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Einstein Relation Calculator (Mobility ↔ Diffusivity)

Convert carrier or ion mobility to diffusion coefficient (and back) with the Einstein relation D = μkT/q. Includes thermal voltage. For semiconductors, electrolytes and ions.

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    Einstein Relation (Nernst–Einstein) formula

    D = μ kBTq = μ · VT
    SymbolQuantitySI unitMeaning
    DDiffusion coefficientm²/sDiffusion coefficient of the charge carrier or ion
    μMobilitym²/(V·s)Electrical mobility (drift velocity per unit field)
    TTemperatureKAbsolute temperature
    zCharge number—Number of elementary charges carried (1 for electrons and holes)

    What is Einstein Relation (Nernst–Einstein)?

    Einstein Relation (Nernst–Einstein): For particles in thermal equilibrium, the diffusion coefficient is proportional to the mobility, with the thermal voltage kT/q as the constant. — Albert Einstein, 1905

    The Einstein relation is a fluctuation–dissipation result. The same microscopic collisions that limit how fast a charged particle drifts in an electric field (mobility μ) also set how fast it spreads out by random motion (diffusivity D). The ratio D/μ equals the thermal voltage VT = kT/q, which is about 25.85 mV at 300 K.

    In semiconductor device physics it connects the drift and diffusion terms of the current equations. In electrochemistry it is called the Nernst–Einstein relation and links ionic conductivity to diffusion.

    How to use this calculator

    1. Choose the quantity you want to find in the Calculate menu. It is highlighted as the result field.
    2. Enter the known values. Scientific notation such as 1.6e-11 is accepted.
    3. Select a unit for each value. Units are converted to SI automatically.
    4. The answer updates as you type. Open Show calculation steps to see the working, or use Copy result to share a link with your inputs.

    Worked example

    Electrons in lightly doped silicon have μ ≈ 1400 cm²/(V·s) at 300 K.

    1. VT = kT/q = 1.380649 × 10⁻²³ × 300 / 1.602 × 10⁻¹⁹ = 0.02585 V.
    2. D = 1400 × 0.02585 = 36.2 cm²/s.

    Frequently asked questions

    Does the Einstein relation hold in heavily doped semiconductors?

    Strictly, it applies to non-degenerate carriers following Boltzmann statistics. In degenerate (very heavily doped) material, D/μ is larger than kT/q and a Fermi–Dirac correction is needed.

    How do I use it for ions in solution?

    Set z to the ion's charge number (e.g. 2 for Ca²⁺) and enter the ionic mobility. The relation holds well in dilute solutions.