-dipole interaction selects bright states through their transition dipoles, producing polarization-dependent absorption in oriented samples. From a singlet ground state, strong electric-dipole absorption normally preserves spin and reaches singlet excitons; direct triplet excitation is weak without spin mixing. The Franck-Condon principle treats excitation as vertical on the nuclear-motion timescale. For one displaced harmonic vibration at low temperature, the displaced-oscillator Franck-Condon progression has relative strengthsHere the Huang-Rhys factor $S$ measures the displacement and $E_{00}$ is the zero-vibration transition energy. Vibrational relaxation before emission produces a Stokes shift. Molecular packing and excitonic coupling shift or split bright features and change their intensities; disorder and vibrations broaden them. Consequently a universal parabolic-band absorption edge is inappropriate for a localized molecular excitation.^[ref]

In quasi-one-dimensional quantum transport, transverse confinement makes discrete subbands while motion along a quantum wire remains continuous. Only occupied propagating subbands conduct. For elastic scattering between equilibrated reservoirs, the Landauer formula isThe sum explicitly counts spin. A perfectly transmitting spin-degenerate subband gives $2e^2/h$ at low temperature; a quantum point contact shows steps as a gate opens additional subbands. The one-dimensional density of states cancels the carrier velocity in the current, giving a conductance quantum rather than a material-dependent velocity factor. Finite two-terminal contact resistance remains even without internal scattering.

Transport is ballistic transport for length $L\ll \ell$, where $\ell$ is the mean free path. In a multichannel wire with $\ell \ll L$ below its localization length, scattering reduces transmission and gives approximately Ohmic behaviour. If the phase coherence length exceeds the sample length, amplitudes interfere; weak localization and sample-dependent universal conductance fluctuations can appear. At greater lengths, generic coherent disordered wires undergo Anderson localization, with localization length of order the channel number times $\ell$, up to symmetry-dependent factors. Thermal averaging and dephasing smear the quantum features.

Strong interactions require more than independent-electron transmission. A clean single-channel interacting wire is described at low energy by a Luttinger liquid, with collective modes and possible spin-charge separation; impurities can generate temperature-dependent power laws. Ideal contacts to noninteracting leads can still fix the two-terminal conductance of a clean wire, so it must not automatically be multiplied by a bulk Luttinger parameter.

Density functional theory replaces the interacting many-electron ground state wavefunction by its spatial density $n(\mathbf{r})$. For fixed interaction and a nondegenerate ground state, the Hohenberg-Kohn theorem fixes the external potential up to an additive constant. If two distinct potentials gave the same density, their strict variational energy inequalities would sum to the contradiction $E_0+E_0^{\prime }<E_0+E_0^{\prime }$.

The density functional variational principle minimizesover admissible densities of the correct particle number. The universal $F$ contains the kinetic and interaction energies. This is exact in principle; an explicit useful functional is the practical difficulty.^[ref]

The Kohn-Sham equations introduce noninteracting orbitals with the same density and decompose $F=T_s+E_H+E_{\mathrm{xc}}$. With $n={\sum }_i{f}_i|{\varphi }_i{|}^2$,The exchange-correlation functional includes exchange, correlation and the interacting kinetic-energy correction relative to $T_s$. Guess a density, construct the potentials, solve and occupy orbitals, then mix and iterate to self-consistent field convergence. The local density approximation uses $E_{\mathrm{xc}}^{\mathrm{LDA}}=\int n{ϵ}_{\mathrm{xc}}^{\mathrm{uniform}}(n)d^{3}r$; a generalized gradient approximation also depends on density gradients.

Kohn–Sham orbital energies are not generally excitation energies, and their sum double counts interactions. The Kohn-Sham energy double-counting correction iswith ion–ion energy added for a total crystal energy. Approximate functionals can miss self-interaction cancellation, band gaps and strong correlation; convergence alone does not prove the global ground state.^[ref]