Past exam of the condensed matter physics major option course of the University of Oxford / 2018 / 6

A sufficiently clean two-dimensional electron gas at low temperature in a perpendicular magnetic field has quantized orbital Landau levels. In the integer quantum Hall regime, the longitudinal resistivity is nearly zero while the Hall resistance has plateauxThe Hall sign depends on carrier charge and lead convention. Fractional plateaux can occur in strongly interacting states, but the following single-particle counting and calibration use the integer effect.

One resolved spin branch of a Landau level has one state per flux quantum $h/e$: in area $A$, its guiding-centre degeneracy is $Ae{B}_z/h$ for positive $B_z$. Defining the nominal filling factor as the number of electrons per available orbital state givesThe same count follows in Landau gauge: a $2\pi /L_y$ increment in $k_y$ moves the orbit centre by $2\pi {\ell }_B^2/L_y$, giving $A/(2\pi {\ell }_B^2)=Ae{B}_z/h$ states. Include each spin branch separately; an unresolved spin-degenerate orbital level contains twice this count, and common plateau indices are then even.

Plateau qualification: $\nu$ as defined by density is continuous when $B_z$ changes at fixed $N_S$. The integer plateau index $p$ can remain unchanged over a field interval because localized states accommodate the changing occupation. Therefore $N_S=pe{B}_z/h$ is not an exact identity across an entire plateau. It is the ideal filled-level relation, often used approximately at a suitable plateau centre; confusing the nominal filling with the plateau index would contradict fixed density.

The plateau resistance realizes the von Klitzing constant $R_K=h/e^2\simeq 25812.8\Omega$, divided by a known integer. A four-terminal Hall measurement with negligible longitudinal voltage and contacts in the quantized regime supplies a reproducible resistance reference, which can be compared to other resistors using precision bridges. Its robustness comes from the quantized transport state, not from measuring the Hall bar's width accurately.

The Hall-plateau voltmeter calibration is obtained as follows. Let $D_p$ be a plateau height in displayed units after any voltage offset is removed, with identified integer plateau index $p$. The true Hall voltage is $I_x{R}_K/p$, so the calibration factor would beThe nominal current is $15\mathrm{nA}$. If several plateau heights are available, their inverse-integer ratios help identify indices and a fit of $D_p$ against $1/p$ can separate a constant offset from the gain. Reversing magnetic field and antisymmetrizing the voltage also removes an even offset. A unique absolute index should not be asserted solely from an unspecified sequence of steps.

For an established exact integer $p$ and independent current and height errors,The current alone contributes $2\%$ relative uncertainty; it is a common calibration uncertainty and does not average away by fitting more plateaux. Voltage-offset uncertainty and deviations from quantization must also be included where relevant. The $25\pm 2\mu \mathrm{m}$ width cancels because the Hall voltage divided by current is a sheet Hall resistance; it would enter the transverse electric field, not this calibration.

TODO — missing source data: the held PDF's last-page stream refers to an absent /Im6 graphic, and its recovery log reports an unknown XObject. The supplied TeX records the same source gap. No plateau heights, labels or axis values survive, so neither a numerical $D_p$ nor a unique $p$ can be read. The numerical voltmeter calibration and its complete graph-reading error cannot be supplied without that graph; the formula and known $2\%$ current contribution above are determined.

As magnetic field grows from zero, a clean two-dimensional electron gas first has an ordinary finite longitudinal resistivity, then develops Shubnikov–de Haas oscillations as broadened Landau levels pass the chemical potential. Thermal and disorder broadening obscure the low-field oscillations; their visibility grows when the level spacing exceeds those broadening scales. In a well-developed integer quantum Hall regime, ${\rho }_{xx}$ has minima close to zero on Hall plateaux and maxima during transitions between them, when extended states cross the chemical potential. Detailed peak heights and envelopes depend on disorder and scattering and are not fixed by quantization alone.

At fixed density the nominal filling is $\nu (B)=N_{S}h/(eB)$, so level-filling features are approximately periodic in $1/B$, with period $e/(N_{S}h)$ for resolved spin levels or $2e/(N_{S}h)$ for an unresolved spin pair. Increasing $B$ lowers the filling; spin splitting can create additional odd-index features. The original schematic below shows these relations without pretending to reconstruct the missing data.

TODO — missing source graph: the actual $0$–$12\mathrm{T}$ positions and integer plateau labels require $N_S$ or surviving plateau fields from part (b). Neither is provided independently. Thus a uniquely labelled field-axis sketch for this sample cannot be completed. The following DOS diagram distinguishes extended centres and localized tails generically; its chemical-potential markers are illustrative and do not stand for the requested 8 T or 10 T assignments.

At $B_z=10\mathrm{T}$, the nominal filling isA labelled density-of-states picture has disorder-broadened peaks at these orbital and spin energies, localized tails and extended states near their centres. A chemical potential in the localized region corresponds to a Hall plateau and small longitudinal resistivity; passing through an extended-state centre corresponds to a transition.

TODO — missing density/plateau assignment: there is no basis for choosing one of those positions at 10 T without the missing Hall graph. For example $N_S=4.84\times 10^{15}{\mathrm{m}}^{-2}$ gives nominal ${\nu }_{10}\simeq 2$, whereas $N_S=6.04\times 10^{15}{\mathrm{m}}^{-2}$ gives ${\nu }_{10}\simeq 2.5$. In an ideal resolved-level picture the first can put the chemical potential between filled levels, while the second leaves a level partly filled. Both are compatible with the surviving current and width, demonstrating why the requested placement is undetermined.

At $B_z=8\mathrm{T}$,The cyclotron spacing is $\hslash e(8\mathrm{T})/m^{\ast }$, smaller than at 10 T, and each resolved level's degeneracy per area is also reduced by $4/5$. Thus the same carrier density occupies more levels. This ratio is determined without the missing graph, but it does not specify whether the chemical potential lies in a localized tail or an extended-state centre.

TODO — missing density/plateau assignment: a numerical index and chemical-potential marker at 8 T require the unavailable density information. For example the first illustrative density in part (i) gives ${\nu }_8\simeq 2.5$, while $N_S=3.87\times 10^{15}{\mathrm{m}}^{-2}$ instead gives ${\nu }_8\simeq 2$. These produce different schematic placements even though both satisfy every surviving sample parameter. Unresolved spin degeneracy and disorder add further qualifications; no unique 8 T label should be invented.

A perfectly clean two-dimensional system has sharply degenerate Landau levels and integer filling only at isolated fields for fixed density. Real disorder broadens each level into a band, with states localized in the tails and extended states near its centre. When the chemical potential moves through a localized region, changing field can change the occupation of those localized states without changing the number of conducting extended channels. The Hall conductance then remains $p{e}^2/h$ and the Hall voltage at fixed current remains $I_{x}h/(p{e}^2)$ over a finite field interval.

In the same regime, separated edge channels and localized bulk states suppress dissipative backscattering, so ${\rho }_{xx}$ is very small. At a transition the chemical potential reaches the next band of extended states; longitudinal conduction rises and the Hall conductance changes to another integer. This is a mobility-gap explanation: a plateau does not require the density of states itself to vanish, nor does it require the carrier density to grow linearly with field. Low temperature, sufficient mobility and suitable contacts are needed to resolve it.

This physical explanation is determined by the surviving question text. Particular measured plateau heights and widths cannot be quoted because the source graph is absent, as recorded in part (b).