Electrical Mobility Calculator
Electrical mobility tells you how fast a charged particle accelerates through a medium when an electric field is applied. Enter the electric field strength and drift velocity to solve for mobility, or enter mobility and temperature to find the diffusion constant via the Einstein-Smoluchowski relation. The calculator works for electrons in metals, charge carriers in semiconductors, and ions in solution. All results update as you type.
What is electrical mobility?
Electrical mobility (symbol mu, unit m2/(V s)) is the ratio of a charged particle's drift velocity to the electric field that drives it: mu = v_d / E. A particle with higher mobility reaches a greater speed for the same applied field. Mobility depends on the carrier's charge, its mass or size, and how often it collides with the surrounding medium. In metals and semiconductors these collisions are with lattice vibrations and defects; in solution they are with solvent molecules. H+ and OH- ions in water show anomalously high mobility because charge hops along hydrogen-bond chains (the Grotthuss mechanism) rather than the ions physically diffusing through the liquid.
The Einstein-Smoluchowski relation
Albert Einstein and Marian Smoluchowski independently showed in 1905 that drift and diffusion are two faces of the same thermal agitation. Their relation is D = mu k_B T / q, where D is the diffusion constant (m2/s), k_B is the Boltzmann constant (1.381e-23 J/K), T is temperature in kelvin, and q is the carrier charge. The ratio k_B T / q is called the thermal voltage (symbol V_T), which equals about 25.7 mV at 298 K for a singly charged carrier. Because of this link, measuring mobility in an electric field is often the easiest route to finding the diffusion constant of an ion, which is otherwise hard to measure directly. The relation holds in thermal equilibrium; at very high fields, or in semiconductors with inter-valley scattering, it can break down.
Drift velocity and current
Drift velocity v_d = mu E is typically tiny compared with the thermal (random) velocity of carriers. An electron in copper under a 1 V/m field drifts at only about 4e-3 m/s, while its thermal velocity is around 1.57e6 m/s. The current appears because millions of carriers all drift in the same direction at once, so even a small drift velocity carries significant charge per second. In semiconductors the picture is the same: a higher mobility means a larger current for a given field, which is why high-mobility materials such as gallium arsenide are prized for fast transistors and radio-frequency devices.
Applications of electrical mobility
Electrical mobility underpins a wide range of technologies. In semiconductor design, electron and hole mobilities determine the speed and power consumption of transistors; strain engineering and III-V compounds are used to push mobility higher. In electrolyte solutions, ion mobility controls the conductivity of batteries, fuel cells, and biological membranes. In air-quality engineering, electrostatic precipitators exploit the mobility of aerosol particles to remove them from exhaust gas. Differential mobility analyzers use an electric field to sort aerosol particles by size, and the same principle is used in ion-mobility mass spectrometry to separate molecules by shape and charge.
Electrical mobility of common charge carriers
| Carrier | Medium | Mobility (m²/(V·s)) |
|---|---|---|
| H+ (proton) | Water | 3.623e-7 |
| OH- | Water | 2.064e-7 |
| K+ | Water | 7.62e-8 |
| Na+ | Water | 5.19e-8 |
| Cl- | Water | 7.92e-8 |
| SO4 2- | Water | 8.29e-8 |
| Li+ | Water | 4.01e-8 |
| Ca2+ | Water | 6.17e-8 |
| Electrons (e-) | Silicon (300 K) | 1.35e-1 |
| Holes (h+) | Silicon (300 K) | 4.80e-2 |
| Electrons (e-) | Germanium (300 K) | 3.90e-1 |
| Holes (h+) | Germanium (300 K) | 1.90e-1 |
Approximate values at 298 K (25 C) and 1 atm. Ion values are in aqueous solution. Semiconductor values are for pure (intrinsic) material at 300 K.
Frequently asked questions
What is the unit of electrical mobility?
The SI unit is m2/(V s), which can also be written m2 V-1 s-1. In older literature you may see cm2/(V s), which is 1e-4 m2/(V s). For ions in solution the values are often quoted in units of 1e-8 m2/(V s).
How does temperature affect electrical mobility?
The answer depends on the medium. In metals and semiconductors, higher temperature means more lattice vibrations (phonon scattering), which reduces mobility - roughly as T^(-3/2) in the phonon-scattering regime. In liquids, higher temperature lowers viscosity, so ions move more easily and mobility increases. The diffusion constant always increases with temperature because D = mu k_B T / q, even if mu itself decreases.
What is the thermal voltage and why does it matter?
The thermal voltage V_T = k_B T / q equals about 25.7 mV at 298 K (room temperature) for a singly charged carrier. It appears in the Einstein relation (D = mu V_T) and in the diode equation. It sets the scale at which thermal energy competes with electrical energy: fields that produce much less than V_T per mean-free-path barely affect carrier statistics, while fields much larger than that push carriers far from equilibrium.
Why do H+ and OH- ions have such unusually high mobility in water?
In most ions, the particle itself has to diffuse through the liquid, limited by its size and the solvent viscosity. For H+ and OH-, charge is transferred by the Grotthuss mechanism: a proton hops along a chain of hydrogen bonds from one water molecule to the next without the original ion moving far. This makes the effective mobility roughly five to ten times higher than comparably sized ions like Na+ or K+.
What is the difference between electrical mobility and Hall mobility?
Standard (drift) mobility is measured along the electric field direction. Hall mobility is measured from the Hall coefficient in a transverse magnetic field and equals |R_H| sigma, where R_H is the Hall coefficient and sigma is the conductivity. The two agree in the simplest scattering models but diverge when the scattering mechanism depends on carrier energy, so comparing them gives information about the dominant scattering process in a semiconductor.
Can I use this calculator for semiconductor electrons and holes?
Yes. Enter the tabulated mobility for the material (for example 0.135 m2/(V s) for electrons in silicon at 300 K) and the applied electric field to get the drift velocity. For the diffusion constant, use diffusion mode with the same mobility and T = 300 K, and set the carrier charge to 1.602e-19 C (one elementary charge) for electrons or holes. The reference table above lists representative values.