= Solution
A <modulation-doped GaAs-AlGaAs electron gas> is confined on the <GaAs> side of the interface. The larger-gap <AlGaAs> supplies a conduction-band barrier; charge transfer from remote donors and the resulting <electric field> bend the bands into an approximately triangular well. <transverse electronic mode> wavefunctions are confined normal to the interface but free in its plane. <electrons> settle into the well until their electrochemical potential agrees with the donor/contact environment. An undoped spacer separates mobile <electrons> from ionized donors, reducing impurity scattering. Equilibrium has one <Fermi level> across the structure; the conduction-band edge is not flat. Surface depletion and the cap also affect the profile. The original drawing illustrates one occupied <confinement> <transverse electronic mode>; a single spatial <electron> gas need not in general imply only one occupied vertical <transverse electronic mode>.
<molecular beam epitaxy> can grow a clean <GaAs> buffer/channel, an undoped <AlGaAs> spacer, a Si-doped <AlGaAs> supply region and a protective <GaAs> cap on a suitable substrate. Control composition, interface quality and background impurities during growth; placing dopants directly in the channel defeats the principal mobility advantage. Define a mesa or isolation region, fabricate low-resistance <ohmic contacts> to the buried gas, and pattern fine <metal> <split gates> by <electron beam lithography> and the <lift-off process>. Negative gate bias depletes the underlying gas, leaving a narrow channel between the gates. Check contact resistance, leakage, <carrier density> and mobility; operation with <ballistic electron transport> requires channel <length> shorter than the relevant scattering <lengths> and smooth <confinement> on the wavelength scale. The lithographic gap does not equal the electrical width because of lateral depletion.
\Image[/past-exam-of-the-physics-of-nanoelectronic-systems-course-of-the-university-of-cambridge/2013/paper-1-heterostructure.png]
{title=Original qualitative conduction-band profile, epitaxial layer order and electrostatic split-gate channel}
For a long uniform channel supporting <ballistic electron transport>, transverse <confinement> creates thresholds $E_n$ with continuous longitudinal propagation above each threshold. One directed <spin> branch of a <transverse electronic mode> has states per <length> $dk/(2\pi)$ and <group velocity> $v=\hbar^{-1}dE/dk$. Hence $v\,dk/(2\pi)=dE/h$: <velocity> cancels the one-dimensional <density of states>. Including two unresolved <spin> states,
$$
I=\frac{2e}{h}\sum_n\int T_n(E)[f_S(E)-f_D(E)]\,dE.
$$
For low <temperature> and small bias, a fully open <transverse electronic mode> has $T_n(E_F)=1$, giving
$$
\boxed{G=\frac{2e^2}{h}N.}
$$
The assumptions are ideal reservoirs/contacts, negligible reflection and interactions that would invalidate the independent-subband description, and unresolved <spin> splitting. A resolved <spin> branch contributes $e^2/h$. The finite two-terminal <electrical resistance> is a contact/mode-injection effect, despite the absence of dissipative scattering inside the ideal channel.
For <voltage partition in a ballistic constriction>, a short saddle has strong local electric-field and <electrostatic screening> variation near its entrances and barrier, rather than the uniform Ohmic <voltage> gradient of a diffusive wire. Right-moving incident states inherit the source distribution and left-moving states the drain distribution; there is no unique local equilibrium <chemical potential> inside an ideal ballistic saddle. The electrostatic potential must be determined self-consistently. In a symmetric rigid-barrier approximation, the barrier follows the mean lead potential and half the electrochemical bias lies on each side. This is a stated approximation, not a universal spatial <voltage> profile.
For a saddle $U=U_0-m^*\omega_x^2x^2/2+m^*\omega_y^2y^2/2$, <transverse electronic mode> thresholds are $E_n=U_0+\hbar\omega_y(n+1/2)$. The smooth-barrier transmission is $T_n(E)=[1+\exp(-2\pi(E-E_n)/(\hbar\omega_x))]^{-1}$, approaching a step for a sharp threshold on the bias/thermal scale.<Quantized transmission of a saddle-point constriction by M. Büttiker>[Phys. Rev. B 41, 7906–7909, 1990: saddle transmission model]{ref}
Let $\mu_{S,D}=E_F\pm eV/2$ and assume no additional bias dependence of $T_n$. At zero <temperature>,
$$
I=\frac{2e}{h}\sum_n\int_{\mu_D}^{\mu_S}T_n(E)\,dE,\qquad
\boxed{\frac{dI}{dV}=\frac{e^2}{h}\sum_n[T_n(\mu_S)+T_n(\mu_D)].}
$$
When the source and drain sample $N+1$ and $N$ thresholds, the differential <electrical conductance> is $(N+1/2)2e^2/h$. These half plateaus arise from the two ends of the bias window, not fractional carriers. The dc ratio $I/V$ need not share their plateau values. Heating, <spin> splitting and interaction-induced features must be distinguished experimentally.
In <bias spectroscopy of one-dimensional subbands>, plot differential <electrical conductance> or its gate derivative against gate bias and source-drain bias. Threshold branches occur where $E_n(V_g)=\mu_S$ or $\mu_D$. When one bias-window edge meets $E_n$ and the other $E_{n+1}$,
$$
\boxed{\Delta E_n=e|V|.}
$$
Thus the intersections of adjacent threshold branches calibrate the one-dimensional spacing directly in <energy>. A gate lever arm $\alpha=-e^{-1}\partial E_n/\partial V_g$ can then convert gate intervals to <energy>; it should be measured rather than guessed. Asymmetric bias partition, a moving barrier or a gate-dependent spacing modifies slopes and plateau weights, requiring the corresponding self-consistent bias model.
\Image[/past-exam-of-the-physics-of-nanoelectronic-systems-course-of-the-university-of-cambridge/2013/paper-1-bias-spectroscopy.png]
{title=Original rigid-saddle bias spectroscopy and differential half plateaus}
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