Past exam of the physics of Nanoelectronic Systems course of the University of Cambridge / 2016

Original public PDF, physical pages 832–834; exam dated 19 April 2016.

All three questions and all alternatives in Question 3 are covered. Questions 1 and 2 contain unlabelled requests, retained within their respective question Solutions.

A high-mobility modulation-doped GaAs-AlGaAs electron gas is grown by molecular beam epitaxy as a clean layered semiconductor heterostructure. An undoped gallium arsenide region is followed by an undoped aluminium gallium arsenide spacer and a donor-doped barrier region, commonly with a protective cap. The closely matched crystal structures allow a high-quality heterojunction. The spacer supplies a crucial separation: the mobile electrons reside away from the ionized donors, reducing impurity scattering compared with putting the donors directly in the conducting layer.

The larger conduction band energy on the aluminium gallium arsenide side permits donor electrons to transfer into gallium arsenide. The resulting charge separation creates band bending; together with the band offset, it forms a roughly triangular quantum well at the interface. The normal-direction wavefunction is confined, while motion parallel to the interface remains approximately free:When only the lowest confinement subband is occupied, this is a two-dimensional electron gas. It is the quantized normal motion, not a physically zero layer thickness, that makes the low-energy system two-dimensional.

At zero magnetic field, one occupied spin-degenerate parabolic subband has constant density of states per area per energy, $D_0=m^{\ast }/(\pi {\hslash }^2)$ above its edge. A perpendicular field quantizes the in-plane orbital motion into Landau levels:Here $g^{\ast }$ is the material's effective spin factor and $e$ is the positive elementary charge; no free-electron $g=2$ assumption is made for this semiconductor. Each resolved spin branch contains $e|B|/h$ states per unit area. The ideal density of states becomeswith Landau-level broadening in a real sample. The Landau filling factorcounts the carrier density in units of one spin-resolved level's degeneracy. Two unresolved spin branches have twice that degeneracy; resolved branches must not each be counted twice.

Disorder localizes states in the broadened level tails, while extended states occur near level centres. As $B$ varies, occupation of localized states can change without changing the number of current-carrying edge channels. This produces a finite-width quantum Hall plateau, rather than quantization only at isolated fields. In a mobility gap, longitudinal resistivity approaches zero and the Hall magnitude is $R_K/N=h/(N{e}^2)$ for $N$ occupied transmitting branches. When the chemical potential passes through extended states, longitudinal dissipation peaks and the Hall value changes to another plateau. A spectral density of states can remain nonzero where transport is localized. Very strong disorder can destroy the resolved quantum Hall regime; the statement is not that any amount of disorder improves it.

The original plot is schematic, not fitted data. At fixed sheet density, increasing $|B|$ decreases Landau filling factor, so the Hall magnitude climbs through $R_K/N$ plateaus while longitudinal peaks separate the plateaus. The signed Hall voltage depends on carrier and probe conventions.

For the gate calculation, the target sheet density is fixed by the desired Landau filling factor:Use top-gate density tuning in the parallel-plate approximation. The geometric capacitance per unit area isThereforeThe gate must become more negative relative to the electron gas, reducing its density. The voltage is the change from the gate setting associated with the stated initial density; if that setting is defined as zero, it is the required gate voltage. Work-function offsets, trapped charge and quantum capacitance are neglected. An absolute voltage referenced to some other uncalibrated potential cannot be inferred from geometry alone. Numerical values use exact SI $h=6.62607015\times 10^{-34}\mathrm{J}\mathrm{s}$ and $e=1.602176634\times 10^{-19}\mathrm{C}$, and ${ϵ}_0\simeq 8.8541878188\times 10^{-12}\mathrm{F}{\mathrm{m}}^{-1}$.^[ref]

For the subsequently specified five-channel plateau, the chemical potential between the fifth and sixth resolved branches means $N=5$, not ten. The physical contact order in the PDF is $1,2,3,4$ along one side of the mesa. Use ideal equilibrating Ohmic contacts and the Landauer-Büttiker formula, rather than inferring a longitudinal measurement merely from the contacts being drawn on one edge.

Choose one field polarity so the chiral order is $1\to 2\to 3\to 4\to 1$. With $G_H=5e^2/h$,Contacts 2 and 4 are voltage probes, so $I_2=I_4=0$. With $V_3=0$, this gives $V_2=V_1$, $V_4=V_3=0$, and $I_1=G_H{V}_1$. Hencefor this circulation. Reversing the field reverses the contact order and gives $R_{13,24}=-h/(5e^2)$ with the same voltage ordering. The resistance magnitude is $5.16\mathrm{k\Omega }$; it is not zero. Contact 2 samples the outgoing source potential and contact 4 the outgoing drain potential for the chosen chirality.

The antidot is a depleted island supporting closed quantum Hall edge states. Tunnelling between those states and opposite propagating edges can produce resonant backscattering and resistance peaks as $B$ varies. One electronic Aharonov-Bohm phase winds by $2\pi$ when flux through a fixed effective orbit area $S$ changes by $h/e$. Using the supplied area as that effective area,With the two spin-split branches of the lowest orbital level occupied, the common Coulomb-dominated antidot oscillation model has two charge rearrangements per added $h/e$ flux. Both branches contribute to charge buildup; screening and their capacitive coupling can make tunnelling through the outer edge occur twice during the cycle. It then givesThis is two electron-tunnelling resonances per flux cycle, not evidence that the mobile particles have charge $2e$. The primary charging model specifically addresses this two-spin regime.^[ref]

The sketch shows resistance peaks in a weak-backscattering configuration, with arbitrary height and linewidth because tunnel couplings and temperature are unspecified. A two-spin count alone does not force two independent noninteracting resonance series to be exactly half a flux period apart: their relative positions also depend on spin splitting and confinement. Thus $51.7\mathrm{mT}$ describes the stated charging-model interpretation; $103.4\mathrm{mT}$ is the single-electron fixed-area flux period. The actual enclosed edge-orbit area can differ from the geometrically depleted area.

For bulk filling two and total forward transmission $\mathcal{T}$ in spin-resolved units, ideal chiral probes give the longitudinal four-terminal resistanceResonant loss of transmission therefore gives peaks. Their size, background and detailed shape cannot be calculated from the supplied area and bulk filling alone. If the gates instead make a tunnelling path that enhances forward transmission, the resonant contrast can be reversed; the plotted peak convention is explicitly the backscattering one.

Use two collinear quantum point contacts in a clean two-dimensional electron gas. Surface split gates deplete the electron gas beneath them, leaving two short constrictions facing one another across an open region. The first injects a directional beam from a biased reservoir; the second collects carriers into a downstream reservoir, with electric current flowing in the same spatial direction through both constrictions. Additional surrounding contacts can absorb carriers that miss the collector. A direct collector signal at zero magnetic field, sensitive to alignment and to obstruction of the intervening path, demonstrates ballistic electron transport across the separation.

An illustrative GaAs device uses gate gaps of a few hundred nanometres and electrically tunable constriction widths of tens to a few hundred nanometres, with injector–collector separation of a few micrometres. These are typical design scales, not dimensions supplied for a particular chip in the question. The electrical width can be much smaller than the lithographic gap because depletion extends laterally. A one-mode opening has width comparable to roughly half a Fermi wavelength; it is not simply any opening below one micrometre. Low-temperature operation suppresses inelastic scattering and resolves transverse modes. The direct trajectory length should be comfortably below the relevant transport mean free path.

Collimation is provided by the smooth exit from a constriction. In adiabatic electron collimation, transverse mode action is approximately conserved, giving $p_{\perp }W\simeq \mathrm{constant}$. As the channel opens from width $W_0$ to $W_{\mathrm{out}}$, $p_{\perp ,\mathrm{out}}\simeq p_{\perp ,0}{W}_0/W_{\mathrm{out}}$. At the same time the bottom of the electrostatic potential falls, increasing longitudinal kinetic energy. Thusand the angular distribution narrows. The reciprocal collector selects this forward-directed set of trajectories. A narrow abrupt aperture by itself may diffract into a broad fan; the smooth funnel and its potential profile are important. Here “focusing” refers primarily to collimation and efficient injector–collector coupling, not a claim that every emitted trajectory converges to one exact geometric point.

In scanning gate microscopy, a conducting tip biased sufficiently negatively creates a local repulsive potential and, when strong enough, a small depleted region in the electron gas beneath it. Raster the tip and record the collector current or differential conductance. Placing the perturbation on a strongly contributing path deflects or backscatters electrons and changes collection, producing spatial contrast. A primary scanning-probe experiment records conductance changes as a negatively charged tip depletes and backscatters the two-dimensional gas.^[ref]

The resulting image is a map of transport sensitivity to the tip, not a literal nonperturbing photograph of an electron trajectory. Broad electrostatic gating of either constriction should be distinguished from local beam disruption. Weak smooth disorder can create branches even in a high-mobility sample. Where electron phase coherence is preserved, interference between alternative paths can add fringes, often with a scale comparable to half a Fermi wavelength for a round-trip path-length change. Ballistic motion and phase-coherent interference are related but distinct requirements.

A small perpendicular magnetic field curves the imaged paths through the Lorentz force, $\mathbf{F}=-e\mathbf{v}\times \mathbf{B}$. Reversing $B$ reverses the bending, and a fixed aligned collector generally loses the initially straight beam unless a curved path connects it. For a circular orbit at the Fermi velocity,For one valley with two nearly equally occupied spin species at small field, state counting in the Fermi disc gives $n_s=2\pi k_F^2/(2\pi {)}^2=k_F^2/(2\pi )$. HenceSpin polarization or extra valley degeneracy changes the density-to-wavevector relation; the stated low-field GaAs estimate uses the ordinary spin-degenerate case.

The electron mobility gives the momentum relaxation time in a parabolic-band Drude model, $\mu =e{\tau }_{\mathrm{tr}}/m^{\ast }$. ThereforeThe effective mass cancels from both ${\ell }_{\mathrm{tr}}$ and $r_c$ when expressed in terms of density and mobility. For the supplied density,With $m^{\ast }=0.067m_0$, $m_0\simeq 9.1093837139\times 10^{-31}\mathrm{kg}$, one also obtains $v_F\simeq 1.937\times 10^5\mathrm{m}{\mathrm{s}}^{-1}$ and ${\tau }_{\mathrm{tr}}\simeq 38.1\mathrm{ps}$. The requested lengths areA useful independent ratio is ${\ell }_{\mathrm{tr}}/r_c=\mu |B|=3$. The standard constants used here are explicitly tabulated, rather than inferred from the sample.^[ref]

Mobility measures momentum relaxation. Predominantly small-angle elastic scattering can occur more frequently than that transport time, and the phase-coherence time is different again. Thus the calculation supports a several-micrometre transport scale, without proving that every electron experiences literally no perturbation over that distance.

As temperature rises, the directed ballistic collector signal decreases. Increased electron-phonon scattering and other inelastic processes reduce survival of the directed beam; thermal averaging broadens the mode and angular selection, and interference contrast also diminishes. A diffuse background current can remain, so disappearance of beam contrast need not mean all current through the device vanishes.

The fractional quantum Hall effect occurs when interactions produce an incompressible correlated state within a partially occupied Landau level. The orbital kinetic energy is already quantized; the important degeneracy is among many possible arrangements of electrons within the level. Repulsion can lift that degeneracy and create a many-body excitation gap. The interaction scale is of order $e^2/(4\pi ϵ{\ell }_B)$, where magnetic length ${\ell }_B=(\hslash /(e|B|){)}^{1/2}$; it is not a gap predicted by independent-electron fractional occupation alone.

A composite fermion description incorporates strong correlations by attaching $2p$ statistical flux quanta to each electron, with positive integer $p$. The attachment is a representation of correlated wavefunction phases and zeros, not a literal solenoid attached to a particle. Attaching an even number preserves fermionic exchange character. In a mean-field approximation for positive applied field,The particles consequently experience a reduced effective field. Define the signed effective filling ${\nu }^{\ast }=n_s{\varphi }_0/B^{\ast }$ and electronic filling $\nu =n_s{\varphi }_0/B$. Their relation isWhen $q$ effective levels are filled with the same field orientation, ${\nu }^{\ast }=q$, giving $\nu =q/(2pq+1)$. Reversed effective-field orientation has ${\nu }^{\ast }=-q$, giving $\nu =q/(2pq-1)$. Thus the Jain sequence isFor $p=1$, examples are $1/3$, $2/5$, $3/7$ on one branch and $2/3$, $3/5$ on the reversed branch. Fractional electronic plateaus can be understood as integer effective Hall states of these composite objects. The original proposal explicitly makes this connection.^[ref]

At a gapped fractional state, longitudinal dissipation becomes very small and the Hall magnitude approachesDisorder can localize quasiparticles and allow the plateau to persist over a filling interval, provided it does not overwhelm the interaction gap. The Laughlin wavefunction at $\nu =1/(2p+1)$ is a particularly simple correlated example; its charged excitations carry fractions of the electron charge. More generally the basic quasiparticle charge in the simple Jain sequence has magnitude $e/(2pq\pm 1)$. Such excitations are distinct from noninteracting bare electrons occupying a fraction of a level.

At $\nu =1/(2p)$, the mean effective field $B^{\ast }$ vanishes. This permits a compressible composite-fermion Fermi-sea description, rather than an ordinary integer effective-level gap. The external field remains large: cancellation is in the effective description. Not every fractional state is explained by the simplest integer composite fermion sequence, and the effective masses and gaps reflect interactions rather than simply the GaAs band mass. The utility of the construction is that it organizes many observed fractions and their nearby excitations into one understandable pattern.

A quantum point contact is made by using negative surface split gates to leave a short narrow channel in a clean two-dimensional electron gas. Its length should be smaller than the transport mean free path; transverse confinement should be strong enough that subband spacing exceeds thermal and lifetime broadening. Low-resistance Ohmic contacts connect large reservoirs. Apply a small source–drain bias, measure current and differential conductance, and sweep the gate to change the channel width or barrier height. Contact and wiring series resistance must be accounted for when comparing plateau values.

The Landauer formula explains the conductance without treating the short channel as a diffusive wire. For each propagating one-dimensional mode, the density of states and group velocity cancel in the incident flux, leaving $e/h$ per spin per energy interval. The reservoir occupation difference supplies the net current:The factor two assumes unresolved spin degeneracy. In zero-temperature linear response,A well-transmitted channel has $T_n\simeq 1$, giving steps $G=N(2e^2/h)$ as modes open. A finite two-terminal resistance persists even for a perfectly ballistic channel because reservoirs inject only a finite number of modes; it is not evidence of resistive scattering inside the channel. Gate-controlled point-contact conductance quantization was demonstrated in a two-dimensional electron gas.^[ref]

The saddle-point model of a quantum point contact resolves how modes open. Near the bottleneck, choose the channel direction as $x$ and writeTransverse harmonic confinement gives barrier tops $E_n=U_0+\hslash {\omega }_y(n+1/2)$. Scattering through the inverted longitudinal harmonic barrier givesSmall longitudinal curvature makes a sharp transmission threshold compared with the transverse subband spacing; finite curvature rounds each conductance step. This is the explicit smooth-barrier model developed for a saddle constriction.^[ref]

At larger source–drain bias, the current samples a finite energy window. For a rigid transmission spectrum at zero temperature,With symmetric bias, ${\mu }_{S,D}=E_F\pm eV/2$, the nonlinear conductance of a quantum point contact isIf both window edges sample $N$ open modes, the usual plateau remains. If one edge samples $N+1$ and the other $N$, differential conductance approaches $(N+1/2)(2e^2/h)$: a half-integer plateau relative to the zero-bias step. It is a bias-window effect, not half an electron spin degeneracy. Ridges in gate–bias transconductance occur when a reservoir chemical potential crosses a subband threshold; their separations can measure subband energies.

The half-plateau expression assumes symmetric bias and an approximately bias-independent barrier. An asymmetric voltage drop or self-consistent shift of $U_0$ changes the numerical relation. At still larger biases, heating and inelastic scattering can invalidate a rigid elastic saddle model. Thus the zero-bias quantization counts transmitted modes, while nonlinear spectroscopy compares mode counts at the two ends of the bias window.

An electrostatically defined quantum dot uses depletion gates to confine a small island laterally in a two-dimensional electron gas. Two narrow barriers allow weak tunnelling to source and drain; a plunger gate tunes the island's electrochemical levels, and additional gates can form a double quantum dot. Adjusting the gates resolves successive charge states and eventually reaches the single-electron regime. An electrical dot is defined by confinement and tunnel coupling, not simply by the drawn metal area.

In a constant interaction model,Here $Q_g$ is gate-induced charge and $C_{\Sigma }$ total island capacitance. Successive additions cost $E_{\mathrm{add}}=e^2/C_{\Sigma }+{ϵ}_{N+1}-{ϵ}_N$, with spin, exchange and correlations modifying the orbital term. Coulomb blockade appears when no allowed charge transition lies in the source–drain energy window. Gate sweeps give charge steps and transport peaks; finite-bias Coulomb diamonds constrain charging and orbital energies under a stated bias convention. Resolved single-electron operation requires thermal and tunnel broadening smaller than the relevant addition energies.

Quantum-dot charge sensing avoids relying on a measurable current through a nearly isolated dot. A nearby quantum point contact biased on a sensitive conductance slope, or a single-electron transistor, responds capacitively when one electron enters or leaves the dot. A time trace of sensor current can therefore distinguish $N$ from $N-1$ and resolve individual tunnelling events. The sensor measures charge; a spin-dependent step is needed to measure spin.

For one electron, Zeeman splitting gives a lower and a higher spin level separated by $E_Z=|g^{\ast }|{\mu }_B|B|$. In energy-selective spin readout, initialize an empty dot and pulse both levels below the reservoir chemical potential to load one electron. Then move to a readout configurationAn electron in the higher state can tunnel out, temporarily leaving the dot empty, after which a lower-state electron can refill it. An electron initially in the lower state remains trapped. The charge sensor records a pulse for the first outcome and no escape pulse for the second. This is spin-to-charge conversion; it measures the spin projection along the field through a state-dependent tunnelling probability. A primary single-shot experiment uses this conversion with a quantum-point-contact charge detector.^[ref]

The readout window must be long enough to detect escape but short compared with excited-spin relaxation:Detector bandwidth must also resolve the empty-state dwell time before refilling. Thermal activation, relaxation before escape, missed fast events and sensor backaction cause errors. Since the excited electron can leave and be replaced, this simple sequence does not preserve an arbitrary initial spin state.

For coupled dots, label charge occupation by $(n_L,n_R)$. Gate-controlled detuning can make $(1,1)$ and $(0,2)$ accessible. Two electrons in the lowest orbital of one dot must form a singlet state; a triplet state requires a different orbital or another higher-energy configuration. Choose a measurement window where the $(0,2)$ singlet is available but the $(0,2)$ triplet is not. Thenunder spin-conserving tunnelling. The second process is forbidden by spin symmetry and the Pauli exclusion principle, producing Pauli spin blockade. Charge sensing distinguishes transfer from a blocked $(1,1)$ state. This measures singlet versus triplet content; without a reference spin or additional control it is not a measurement of each electron's absolute spin direction separately.

The exchange interaction in a double quantum dot provides control of these states. In an effective spin description,For a symmetric weak-tunnelling two-site model with onsite charging cost $U$, $J\simeq 4t_c^2/U$. Gates change tunnel coupling $t_c$ and detuning, allowing preparation and evolution followed by a charge-conversion measurement. Spin mixing from local magnetic-field differences or spin-orbit effects can lift blockade, so the sensing interval must be chosen before unwanted relaxation obscures the initial state. Single-dot readout uses spin-dependent escape; double-dot readout can use spin-dependent charge transfer. In both cases the measurable signal is a resolved charge change.