Past exam of the physics of Nanoelectronic Systems course of the University of Cambridge / 2013 / Paper 1 / 1

Use $e>0$, $B>0$ along $+z$, and $U(y)$ for the electron's confinement potential energy. The spin-resolved Landau level branches in the smooth-edge approximation areThe spin labels denote upper and lower energies; the sign of a material's effective $g$ factor is not inferred from the stated positive splitting. In units of $\hslash {\omega }_c$, the four occupied bulk energies are $1/3,2/3,4/3,5/3$. A chemical potential in $(5/3,7/3)$ therefore gives filling factor four. The branches rise at both confining edges and cross the common Fermi energy once at each edge. Each crossing is one spin-resolved quantum Hall edge state; there are four on either side. The two spin branches within an orbital level are separated by $\hslash {\omega }_c/3$, while the intervening gap to the next orbital pair is $2\hslash {\omega }_c/3$.

In Landau gauge for a uniform magnetic field $\mathbf{A}=(-By,0,0)$, the guiding centre is $Y=\hslash k_x/(eB)$. The group velocity isThus the upper edge in the original trajectory diagram carries electrons to the right and the lower edge to the left. Reversing the magnetic field reverses both arrows. Semiclassically these are skipping trajectories; quantum mechanically the guiding centres label propagating quantum Hall edge states. The larger orbital-plus-spin energy crosses the Fermi energy farther toward the interior, because less confinement energy is needed. Opposite spins in one orbital branch are therefore spatially separated.

In a clean bulk with the Fermi energy between Landau levels, there are no extended bulk states at that energy. Occupied bulk cyclotron orbits lie below it; one must not draw them as extra conducting Fermi-level channels. Smooth disorder can instead produce localized bulk contours without adding a direct source-to-drain channel.

For the spin separation at a smooth quantum Hall edge, linearize the confinement energy at a crossing. Compensating a spin energy difference by a potential-energy change givesHere $V^{\prime }$ is interpreted as the specified slope of electron potential energy. If it denotes electrostatic voltage per length instead, the denominator is $e|V^{\prime }|$. The two edges of a neck and the two sides of an antidot have oppositely directed slopes; the formula is a positive local separation on any one edge. For local filling two, only the $n=0$ spin pair is occupied at the neck minimum: $E_{0,+}+U_{\min }<E_F<E_{1,-}+U_{\min }$. Raising both branches with the confinement produces two crossings on each side, as shown in the original neck sketch. This fixed-slope estimate precedes self-consistent electrostatic screening.

An antidot is a depleted potential hill. guiding centres circulate around its closed equipotentials, with chirality opposite to the outer sample edge. A phase-coherent round trip contributes magnetic phase $2\pi BS/{\Phi }_0$, with ${\Phi }_0=h/e$. A localized orbit resonant with the incident energy can transfer an electron between extended channels. If the outgoing channel returns toward the injecting reservoir, the process is resonant reflection and reduces electrical conductance. If coupling instead connects an otherwise blocked incident channel to the far reservoir, it is resonant transmission and adds electrical conductance. Transmission peaks require that forward connectivity; a closed orbit by itself does not guarantee them. Interchannel scattering can change which orbit participates.^[ref]

For the antidot resonance conductance bounds with one active spin-resolved resonance, writeHere ${\Gamma }_L,{\Gamma }_R$ are energy broadenings; the corresponding escape rates are ${\Gamma }_L/\hslash ,{\Gamma }_R/\hslash$. In a reflection-dominated configuration with two otherwise transmitted channels, $G=(e^2/h)(2-\mathcal{L})$: a $2e^2/h$ background can dip to $e^2/h$. In a transmission-dominated configuration with one spectator channel and the second otherwise blocked, $G=(e^2/h)(1+\mathcal{L})$: peaks rise from $e^2/h$ toward $2e^2/h$. These are the two ideal channel-count examples plotted below. More generally, $G=(e^2/h)(N_{\mathrm{open}}-R)$ or $(e^2/h)(N_{\mathrm{dir}}+T)$; a different spectator count changes the integer offset. The constriction filling alone does not specify every tunnel coupling or resonance height. Equal couplings permit unit resonant transfer in the isolated-level model; asymmetry, temperature and dephasing reduce the contrast. Holding the effective orbit area approximately fixed gives $\Delta B\simeq h/(eS)$ for one branch, not a universal charging-corrected period.

In quantum-dot versus antidot resonance topology, a quantum dot is a potential well holding a finite electron island. Its closed guiding-centre circulation is reversed relative to an antidot at the same field. With two barriers in series, current follows source $\to$ dot $\to$ drain; off resonance there need not be a direct conducting path. A gate shifts a level or an addition electrochemical potential into alignment with the reservoirs and produces a electrical conductance peak. For an isolated coherent spin-resolved level, the zero-temperature peak is at most $e^2/h$ and reaches it only for equal couplings; unresolved noninteracting spin degeneracy doubles this bound. In weak sequential tunnelling, peak height depends on rates and temperature rather than being quantized. Charging produces Coulomb blockade peaks with spacing set by addition energy and gate lever arm. An antidot also supports discrete orbit/charging resonances, but can be side-coupled to already open channels and produce dips. Gate voltage changes both orbit area and confinement, whereas a magnetic sweep directly changes enclosed flux.

Coulomb interactions require a self-consistent confinement potential. A gradual decrease in electron density cannot leave arbitrary uncompensated donor charge at no cost. Partly occupied Landau levels have a large charge response: electrostatic screening flattens their effective energy near the Fermi energy over compressible quantum Hall edge strips. Between them, integer-filled incompressible quantum Hall edge strips have a gap, weak density response and most of the potential variation. In an idealized cross-section the effective branch thus alternates between flat, partially occupied portions and rising portions crossing a gap; its carrier density varies oppositely. The bare levels still have Landau level and spin splittings locally, but the common electrostatic potential is no longer an imposed straight slope. exchange interaction can additionally renormalize spin gaps. Widths must follow electrostatics and the density gradient, so the unscreened $E_Z/|V^{\prime }|$ separation is not a quantitative screened-strip width.^[ref]