Hall Effect Simulator — Charge Carrier Density Measurement
Compute the Hall voltage V_H in real time from current, magnetic field, carrier density and sample thickness using the Lorentz-force balance. Learn the principle behind semiconductor carrier-density measurement with a Hall bar schematic and a log-log graph.
Parameters
Current I
mA
Magnetic field B
T
log10 n
log
n = 1.00e+23 /m^3
Sample thickness t
μm
Sample width is fixed at w = 10 mm; elementary charge q = 1.602e-19 C. Carrier density uses a log-scale slider, and only the magnitude of the Hall voltage is shown.
Results
—
Hall voltage V_H
—
Hall coefficient R_H
—
Hall field E_H
—
Current density j
Hall bar schematic
Rectangular sample with vertical current I, magnetic field B out of the page, and Hall voltage V_H developing transversely
Hall voltage vs carrier density (log-log)
X = log10(n), Y = |V_H| in log scale. V_H is proportional to 1/n, a straight line. The yellow dot marks the current operating point.
Theory & Key Formulas
In steady state on a Hall bar, the Lorentz force on the carriers is balanced by the electric field from the charge that has piled up at the sides. This gives the Hall voltage $V_H$ in terms of the current $I$, the magnetic field $B$, the carrier density $n$, and the sample thickness $t$.
Here $q = 1.602\times10^{-19}$ C is the elementary charge, $w$ the sample width, and $t$ the sample thickness. The simulator reports magnitudes only and does not infer carrier sign from $R_H$.
What is the Hall Effect Simulator?
🙋
How do you actually measure the carrier density of a semiconductor? You cannot count them under a microscope, right?
🎓
Good question. The standard technique uses the Hall effect. Drive a current I through the sample and put a perpendicular magnetic field B across it; a transverse voltage V_H appears, given by $V_H = IB/(nqt)$. Since you measure V_H and know I, B and t, you can solve for n directly. Move the sliders in the tool and notice that V_H is inversely proportional to n, i.e. a straight line with slope -1 on a log-log plot.
🙋
Why does a voltage develop in the direction perpendicular to both the current and the field?
🎓
Each moving carrier feels a Lorentz force $F = qv\times B$. Carriers moving along the current are pushed sideways and pile up on one face of the sample. The accumulated charge sets up an electrostatic field that opposes further deflection; when the two cancel, you have a steady-state transverse field. Multiply it by the sample width w and that is the Hall voltage.
🙋
With the defaults I get V_H about 3 mV. Is that actually measurable?
🎓
Easily. Lab-grade lock-in amplifiers reach the nanovolt floor, so microvolts and below are routine. If you push log10(n) to 28 or 29 (metallic densities), V_H drops below a microvolt and the measurement becomes much harder. The fact that semiconductors (log10 n between 22 and 25) give millivolt signals is precisely why Hall characterization is an industry standard.
🙋
What is the Hall coefficient R_H for?
🎓
$R_H = 1/(nq)$ is a geometry-independent material parameter, so it is what papers report when comparing semiconductors. Its sign also distinguishes electron conduction (negative) from hole conduction (positive). This tool reports magnitudes only, but in a real measurement the sign of R_H is essential information for identifying n-type versus p-type material.
Frequently Asked Questions
For a fixed current I, the current density j = I/(wt) is larger in a thinner sample. The Hall voltage is effectively determined by the carriers crossing per unit area times the magnetic field, so smaller t gives larger j and therefore larger V_H. In the simulator, reducing t from 100 μm to 1 μm grows V_H by a factor of 100. Commercial Hall sensors use chips only a few microns thick precisely to obtain a strong sensor output.
Yes. Electrons (negative) move opposite to the current direction, while holes (positive) move with it. The Lorentz force $F = qv\times B$ pushes both species to the same face of the sample, but because the charges have opposite sign the measured polarity of V_H is reversed. In practice, the sign of V_H is precisely what tells you whether the material is n-type or p-type. This simulator reports magnitudes only and does not display the polarity.
Metals have carrier densities of about $10^{28}$ to $10^{29}$ /m^3, while semiconductors at typical doping levels sit at $10^{20}$ to $10^{25}$ /m^3 — four to nine orders of magnitude lower. Because $V_H \propto 1/n$, V_H in metals is therefore much smaller for the same I, B, and t. Set log10(n) to 28 or 29 in the simulator and watch V_H drop into the microvolt or nanovolt range; this is the very reason practical Hall sensors are built from low-density semiconductors such as InSb or GaAs.
Not directly, but if you separately measure the sheet resistance $R_s$ (for example using the van der Pauw method) you can combine it with V_H to obtain the Hall mobility $\mu_H = R_H/\rho$, where $\rho$ is the resistivity. The Hall mobility is one of the most important quality metrics for semiconductors. The standard laboratory protocol combines van der Pauw resistance and Hall voltage on the same sample to extract both n and $\mu_H$ simultaneously.
Real-World Applications
Carrier density characterization in semiconductor processing: After ion implantation or diffusion doping, Hall measurements verify whether the target carrier density has been achieved. Combined with the van der Pauw method, sheet resistance and Hall voltage on a few-micron-thick silicon wafer yield n and $\mu_H$ at the same time, exactly the data needed to predict device behavior. Hall metrology stations are integrated into nearly every step of a modern semiconductor cleanroom flow.
Automotive Hall sensors: Crankshaft and camshaft position sensors that read non-contact rotation use Hall elements made of low-density InSb or InAs. Even with GMR and TMR sensors widely available, Hall devices remain dominant for their temperature stability and noise immunity, and a single hybrid vehicle can carry several dozen of them — in ABS, electric power steering, battery current monitoring, and many other systems.
Magnetic sensors and current measurement: Clamp-on ammeters read the magnetic field around a current-carrying wire with a Hall element, allowing currents up to about 1000 A to be measured without breaking the circuit. The same principle drives the digital compasses in smartphones and the lid-open switches in laptops. With no moving parts, Hall devices have essentially unlimited service life, which is their main advantage over competing sensors.
Quantum Hall effect and the SI: At high fields and low temperatures V_H becomes quantized in integer multiples of $R_H = h/(ne^2)$, the quantum Hall effect. The plateau values are universal constants independent of the sample, and they are used to measure Planck's constant $h$ to high precision — the same constant that, in the 2019 SI redefinition, fixes the kilogram. The classical Hall effect simulated here is the gateway to that quantum world.
Common Misconceptions and Pitfalls
The most common mistake is believing that the Hall voltage does not depend on sample thickness. In fact $V_H = IB/(nqt)$, so V_H is inversely proportional to t. Increase t from 100 μm to 5000 μm in this simulator and V_H drops by a factor of 50. The reason commercial Hall elements are built only a few microns thick is to maximize this signal — it is one of the most effective design knobs for boosting the sensor SNR.
A second pitfall is treating the residual voltage observed at B = 0 as a failure. On a real Hall bar the current and voltage contacts are never perfectly aligned, so at zero field a finite offset voltage (the misalignment voltage) remains, mixing in a small fraction of the longitudinal resistance. Standard practice is to reverse the magnetic field and subtract, a symmetrization that cancels the offset. This simulator is idealized and gives V_H = 0 at B = 0, which differs from a real measurement.
Finally, be careful before concluding the carrier type from the sign of V_H alone. While the sign of R_H tracks the carrier sign in principle, that statement assumes a fixed geometric convention for the current and field directions. Reversing a probe lead or flipping the magnet effectively doubles the sign flips and confuses the interpretation. Real measurements always calibrate the polarity against a reference sample (such as n-type Ge) before recording the device under test. The simulator reports magnitudes only.