= Solution
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 gives
$$
\boxed{\Delta y\simeq\frac{E_Z}{|V'|}
=\frac{\hbar\omega_c}{3|V'|}.}
$$
Here $V'$ is interpreted as the specified slope of <electron> <potential energy>. If it denotes electrostatic <voltage> per <length> instead, the denominator is $e|V'|$. 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.<Electron-electron interactions in antidot-based Aharonov-Bohm interferometers by S. Ihnatsenka and others>[section I and section III.C, especially Fig. 7: localized states, interchannel scattering and resonant transmission]{ref}
For the <antidot resonance conductance bounds> with one active spin-resolved resonance, write
$$
\mathcal L(E)=\frac{\Gamma_L\Gamma_R}{(E-\epsilon_d)^2+(\Gamma_L+\Gamma_R)^2/4},\qquad
0\leq\mathcal L\leq1.
$$
Here $\Gamma_L,\Gamma_R$ are energy broadenings; the corresponding escape rates are $\Gamma_L/\hbar,\Gamma_R/\hbar$. 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, $G=(e^2/h)(N_{\rm open}-R)$ or $(e^2/h)(N_{\rm dir}+T)$; 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>.
\Image[/past-exam-of-the-physics-of-nanoelectronic-systems-course-of-the-university-of-cambridge/2013/paper-1-antidot-resonances.png]
{title=Original neck spectrum, alternative channel connectivities and illustrative antidot conductance dips and peaks}
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>.
\Image[/past-exam-of-the-physics-of-nanoelectronic-systems-course-of-the-university-of-cambridge/2013/paper-1-dot-comparison.png]
{title=Original series-dot paths and gate resonances compared with a side-coupled antidot}
<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 $E_Z/|V'|$ separation is not a quantitative screened-strip width.<Electrostatics of edge channels by D. B. Chklovskii and others>[abstract: steplike self-consistent potential and alternating compressible and incompressible strips]{ref}
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