= Solution
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 $W_{\rm out}$, $p_{\perp,\rm out}\simeq p_{\perp,0}W_0/W_{\rm out}$. At the same time the bottom of the electrostatic potential falls, increasing longitudinal kinetic energy. Thus
$$
\sin\theta_{\rm out}=\frac{p_{\perp,\rm out}}{p_{\rm out}}
\simeq\frac{W_0}{W_{\rm out}}\frac{p_{\perp,0}}{p_{\rm out}},
$$
and 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.
\Image[/past-exam-of-the-physics-of-nanoelectronic-systems-course-of-the-university-of-cambridge/2016/paper-1-ballistic-beam.png]
{title=Aligned quantum point contacts collect a direct beam; a weak field bends it away}
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.<Imaging Coherent Electron Flow by B. J. LeRoy and others>[2002 author abstract: tip depletion, backscattering, conductance imaging and mode-dependent angular patterns]{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>,
$$
\frac{m^*v_F^2}{r_c}=ev_F|B|,\qquad
r_c=\frac{m^*v_F}{e|B|}=\frac{\hbar k_F}{e|B|}.
$$
For one valley with two nearly equally occupied spin species at small field, state counting in the Fermi disc gives $n_s=2\pi k_F^2/(2\pi)^2=k_F^2/(2\pi)$. Hence
$$
\boxed{r_c=\frac{\hbar\sqrt{2\pi n_s}}{e|B|}.}
$$
Spin 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>, $\mu=e\tau_{\rm tr}/m^*$. Therefore
$$
\boxed{\ell_{\rm tr}=v_F\tau_{\rm tr}=\frac{\mu\hbar\sqrt{2\pi n_s}}e.}
$$
The <effective mass> cancels from both $\ell_{\rm tr}$ and $r_c$ when expressed in terms of density and mobility. For the supplied density,
$$
k_F=1.12100\times10^8\,\mathrm{m^{-1}},\qquad
\lambda_F=56.05\,\mathrm{nm}.
$$
With $m^*=0.067m_0$, $m_0\simeq9.1093837139\times10^{-31}\,\mathrm{kg}$, one also obtains $v_F\simeq1.937\times10^5\,\mathrm{m\,s^{-1}}$ and $\tau_{\rm tr}\simeq38.1\,\mathrm{ps}$. The requested lengths are
$$
\boxed{\ell_{\rm tr}=7.38\,\mathrm{\mu m},\qquad
r_c(30\,\mathrm{mT})=2.46\,\mathrm{\mu m}.}
$$
A useful independent ratio is $\ell_{\rm tr}/r_c=\mu|B|=3$. The standard constants used here are explicitly tabulated, rather than inferred from the sample.<Fundamental Physical Constants — Complete Listing by National Institute of Standards and Technology and others>[2022 CODATA: electron mass, elementary charge and Planck constant]{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.
\b[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.
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