AllenDynes

Allen-Dynes refers to a widely used modification of the McMillan equation for estimating the superconducting critical temperature (Tc) of conventional, electron–phonon mediated superconductors. The term originates from the 1975 work of physicists P. B. Allen and R. C. Dynes, who reanalyzed and extended earlier empirical formulas to account more accurately for strong electron–phonon coupling and the detailed shape of the phonon spectrum.

The Allen–Dynes formula expresses Tc in terms of the electron–phonon coupling constant (λ), the Coulomb pseudopotential (μ*), and characteristic phonon frequencies, notably the logarithmic average frequency ω_log. It introduces two multiplicative correction factors, f1 and f2, that adjust the McMillan expression for strong-coupling effects and spectral shape. In common form, Tc is given approximately by:

Tc ≈ (f1 f2) (ω_log / 1.2) exp[ −1.04(1 + λ) / (λ − μ*(1 + 0.62λ)) ].

Typical values for μ* are around 0.1, but depend on material screening.

Applications and limitations

The Allen–Dynes expression is widely used in computational materials science and density functional theory studies to screen and interpret candidate phonon-mediated superconductors. It provides a practical compromise between simplicity and physical realism but remains approximate: it assumes electron–phonon pairing, relies on accurate phonon and electron–phonon calculations, and can be less reliable for very strong-coupling regimes or materials with unconventional pairing mechanisms.