Quantized Transport in Two Dimensions

The quantum Hall effect stands as one of the most striking manifestations of quantum mechanics in macroscopic systems. Discovered in 1980 by Klaus von Klitzing, the integer quantum Hall effect (IQHE) revealed that the Hall resistance of a two-dimensional electron gas subjected to a strong perpendicular magnetic field takes on values quantized in units of $\frac{h}{e}$. Remarkably, these plateaus are extraordinarily precise, enabling the effect to serve as a standard for the resistance unit. The phenomenon arises because electrons in a magnetic field occupy discrete Landau levels with energy spacing $ħ\omega c_{}$, where $\omega c_{}$ is the cyclotron frequency. When the chemical potential resides in a gap between Landau levels and the sample edges support chiral one-dimensional channels, backscattering is suppressed and the Hall conductance becomes topologically protected.

The key dimensionless parameter characterizing the system is the filling factor $\nu$, defined as the ratio of the electron areal density to the density of magnetic flux quanta. In formula form,

$\nu =\frac{n}{\frac{e}{h}}$

where $n$ is the two-dimensional electron density, $e$ the elementary charge, and $h$ Planck's constant. Each Landau level can accommodate one electron per flux quantum, so $\nu$ counts the number of completely filled Landau levels. The Hall conductivity then follows the celebrated relation

${\sigma }_{\mathrm{xy}}=\nu \frac{e}{h}{2}^{}$

while the Hall resistance is the inverse, $R_{\mathrm{xy}}=\frac{h}{\nu }$. In practice, measuring $R_{\mathrm{xy}}$ and observing the plateaus as a function of magnetic field or carrier density provides direct evidence of topological quantization. The calculator on this page implements these relations, enabling the user to explore how variations in magnetic field or electron density shift the filling factor and associated transport coefficients.

To use the calculator, supply the areal electron density in units of m⁻², the perpendicular magnetic field in Tesla, and the sample area in square meters. The script evaluates the filling factor, the Hall conductivity, the Hall resistance, and the Landau level degeneracy within the specified area, given by

$N=\frac{\mathrm{eBA}}{h}$

This degeneracy counts the number of states available in each Landau level for the given area, offering intuition about how many electrons can be accommodated before another level is required. The calculator also reports the nearest integer to the filling factor to indicate the closest quantum Hall plateau. If the filling factor lies close to an integer—say within 0.05—the system is likely in an integer quantum Hall state. Otherwise, it may correspond to a fractional state where electron interactions create new energy gaps characterized by fractional $\nu$ values.

The fractional quantum Hall effect (FQHE) discovered by Tsui, Stormer, and Gossard in 1982 demonstrates the richness of many-body physics in two dimensions. At fractional fillings such as $\frac{1}{3}$ or $\frac{2}{5}$, strong Coulomb interactions cause electrons to form correlated states described by composite fermions or conformal field theories. While the calculator focuses on non-interacting formulas, the filling factor it outputs serves as the starting point for understanding whether such correlated phases might arise under given experimental conditions.

The quantization of Hall conductance has deep topological underpinnings. Each filled Landau level contributes a Chern number of one to the total Hall conductance, linking the IQHE to topological band theory. The robustness against disorder and impurities stems from the fact that the Hall conductance is a topological invariant insensitive to local perturbations. Edge states—chiral one-dimensional modes that propagate along the sample boundary—provide a physical picture of this robustness: since they travel in only one direction, there is no backward channel for scattering. The number of edge states equals the filling factor, so measuring transport gives direct insight into the topology of the bulk electron wave functions.

In practical devices such as GaAs/AlGaAs heterostructures or graphene, controlling the carrier density is achieved via electrostatic gates. As the density is swept, the filling factor passes through successive integers or rational fractions, each associated with its own plateau in $R_{\mathrm{xy}}$. The magnetic field strength required to reach a given filling factor can be approximated from the formula above. For example, a density of 3×10¹⁵ m⁻² at a magnetic field of 10 T yields a filling factor near 1.2, placing the system near the first plateau. The table below summarizes several scenarios calculated using the tool for illustrative densities and fields.

n (m−2)	B (T)	ν	Rxy (Ω)
3×1015	10	1.24	20 Ω
1×1015	5	0.83	31 Ω
5×1015	15	1.38	18 Ω

Beyond providing numerical values, the calculator encourages users to contemplate the hierarchy of energy scales in the quantum Hall regime. The Landau level spacing $ħ\omega c_{}$ must exceed the thermal energy $k{B}_{}T$ to observe quantization, motivating experiments at low temperatures and high magnetic fields. Disorder broadening should remain smaller than the gaps to maintain sharp plateaus. Through these considerations, the quantum Hall effect has fostered profound connections between condensed matter physics, topology, and even high-energy theory via the AdS/CFT correspondence, where similar quantization appears in holographic models.

Although this calculator handles only the basic IQHE formulas, it can be extended to investigate more intricate scenarios. For bilayer systems or materials with spin and valley degeneracies, the filling factor incorporates those degrees of freedom, modifying the Hall conductance steps. In systems exhibiting the anomalous Hall effect, Berry curvature plays a role analogous to magnetic field. By adapting the formulas, one could explore the topological Hall effect in skyrmion crystals or the higher-dimensional quantum Hall states proposed in mathematical physics. Thus, even a simple tool like this serves as a gateway to a vast landscape of quantum phenomena.

Historically, the precision of the IQHE has enabled the determination of the fine-structure constant through electrical measurements, uniting quantum mechanics with metrology. The Hall resistance plateaus are so reproducible that they form the basis of the von Klitzing constant, now defined exactly as $R{K}_{}=\frac{h}{e^2}$. With the advent of graphene, quantum Hall physics has entered new regimes where relativistic-like Dirac fermions produce half-integer shifts in the filling factor, further enriching the theoretical landscape. As researchers continue to discover novel two-dimensional materials and engineer moiré superlattices, the quantized Hall response remains a cornerstone for probing electronic topology.

In summary, the quantum Hall effect exemplifies the interplay between quantum mechanics, topology, and material science. This calculator provides a practical means to compute core quantities—filling factor, Hall conductivity, Hall resistance, and Landau level degeneracy—directly from measurable parameters. By experimenting with the inputs, students and researchers can build intuition about how magnetic field strength and electron density influence transport, guiding the design of experiments and the interpretation of data. The tool thus functions both as a computational aid and as a pedagogical bridge to one of condensed matter physics' most elegant phenomena.