Past exam of the quantum condensed matter course of the University of Cambridge / 2006 / Section B / B4

In an isolated energy band, semiclassical motion satisfies $\hslash \dot{k}=F$ and $v={\hslash }^{-1}dE/dk$. The band is periodic in crystal momentum, so a constant force sweeps the state through equivalent points of the Brillouin zone. Its velocity repeats rather than increasing without bound. For an electron in a constant electric field, $F=-e\mathcal{E}$, and the Bloch oscillation period isFor $E(k)=E_0-2J\cos ka$, $v=(2Ja/\hslash )\sin ka$ and integration gives a bounded position oscillation of amplitude $2J/|F|$ about its mean. A filled band has no corresponding net transport oscillation after summing uniformly over all $k$.

Observation requires scattering times longer than $T_B$ and negligible interband tunnelling. Ordinary crystals often scatter too rapidly before a full zone traversal; larger-period structures and clean atom-optical lattices help. In the cold-atom Bloch oscillation experiment, accelerating a standing-wave lattice supplied a controlled force, and releasing the atoms at different times revealed an oscillating momentum distribution.^[ref]

A field-effect transistor uses a gate voltage to control the conducting channel between source and drain. In a MOSFET, the gate is insulated from the semiconductor: an electric field changes the channel carrier population with little steady gate current. An n-channel device creates an semiconductor inversion layer when the gate sufficiently attracts electrons to a p-type surface. The drain supply provides the output power; the gate controls that flow.

The long-channel MOSFET current law follows from the gradual-channel approximation. With gate capacitance per area $C_{\mathrm{ox}}$, threshold voltage $V_T$, channel width $W$, length $L$, constant charge carrier mobility $\mu$ and $V_{mov}=V_{GS}-V_T>0$, the local sheet charge magnitude is $C_{\mathrm{ox}}(V_{mov}-V(x))$. Integrating the drift current along the channel givesAt $V_{DS}=V_{mov}$, the drain end reaches pinch-off. The ideal long-channel saturation current is $I_D=\mu C_{\mathrm{ox}}W{V}_{mov}^2/(2L)$; this assumes negligible channel-length modulation. A junction field-effect transistor instead changes the channel width through a reverse-biased depletion region.

Experimentally, measure $I_D$ against $V_{GS}$ at fixed drain bias to see gate control, and against $V_{DS}$ at several fixed gate biases to distinguish the approximately linear and saturation regimes. Simultaneously checking gate leakage tests the insulated-gate assumption.

A Wigner crystal is an ordered electron arrangement favored when repulsion overwhelms the kinetic cost of localization. In a three-dimensional electron gas with a neutralizing background, define a mean spacing $a=(3/4\pi n{)}^{1/3}$ and effective Bohr radius $a_B^{\ast }=4\pi \epsilon {\hslash }^2/(m^{\ast }{e}^2)$. The Coulomb-to-kinetic energy ratio of an electron gas scales asThus low density favors the crystal in the degenerate quantum model. Translational symmetry is broken, and excitations include collective lattice vibrations. The melting criterion depends on dimension, screening, disorder and quantum fluctuations, so this scaling argument does not supply a universal numerical transition density.

The electron crystallization on liquid helium experiment provides an important related observation: a dilute two-dimensional electron sheet above liquid helium crystallized on cooling. It was a classical electron system, with thermal rather than Fermi motion competing with Coulomb energy; its coupling parameter is $\Gamma =E_C/(k_{B}T)$. It therefore illustrates Coulomb ordering without being a direct measurement of the zero-temperature three-dimensional quantum transition.^[ref]

In the quantum Hall effect, a two-dimensional electron system at low temperature in a strong perpendicular magnetic field has transverse conductivity of quantized magnitude $|{\sigma }_{xy}|=\nu e^2/h$. On a plateau with negligible longitudinal conductivity, quantum Hall conductivity tensor inversion gives $|{\rho }_{xy}|=h/(\nu e^2)$ and ${\rho }_{xx}\to 0$; the Hall sign depends on the carrier and orientation convention. The filling factor is $\nu =n_{2D}h/(eB)$ when all spin-resolved states are counted.

For the integer quantum Hall effect, filled Landau levels and a mobility gap give integer $\nu$. Disorder-localized states allow the density or field to vary without changing the number of extended occupied channels, producing a quantum Hall plateau rather than quantization only at isolated fields. Quantum Hall edge states provide an equivalent transport picture. In a Hall bar, separate voltage leads measure transverse and longitudinal voltages while a known current flows; sweeping the field reveals both the plateau and the resistance minimum. The von Klitzing-Dorda-Pepper quantum Hall experiment established this integer quantization.

The fractional quantum Hall effect requires electron correlations within a partially filled Landau level. An incompressible correlated state can occur at fractions such as $\nu =1/3$; a noninteracting partially filled level does not account for the associated gap. The Laughlin wavefunction at $\nu =1/m$, odd $m$, contains a factor ${\prod }_{i<j}(z_i-z_j{)}^m$, suppressing close electron approaches while retaining fermionic antisymmetry. Its charged collective excitations can carry magnitude $e/m$; the constituent electrons still have charge $-e$. The Tsui-Störmer-Gossard fractional Hall experiment observed the fractional Hall feature together with a longitudinal-resistance minimum, providing the experimental distinction from an arbitrary rational filling of an ordinary conducting state.