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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.

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 $v={\hslash }^{-1}dE/dk$. Hence $vdk/(2\pi )=dE/h$: velocity cancels the one-dimensional density of states. Including two unresolved spin states,For low temperature and small bias, a fully open transverse electronic mode has $T_n(E_F)=1$, givingThe 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 $U=U_0-m^{\ast }{\omega }_x^2{x}^2/2+m^{\ast }{\omega }_y^2{y}^2/2$, transverse electronic mode thresholds are $E_n=U_0+\hslash {\omega }_y(n+1/2)$. The smooth-barrier transmission is $T_n(E)=[1+\exp (-2\pi (E-E_n)/(\hslash {\omega }_x)){]}^{-1}$, approaching a step for a sharp threshold on the bias/thermal scale.^[ref]

Let ${\mu }_{S,D}=E_F\pm eV/2$ and assume no additional bias dependence of $T_n$. At zero temperature,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 $E_{n+1}$,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.

graphene is a two-dimensional honeycomb lattice with two sublattices. Nearest-neighbour hopping gives two bands $E_{\pm }(\mathbf{k})=\pm t|{\sum }_{j=1}^3{e}^{i\mathbf{k}\cdot {\mathbf{\delta }}_j}|$. They touch at two inequivalent valleys. Expanding near valley $\tau =\pm 1$ givesThe two-component spinor describes sublattice amplitudes, not real spin. This is a massless two-dimensional Dirac Hamiltonian with material velocity $v_F$ in place of $c$. The emergent band quasiparticles are relativistic in this mathematical sense; the electron's vacuum rest mass is not set to zero. Real spin and valley each contribute a degeneracy of two in the ideal low-energy model.

A single-walled carbon nanotube is a rolled graphene sheet with carbon nanotube chiral vector ${\mathbf{C}}_h=n{\mathbf{a}}_1+m{\mathbf{a}}_2$. Periodicity requires $\mathbf{k}\cdot {\mathbf{C}}_h=2\pi \ell$, selecting lines through graphene momentum space and producing one-dimensional transverse electronic modes. Near a valley,A quantization line through one of the Dirac valleys in graphene gives a metallic branch; ideal nanotube zone folding gives the criterion $n-m\equiv 0(\mathrm{mod}3)$. Otherwise the nearest line misses the valley and opens $E_g=2\hslash v_F|q_{\perp }|$. The transverse momentum gives a carbon-nanotube Dirac mass, with $m_{\mathrm{eff}}=\hslash |q_{\perp }|/v_F$. For a large-radius ideal semiconducting tube, $|q_{\perp }|\simeq 2/(3d)$ and $E_g\simeq 4\hslash v_F/(3d)$, showing the inverse-diameter gap. Curvature, strain, spin-orbit coupling and interactions can change the simple classification; nominally metallic non-armchair tubes can acquire small gaps. One-dimensional transverse electronic mode thresholds give Van Hove singularities, unlike graphene's low-energy density of states proportional to $|E|$.

A gate changes occupation and the Fermi level in both systems. Ideal monolayer graphene can be tuned through charge neutrality from holes to electrons, but an ordinary scalar gate does not open a bulk band gap. Including spin and valley,The gate relation includes geometric and quantum capacitances, so one should not equate gate voltage to Fermi energy without the capacitance model. A semiconducting carbon nanotube can instead be depleted when its Fermi level lies in a gap and turned on in electron or hole bands. Contact Schottky barriers can strongly influence the observed nanotube field effect. A truly gapless ideal metallic tube cannot be switched off simply by moving the Fermi level through its band crossing. These gate responses reflect band structure and contacts, not a change of wrapping indices.