= Solution
\b[A trajectory-resolving ballistic experiment.] Use a high-mobility modulation-doped GaAs-AlGaAs heterostructure with a <two-dimensional electron gas>. Grow the layers by molecular-beam epitaxy, separate the donor layer from the conducting interface with an undoped spacer, and pattern an isolated mesa with ohmic reservoir contacts. Electron-beam lithography defines metal <split gates>; negative gate <voltage> depletes the underlying <electron> gas, leaving narrow adjustable injector and collector <quantum point contacts> along an edge. Keep the relevant path shorter than the momentum-relaxation <mean free path>, cool to suppress <phonon scattering> and thermal smearing, and use a small excitation to avoid heating.
Inject a low-frequency current through one contact to a distant drain and measure the collector's open-circuit <voltage> with a high-input-impedance amplifier. Sweep perpendicular <magnetic field> and gate settings. The <cyclotron radius> is $r_c=\hbar k_F/(e|B|)$, with $k_F=\sqrt{2\pi n_s}$ for a spin-degenerate single-valley gas. For contact separation $L$, semicircular or skipping trajectories focus into the collector when
$$
\boxed{L=2jr_c,\qquad |B_j|=\frac{2j\hbar k_F}{eL},\quad j=1,2,\ldots.}
$$
The appropriate field sign bends carriers toward the collector; the opposite sign suppresses the peaks. Density-dependent focusing positions, suppression with increased scattering and reproducible gate control distinguish directional ballistic propagation from an ordinary diffusive <voltage>. <Quantum interference> may add fine structure, but the basic focusing signal does not require phase coherence over every path. Split-gate injection and collection in a two-dimensional gas were directly demonstrated in the original focusing experiment.<Coherent electron focusing with quantum point contacts in a two-dimensional electron gas by H. van Houten and others>[1989 primary publication, institutional abstract: split Schottky gates, injector/collector, skipping orbits and field-dependent focusing]{ref}
\b[Multi-probe current bookkeeping.] For one directed spin-resolved <mode>, velocity times one-dimensional <density of states> gives a particle flux $dE/h$. A small reservoir electrochemical-energy shift $\mu_j$ therefore supplies excess flux $\mu_j/h$. Let $N_i$ be the number of propagating <modes> at lead $i$, $R_{ii}$ their summed return probability, and $T_{ij}$ the total transmission from reservoir $j$ into lead $i$. Multiplying outgoing-minus-incoming <electron> flux by charge $-e$ gives the source's convention,
$$
\boxed{I_i=-\frac eh(N_i-R_{ii})\mu_i
+\frac eh\sum_{j\ne i}T_{ij}\mu_j.}
$$
The reservoirs equilibrate arriving carriers; an ideal <voltage> probe adjusts its <chemical potential> until its net current vanishes. Here $\mu_i$ is an energy, not a <voltage>. Choose the lead-voltage convention $\mu_i=-eV_i$ and positive <conventional current> injected into the junction. The working equation is then
$$
\boxed{I_i=g_0\left[(N_i-R_{ii})V_i-\sum_{j\ne i}T_{ij}V_j\right],\qquad g_0=e^2/h.}
$$
Conservation of scattering probability and <gauge invariance> require both column and row balance: every outgoing channel is accounted for and equal <voltages> give zero current. The counts $M,N$ below include every occupied internal channel; an extra factor of two must not be inserted if <spin> is already counted. If one instead counts orbital channels with unresolved <spin degeneracy>, double all transmission counts consistently.
\b[Zero-field routing.] The original diagram labels lead 1 at the left, 2 above, 3 at the right and 4 below. With columns denoting injectors and rows collectors, straight-through ballistic routing and the stipulated excess-wide-channel rule give
$$
\mathsf T_0=\begin{pmatrix}
0&0&M&0\\
0&0&0&M\\
M&0&0&N-M\\
0&M&N-M&0
\end{pmatrix},\qquad (N_1,N_2,N_3,N_4)=(M,M,N,N).
$$
There is no reflection in this ideal routing model. Set $V_4=0$, $I_2=I_3=0$, and $I_1=I=-I_4$. The two probe equations give
$$
V_2=0,\qquad V_3=\frac MN V_1,
\qquad I=g_0M\left(1-\frac MN\right)V_1.
$$
Thus
$$
\boxed{R_{14,23}(0)=\frac{V_2-V_3}{I}=-\frac{h}{e^2(N-M)}.}
$$
The derivation assumes $N>M>0$. At $N=M$, the ideal network splits into two disconnected opposite-lead pairs and cannot carry the imposed 1-to-4 current at finite <voltage>; the divergence diagnoses failure of that limiting transport configuration, not an infinite dissipative resistance of each channel. Negative <bend resistance> is a nonlocal <voltage> sign, compatible with positive total dissipated power.
\b[Positive field.] At $+B_0$, left-deflected incoming carriers go $1\to2$, $2\to3$, $3\to4$. Lead 4 supplies $M$ <modes> to narrow lead 1; the remaining $N-M$ enter wide lead 3. The balanced transmission matrix is
$$
\mathsf T_+=\begin{pmatrix}
0&0&0&M\\
M&0&0&0\\
0&M&0&N-M\\
0&0&N&0
\end{pmatrix}.
$$
Therefore $V_2=V_1$, $V_3=MV_1/N$ and $I=g_0MV_1$, giving
$$
\boxed{R_{14,23}(+B_0)=\frac h{e^2}\frac{N-M}{MN}
=\frac h{e^2}\left(\frac1M-\frac1N\right)>0.}
$$
\b[Negative field.] Reversing the chirality gives $\mathsf T_-=\mathsf T_+^{\mathsf T}$, as expected from reciprocity for this spin-independent routing model. With $V_4=0$, the zero-current conditions become $V_3=V_4=0$ and $V_2=V_3=0$. Thus
$$
\boxed{R_{14,23}(-B_0)=0.}
$$
This does not contradict reciprocity: the general relation interchanges the current and <voltage> lead pairs when reversing field; it does not require the same fixed-contact four-terminal resistance to be even in $B$.
\Image[/past-exam-of-the-physics-of-nanoelectronic-systems-course-of-the-university-of-cambridge/2015/paper-1-bend-routing.png]
{title=Original transmission-matrix bookkeeping for the three ideal field limits; rows collect modes injected by columns}
\b[Comparison with the measured trace.] The PDF shows a negative region around zero field, approximately zero signal at sufficiently negative field, and a positive rising resistance near $+0.4\ {\rm T}$ and above. These signs match the three routing limits. Near zero field the depth is roughly a few tenths of a kilohm; the graph does not supply $M,N$, so there is no justified exact amplitude fit. Rounded corners, collimation and finite <cyclotron radius> continuously change which outgoing lead is reached between the ideal limits. Skipping trajectories and geometrical focusing can produce shoulders or resonances. At larger positive fields magnetic depopulation and <spin> resolution can change the <mode> counts, increasing the positive resistance or producing structure; disorder, incomplete equilibration and finite temperature round ideal features. These are candidate mechanisms, not uniquely established assignments of every small feature from one trace.
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