| Morel–Anderson pseudopotential | |
|---|---|
| Name | Morel–Anderson pseudopotential |
| Field | Condensed matter physics |
| Introduced | 1962 |
| Inventor | Pierre Morel and Philip W. Anderson |
| Related | Bardeen–Cooper–Schrieffer theory, Eliashberg theory, Coulomb interaction, Electron–phonon interaction |
Morel–Anderson pseudopotential
The Morel–Anderson pseudopotential is an effective screened Coulomb parameter used to describe the residual repulsion between electrons in the presence of an attractive electron–phonon interaction in superconductors. It quantifies how high-energy electronic processes renormalize the bare Coulomb interaction to a smaller, frequency-dependent effective interaction that enters pairing theories such as Eliashberg theory and extensions of Bardeen–Cooper–Schrieffer theory. The concept is important for predicting critical temperatures and pairing symmetries in conventional superconductors and in computational studies of materials.
The Morel–Anderson pseudopotential was introduced to reconcile the competing roles of the instantaneous Coulomb repulsion and the retarded attractive interaction mediated by phonons in metallic superconductors. In a metal, the bare Coulomb parameter is large, but retardation effects associated with the phonon energy scale and screening by other electrons reduce its effective value for pairing. Early work by Philip W. Anderson and Pierre Morel built on experimental and theoretical foundations provided by Bardeen–Cooper–Schrieffer theory and developments in many-body techniques at institutions such as Bell Labs and research groups around MIT and Princeton University. The pseudopotential serves as a compact phenomenological parameter that preserves stability and continuity between microscopic electronic structure and macroscopic superconducting properties.
Formally, the Morel–Anderson pseudopotential, commonly denoted μ* (mu-star), is defined from the bare Coulomb parameter μ and renormalized by a logarithmic reduction factor involving a characteristic electronic energy (often the Fermi energy E_F) and a characteristic phonon frequency (Debye frequency ω_D or an average phonon frequency). In one frequently used form: μ* = μ / [1 + μ ln(E_F/ω_c)], where μ = N(0) V_C is the dimensionless product of the density of states at the Fermi level N(0) and the screened Coulomb matrix element V_C, and ω_c is a cutoff frequency related to the phonon spectrum. This expression is derived within a renormalization group-like separation of scales and appears in linearized gap equations of Eliashberg theory. The parameter μ* is explicitly energy- and material-dependent and often treated as an adjustable input in phenomenological analyses and first-principles calculations using density functional theory-based approaches.
The derivation proceeds by integrating out electronic degrees of freedom above a phonon cutoff and resumming logarithmic divergences associated with particle–hole excitations. Starting from a microscopic Hamiltonian containing a screened Coulomb term and an electron–phonon coupling term, perturbative diagrammatic methods or functional renormalization yield an effective low-energy interaction. The retardation effect—phonon-mediated attraction acting over time scales set by ω_D—means Coulomb repulsion is reduced when projected onto the low-energy pairing window. The formalism connects with the frequency-dependent pairing kernel in Eliashberg equations and can be cast in the language of Matsubara Green's functions, self-energies, and vertex corrections. Works citing systematic treatments include those founded on Migdal's theorem and extensions that account for vertex corrections in low-carrier or strong-coupling regimes.
In materials where superconductivity is predominantly driven by electron–phonon coupling—such as elemental metals (e.g., Nb, Pb) and conventional compounds—the competition between μ* and the electron–phonon coupling constant λ largely determines the transition temperature T_c via formulas derived from Eliashberg theory or empirical approximations like the McMillan formula and the Allen–Dynes equation. A small μ* enhances the effective pairing strength (λ − μ*), favoring superconductivity and singlet Cooper pair formation as in s-wave superconductivity. Accurate estimation of μ* is therefore crucial for materials design and interpretation of tunneling spectroscopy, isotope effect measurements, and phonon-mediated pairing studies performed at facilities such as national laboratories and university condensed-matter groups.
The standard Morel–Anderson expression relies on approximations: weak to moderate coupling, Migdal's theorem validity, an assumed separation of energy scales (E_F ≫ ω_D), and isotropic Fermi surfaces. In low-carrier-density systems, low-dimensional materials, or strongly correlated compounds (e.g., proximate to Mott insulator physics or unconventional superconductors), μ* may be ill-defined or require extensions. Corrections include frequency-dependent Coulomb kernels, anisotropic pseudopotentials, and treatments incorporating dynamically screened interactions via the random phase approximation or time-dependent density functional theory. Advanced studies sometimes replace a single μ* by a matrix of Coulomb parameters resolved by orbital or momentum channels, connecting with multi-band approaches used for materials like MgB2 and iron-based superconductors.
Practical computation of μ* occurs in workflows coupling density functional theory for electronic bands and phonons (e.g., using Quantum ESPRESSO, VASP, or ABINIT) with Migdal–Eliashberg solvers and many-body perturbation theory codes (e.g., EPW). Methods estimate the screened Coulomb interaction via constrained random-phase approximation or model dielectric functions, producing μ or directly μ*. Applications span predicting T_c for elemental superconductors, evaluating isotope effects, and screening candidate materials in high-throughput searches conducted by research centers and consortia. Reliable use of μ* supports conservative, stability-focused materials design by ensuring theoretical predictions of superconducting properties remain tied to robust, physically motivated approximations and established experimental benchmarks.
Category:Superconductivity Category:Condensed matter physics