The Hall coefficient calculator above converts the four numbers you actually measure on a Hall bar — current, magnetic field, sample thickness and the transverse voltage that appears — into the two numbers you actually want: the Hall coefficient of the material and the density of the charge carriers inside it. Add a resistivity and it also returns the Hall mobility, which is how fast those carriers drift for a given electric field.
Arb Digital builds free physics calculators that each own one measurement properly. This page owns the Hall geometry, where a magnetic field pushes carriers sideways until the transverse electric field they build up exactly balances the magnetic force. The live Lorentz force calculator handles the force on a single moving charge in free space, and the drift velocity calculator handles how fast carriers creep along a conductor under an applied field. Neither of those solves for the carrier density, which is the quantity the Hall measurement exists to deliver.
What This Hall Coefficient Calculator Does
Pass a current along a thin bar and put a magnetic field through its face. The moving carriers feel a magnetic force at right angles to both, so they pile up on one edge. That pile-up creates a transverse electric field, and it grows until the electric force on a carrier exactly cancels the magnetic one. At that point the sideways motion stops and a steady voltage sits across the bar. That voltage is the Hall voltage, and it is small — microvolts to millivolts is normal.
The Hall coefficient is defined so that this voltage becomes a material property independent of your particular sample and probe current. The tool reports it in cubic metres per coulomb, along with the carrier density it implies, the sheet carrier density for thin films, and the Hall mobility if you supplied a resistivity. It also states in words whether the sign indicates electrons or holes.
The second mode runs the calculation backwards. Give it a carrier density and a carrier type and it predicts the Hall voltage a sample of that thickness would produce at that current and field, which is the calculation you do before building an experiment to check whether your instrument can even see the signal.
How to Use It
- Choose the direction you are working in. Start from a measured Hall voltage to get a material property, or from a known carrier density to predict a voltage.
- Enter the probe current and the field. Both appear linearly in the Hall voltage, so doubling either doubles the signal. That is the practical route to a measurable voltage in a low-mobility sample.
- Enter the thickness measured along the field. This is the dimension the field passes through, not the width between the voltage probes. Getting the wrong dimension here is the single most common error in the whole measurement.
- Add a resistivity if you want mobility. The conductivity and resistivity calculator converts between resistivity, conductivity and a measured resistance if that is what you have.
- Read the sign, not just the magnitude. The sign of the Hall coefficient is the result that a resistance measurement cannot give you at all.
The Formula: How The Hall Coefficient Is Calculated
For a bar of thickness t carrying current I in a perpendicular field B, the Hall voltage is VH = IB ÷ (nqt), where n is the carrier density and q the carrier charge. Georgia State University's HyperPhysics page on the Hall effect gives this in the equivalent form VH = IB ÷ ned and traces it back to E. H. Hall's 1879 experiment.
Rearranged, the Hall coefficient is
RH = VHt ÷ (IB) = 1 ÷ (nq)
and the carrier density follows as n = 1 ÷ (q RH). The elementary charge used here is 1.602176634 × 10−19 coulombs, which is exact by definition in the SI as published in the NIST table of fundamental physical constants. Hall mobility is μH = |RH| ÷ ρ, where ρ is the resistivity, and the calculator reports it in the customary square centimetres per volt-second.
Work the defaults by hand. A current of 1 mA is 0.001 A, a thickness of 0.1 mm is 1 × 10−4 m, the field is 0.5 T and the measured Hall voltage is 0.6 mV, which is 6 × 10−4 V. Then RH = (6 × 10−4 × 1 × 10−4) ÷ (0.001 × 0.5) = 6 × 10−8 ÷ 5 × 10−4 = 1.2 × 10−4 m³/C. The carrier density is 1 ÷ (1.602176634 × 10−19 × 1.2 × 10−4) = 5.201 × 1022 per cubic metre. The sheet density is that times the thickness, 5.201 × 1018 per square metre. With a resistivity of 0.01 Ω·m the mobility is 1.2 × 10−4 ÷ 0.01 = 0.012 m²/V·s, which is 120 cm²/V·s.
Why The Sign Was A Genuine Puzzle
Before the Hall effect there was no experiment that could tell you what was carrying the current. Resistance is blind to the sign of the charge: negative carriers going one way and positive carriers going the other produce identical currents and identical resistances. The Hall effect breaks that symmetry, because the magnetic force pushes both kinds of carrier to the same edge, which means the resulting voltage has opposite polarity in the two cases.
That made the effect enormously important, and it also produced one of the great anomalies of early solid state physics. Several metals — zinc, cadmium, aluminium — give a positive Hall coefficient, as if the current were carried by positive particles. Nothing positive is mobile in a metal. The resolution had to wait for band theory, which showed that a nearly full band behaves as though the missing states are positive carriers. The Hall effect measured holes decades before anyone had a language for them.
This is why a semiconductor characterisation lab measures the Hall coefficient before almost anything else. It says n-type or p-type in one number, and combined with resistivity it gives the mobility as well. The number density calculator handles the general counting problem of particles per unit volume, but only a Hall measurement attaches a sign to it.
Why Metals Give Such Tiny Voltages
The Hall voltage is inversely proportional to carrier density, and metals have an enormous one. Copper carries roughly 8.5 × 1028 free electrons per cubic metre, six or seven orders of magnitude above a typical doped semiconductor. Its Hall coefficient is around −5.5 × 10−11 m³/C.
Run the copper preset and the consequence is stark: even at half a tesla with a milliamp through a very thin foil, the Hall voltage lands in the tens of nanovolts. Measuring that demands a lock-in amplifier, field reversal to subtract the offset, and careful attention to thermoelectric voltages that are larger than the signal. The same measurement on a doped semiconductor gives millivolts and works with an ordinary meter.
The practical lesson is that thickness is your main lever. The voltage scales as one over t, so a foil ten times thinner gives ten times the signal. This is exactly why Hall sensors are built as thin films, and it is why sheet carrier density — carriers per unit area rather than per unit volume — is the natural quantity for a two-dimensional sample where the thickness is ill-defined.
Where The Simple Formula Stops Being Right
Every expression on this page assumes a single type of carrier, a scattering time that does not depend on energy, and a field low enough that the carriers do not complete a significant fraction of a cyclotron orbit between collisions. Those assumptions cover most teaching and most routine semiconductor characterisation, and they fail in identifiable ways.
When both electrons and holes are mobile in the same sample — an intrinsic or a compensated semiconductor — the two contributions partly cancel, and the measured coefficient is a mobility-weighted combination rather than one over either density. The sign can even flip with temperature as one carrier type overtakes the other. Treating such a result as a simple carrier density gives a number that is not the density of anything.
A more careful treatment introduces a Hall scattering factor, typically between one and two, that corrects for the energy dependence of the relaxation time. It means Hall mobility and drift mobility differ by that factor, so quoting one as the other is a small but real error. At high fields and low temperatures the classical picture fails entirely and the quantum Hall regime takes over, where the transverse resistance quantises into plateaux at values set only by fundamental constants. The cyclotron frequency calculator gives the orbital frequency that decides whether you are near that regime.
Getting The Geometry Right
Three directions matter and they are mutually perpendicular: current along the bar, field through the face, voltage across the width. Thickness in the formula is the dimension the field travels through. Confusing it with the width between the voltage probes is the classic error, and because both are small numbers in millimetres the mistake produces a plausible-looking wrong answer.
The field must also be genuinely perpendicular. A tilt of angle θ from the normal reduces the effective field by cos θ, so a ten-degree misalignment costs about 1.5 per cent and a thirty-degree one costs 13 per cent. If you need to convert between field units before entering a figure, the magnetic field converter handles gauss and tesla, and the solenoid magnetic field calculator or the magnetic field of a wire calculator will tell you what field your own coil actually produces.
Finally, the voltage probes are never perfectly opposite each other. Any offset puts part of the longitudinal voltage into your reading, which is usually far larger than the Hall signal. The standard fix is to reverse the magnetic field and take half the difference of the two readings, because the Hall voltage reverses with the field and the misalignment offset does not.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Using the sample width instead of the thickness — the thickness in the formula is the dimension along the magnetic field. Substituting the probe separation gives a wrong coefficient with no warning sign in the output.
- Discarding the sign of the measured voltage — the sign is the result. Taking the magnitude throws away the n-type or p-type answer, which is the main reason the measurement is done.
- Applying the single-carrier formula to an intrinsic sample — when electrons and holes are both mobile the contributions partly cancel and the result is not the density of either one.
- Confusing Hall mobility with drift mobility — they differ by the Hall scattering factor, which is typically between one and two, so treating them as identical builds in a systematic error.
- Ignoring the probe misalignment offset — in a metal the offset voltage can be a thousand times the Hall signal. Reversing the field and averaging is the standard way to remove it.
Related Free Tools From Arb Digital
For the force on a single moving charge rather than a whole conductor, use the Lorentz force calculator, and for the orbital frequency in a field the cyclotron frequency calculator. The drift velocity calculator covers how fast carriers move along the bar, and the conductivity and resistivity calculator supplies the resistivity that turns a Hall coefficient into a mobility. Use the number density calculator for particle counting in general, the magnetic field converter for gauss and tesla, and the solenoid magnetic field calculator or magnetic field of a wire calculator to work out the field you are applying. Everything Arb Digital publishes sits on the free online tools hub.
Frequently Asked Questions
It is the material property that links the transverse Hall voltage to the current, field and thickness of a sample, defined as the Hall voltage times thickness divided by current times field. For a single carrier type it equals one over the carrier density times the carrier charge, so it has units of cubic metres per coulomb.
Because it carries the sign of the charge carrier. Electrons give a negative coefficient and holes give a positive one, since the magnetic force pushes both to the same edge and therefore builds opposite polarities. That sign is the reason the Hall effect is the standard test for n-type against p-type.
Divide one by the product of the elementary charge and the Hall coefficient. With a coefficient of 1.2 times ten to the minus four cubic metres per coulomb the density is about 5.2 times ten to the twenty-second per cubic metre, which is a typical doped semiconductor.
The dimension of the sample measured along the magnetic field, meaning the direction the field passes through. It is not the width between the voltage probes. Using the wrong one is the most common error in the whole measurement, and it produces a plausible but wrong answer.
Because the voltage is inversely proportional to carrier density and metals have an enormous one. Copper has roughly 8.5 times ten to the twenty-eighth free electrons per cubic metre, so its Hall voltages land in the nanovolt range and need a lock-in amplifier and field reversal to measure.
It is the magnitude of the Hall coefficient divided by the resistivity, expressed in square centimetres per volt-second. It differs from the true drift mobility by the Hall scattering factor, which is usually between one and two, so the two should not be quoted interchangeably.
It is the carrier density multiplied by the thickness, giving carriers per unit area rather than per unit volume. It is the natural quantity for thin films and two-dimensional systems where the thickness is either very small or not well defined.
No. Everything on this page assumes one dominant carrier type. When both are mobile their contributions partly cancel and the measured coefficient becomes a mobility-weighted combination, which can even change sign with temperature as one type overtakes the other.
This tool is provided for educational and study use. It implements the single-carrier, low-field Hall relation and does not account for two-carrier conduction, the Hall scattering factor, anomalous or quantum Hall behaviour, or geometrical corrections for short samples, so treat its output as a physics result rather than a certified measurement.
