How this instrument works
The Hall coefficient, R_H, measures how strongly a conductor develops a transverse voltage when a magnetic field bends its moving charges off their straight path. Drive current I through a slab of thickness t inside field B, and the deflected carriers pile up against one edge until the electric field they build stops any further sideways drift. That balance voltage, V_H, divided by thickness, current, and field, gives R_H = V_H·t ⁄ (I·B) — arranged so the result depends only on the carriers themselves, not on how large the sample is or how hard it was driven.
Its sign is the real payoff. A slab whose current is carried by negative electrons deflects to build a Hall field of one polarity; one whose current is actually carried by positive holes, as in many p-type semiconductors, deflects the opposite way and R_H comes out positive. Two samples with identical resistivity, cut from opposite carrier types, will read identically on an ohmmeter — but their Hall coefficients will have opposite signs. That distinction is why Hall measurements, not resistivity alone, are how a semiconductor lab confirms whether a freshly doped wafer turned out n-type or p-type.
The formula assumes a single dominant carrier species and a field weak enough to stay in the ordinary, classical regime. Push the field high enough and cool the sample low enough, and the quantum Hall effect takes over: R_H stops varying smoothly with B and instead locks onto plateaus tied to fundamental constants, a behavior this linear algebra cannot capture. Below that regime, at everyday lab temperatures and fields of a few tesla or less, the formula is exact enough to remain the standard route to carrier type and carrier density alike.
- Enter the Hall voltage — the transverse voltage measured across the sample, in mV or V.
- Enter the Sample thickness — the dimension measured along the magnetic field direction, in mm or µm.
- Enter the Current driven lengthwise through the sample, in mA or A.
- Enter the Magnetic field applied perpendicular to the sample's flat face, in tesla.
- Read the Hall coefficient, m³ ⁄ C — its sign names the majority carrier and its magnitude sets the carrier density.
Worked example — characterizing a freshly doped wafer
A materials lab measures a newly doped semiconductor sample to confirm its carrier type before it leaves the cleanroom. Driving 0.1 A through a 1 mm thick slab inside a 0.5 T field produces a measured Hall voltage of 5 mV, which is 0.005 V. Plugging into R_H = V_H·t ⁄ (I·B) gives R_H = (0.005 × 0.001) ⁄ (0.1 × 0.5) = 0.000005 ⁄ 0.05 = 0.0001 m³/C — and because it comes out positive, holes rather than electrons are doing the conducting.
That single figure reaches further than a resistivity reading ever could. Carrier density follows from n = 1 ⁄ (R_H·q), where q is the elementary charge, 1.602176634 × 10⁻¹⁹ C: n = 1 ⁄ (0.0001 × 1.602176634 × 10⁻¹⁹) ≈ 6.24 × 10²² carriers per cubic metre, a doping level typical of a moderately doped semiconductor, several orders below the roughly 10²⁸ carriers per cubic metre packed into an ordinary metal. A technician reading a positive 0.0001 m³/C off this instrument now knows the wafer's carrier type and its concentration from one measurement.
Questions
What does the sign of the Hall coefficient mean?
It names the majority charge carrier. A positive R_H means conduction is dominated by positively charged holes, as in many p-type semiconductors; a negative R_H means electrons dominate, as in n-type semiconductors and most ordinary metals. The sign comes straight from which way the carriers deflect under the magnetic force, so it reads directly off the polarity of the measured Hall voltage — no separate test is needed.
Why does the formula divide by sample thickness?
Because the Hall voltage tracks current density, not total current, and current density is current divided by the cross-section it flows through — width times thickness. A thicker sample spreads the same current over more cross-section and weakens the transverse field for a given R_H, so dividing V_H by t and multiplying back by I and B cancels the geometry and leaves a number that depends only on the material.
How do I get carrier density from the Hall coefficient?
Divide 1 by the product of R_H and the elementary charge, q = 1.602176634 × 10⁻¹⁹ C: n = 1 ⁄ (R_H·q). For the 0.0001 m³/C example above, that works out to about 6.24 × 10²² carriers per cubic metre. This relation assumes one dominant carrier type — it is the standard route a materials lab uses to turn a raw Hall measurement into a doping concentration.
Is this the same Hall effect used in car and phone sensors?
Same physics, different job. A packaged Hall-effect sensor is a calibrated device with fixed geometry that outputs a voltage proportional to an unknown external field it is meant to detect. This calculator runs the relationship the other way: it takes a measured voltage from a sample of known geometry and current to extract R_H, an intrinsic material property used to characterize the sample, not to sense an outside field.
Does the Hall coefficient change with a stronger magnetic field?
Not in the ordinary regime this formula covers — R_H is meant to be a property of the material, so a stronger field should raise the measured Hall voltage proportionally and leave R_H unchanged. At very low temperatures and very high fields the quantum Hall effect takes over, and R_H locks onto quantized plateaus instead of scaling smoothly, a different regime the plain formula does not model.
Who actually measures Hall coefficients in practice?
Semiconductor labs characterizing new wafers, materials scientists studying unfamiliar conductors, and physics students repeating Edwin Hall's 1879 experiment all use versions of this measurement. The usual setup follows the van der Pauw or bridge-type geometry standardized in ASTM F76, applying a known current and field and reading the transverse voltage with contacts placed exactly as this calculator's inputs assume.