The electrical mobility calculator above computes μ, the constant that links how fast charge carriers drift to how hard the electric field is pushing them. It is the quantity that separates what a material does from what a particular circuit is doing to it, and it is the number semiconductor physics is largely built on.
Arb Digital publishes free physics calculators that name the relationship between adjacent quantities rather than duplicating them. The drift velocity calculator returns a velocity, which depends on the current you happen to be passing. This page takes that velocity and divides out the field driving it, leaving a property of the material and its temperature. Same physics, one step further along.
What This Electrical Mobility Calculator Does
It works both ways round. Given a drift velocity and the electric field that produced it, mobility is simply the ratio. Given a bulk conductivity and a carrier density, mobility comes out of the conductivity relation instead. Both are shown in SI units of square metres per volt-second and in the square centimetres per volt-second that semiconductor literature almost always uses.
Alongside the mobility it returns the conductivity and resistivity that follow from your carrier density, and the momentum relaxation time — the average interval between the collisions that limit the drift. That last figure is where mobility stops being a fitted number and starts being a statement about what is happening inside the material.
How to Use It
- Choose which route you have data for. A measured velocity and field, or a measured conductivity and a known carrier density.
- Enter the two quantities for that route. Field is volts per metre, so remember that a volt across a millimetre is a thousand volts per metre.
- Enter the carrier density. It is needed for the conductivity route, and it lets the tool derive conductivity in the velocity route.
- Set the charge per carrier and the effective mass ratio. Leave both at 1 unless you are working with multiply charged ions or a semiconductor band.
- Compare the result in cm²/(V·s), because that is the unit almost every published mobility figure is quoted in.
The Formula: Two Routes to the Same Quantity
The definition is vd = μE, so μ = vd ÷ E. Mobility has SI units of m² V−1 s−1, which look strange until you read them as "metres per second, per volt per metre".
The conductivity route starts from the microscopic form of Ohm's law. Current density is J = nqvd, and substituting vd = μE gives J = nqμE. Comparing that with J = σE identifies the conductivity as σ = nqμ, so μ = σ ÷ (nq). Section 9.2 of OpenStax University Physics Volume 2, on the model of conduction in metals, develops this chain from the drift picture.
The Drude relation then connects mobility to the microscopic collision rate: μ = qτ ÷ m*, where τ is the momentum relaxation time and m* is the effective mass. Rearranged, τ = μm* ÷ q, which is what the fourth grid item shows. The elementary charge used throughout is the exact value 1.602176634 × 10−19 C published by NIST as the CODATA value for the elementary charge.
Work the defaults through by hand. A drift velocity of 135 m/s in a field of 1,000 V/m gives μ = 135 ÷ 1,000 = 0.135 m²/(V·s), which is 1,350 cm²/(V·s) since one m²/(V·s) is 10,000 cm²/(V·s). With n = 1022 m−3 and q = e, the conductivity is σ = 1022 × 1.6022 × 10−19 × 0.135 = 216.3 S/m, so the resistivity is 4.62 × 10−3 Ω·m. With an effective mass of one electron mass, τ = 0.135 × 9.1094 × 10−31 ÷ 1.6022 × 10−19 = 7.68 × 10−13 s, which is under a picosecond.
Why Mobility and Carrier Density Are Independent Levers
Conductivity is the product of three things: carrier density, charge and mobility. Two materials can have the same conductivity for completely different reasons, and knowing only the conductivity cannot tell you which.
A metal has an enormous carrier density and a fairly modest mobility. A high-purity semiconductor has a mobility that can be an order of magnitude higher than a metal's, and a carrier density many orders of magnitude lower, so it ends up far less conductive despite the better mobility. Doping raises the carrier density, which raises conductivity, but the added impurities also scatter carriers, which lowers mobility. The two levers work against each other, and that trade-off is central to semiconductor device design.
This is why measuring conductivity alone is not enough to characterise a material. A separate measurement that fixes the carrier density — classically a Hall measurement — is needed before the conductivity can be split into its two factors.
What Actually Limits Mobility
Mobility is set by how frequently carriers are scattered, and there are two dominant mechanisms with opposite temperature behaviour.
Lattice or phonon scattering comes from thermal vibration of the atoms. It gets worse as temperature rises, because there is more vibration to scatter off, so this mechanism reduces mobility at high temperature. Ionised impurity scattering comes from the charged dopant sites, and it gets weaker as temperature rises, because faster-moving carriers are deflected less by a given Coulomb centre.
Because the two work in opposite directions, mobility in a doped semiconductor typically peaks at an intermediate temperature: impurity-limited and rising below the peak, phonon-limited and falling above it. That is one reason a single published mobility figure means nothing without the temperature and doping level it was measured at.
Where the Linear Relation Stops Holding
Everything above assumes drift velocity is proportional to field, which is what makes mobility a constant. At high fields it is not. Carriers gain enough energy between collisions to excite optical phonons efficiently, which dumps that energy back into the lattice, and the drift velocity levels off at a saturation value instead of continuing to rise.
Once that happens, mobility defined as v ÷ E is no longer a constant — it falls as the field rises, because the numerator has stopped growing. This matters enormously in small devices, where even a modest voltage produces a very large field across a short channel, and it is one of the physical limits on transistor performance. If your field is high, treat the mobility this page returns as an effective value at that field rather than as the material's low-field mobility.
Where This Sits Next to the Other Conduction Tools
The drift velocity calculator is the direct upstream tool: it returns the velocity from a current and a conductor geometry, and this page turns that velocity into a material property by dividing out the field. The conductivity and resistivity calculator is the downstream one, handling the bulk quantity that mobility and carrier density together produce.
To get the field in the first place, the electric field calculator and the voltage drop calculator cover the potential gradient along a conductor, and the Ohm's law calculator and the wire resistance calculator handle the circuit side. None of this applies to a free electron in a vacuum, which has no lattice to scatter from — that is the electron speed calculator, and it is a completely different physical situation.
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
- Mixing m²/(V·s) with cm²/(V·s) — the factor is 10,000, and published figures are nearly always in the centimetre unit.
- Using dopant concentration as carrier density — not every dopant atom is ionised at every temperature, and compensation reduces the net figure further.
- Quoting a mobility without a temperature — scattering is strongly temperature-dependent, so a bare number is not reproducible.
- Applying low-field mobility at high field — drift velocity saturates, so mobility defined as velocity over field falls once saturation sets in.
- Assuming one mobility for a material — electrons and holes have different mobilities in the same semiconductor, often by a factor of two or three.
Related Free Tools From Arb Digital
Pair this with the drift velocity calculator for the velocity it consumes, and the conductivity and resistivity calculator for the bulk property it produces. The electric field calculator and the voltage drop calculator give you the field, while the Ohm's law calculator and the wire resistance calculator cover the circuit. For carrier counting use the excess electrons calculator, and for free electrons in vacuum see the electron speed calculator. Everything Arb Digital publishes is on the free online tools hub.
Frequently Asked Questions
In SI they are square metres per volt-second, m² V−1 s−1, which follows directly from velocity divided by field. Semiconductor literature almost universally uses square centimetres per volt-second instead, and one SI unit equals ten thousand of those, so the conversion is a factor of 10,000 and worth checking every time.
Drift velocity depends on the field you happen to be applying, so it changes from one circuit to another. Mobility divides that field out, leaving a property of the material at a given temperature. The same piece of silicon has one mobility but as many drift velocities as there are conditions you put it in.
Because they move through different bands with different effective masses and different scattering behaviour. In silicon, electron mobility is roughly three times hole mobility at room temperature. That asymmetry is why n-channel and p-channel devices of the same size do not have the same drive strength.
Both, depending on which scattering mechanism dominates. Phonon scattering worsens with temperature and pushes mobility down; ionised impurity scattering weakens with temperature and pushes it up. In a doped semiconductor the two combine to give a peak at an intermediate temperature, with impurity-limited behaviour below it and phonon-limited above.
It is the average momentum relaxation time between scattering events, obtained from the Drude relation as mobility multiplied by effective mass divided by charge. It is typically a fraction of a picosecond, and it is the microscopic quantity that mobility is a macroscopic expression of. It depends on the effective mass you enter.
Because the ionised dopant atoms are charged scattering centres. Adding them raises the carrier density, which raises conductivity, but each one also deflects passing carriers and shortens the mean time between collisions. Conductivity still rises overall, but not in proportion to the doping, because mobility is falling at the same time.
No. Resistivity fixes the product of carrier density and mobility, not either factor separately. You need an independent measurement of carrier density — classically a Hall measurement — before the two can be separated. That is exactly why this page asks for a carrier density in the conductivity route.
The same definition applies, and ionic mobility is defined the same way as drift velocity per unit field. The numbers are far smaller than in solids because ions are heavy and the surrounding solvent drags on them, and the charge per carrier is often greater than one, which is why this page takes the charge state as an input.
This tool is provided for educational use. It applies the low-field Drude relations with a single carrier type and does not model velocity saturation, two-carrier conduction, degeneracy or band structure.
