= Solution
The <electron-phonon interaction> can produce a retarded attraction. A carrier distorts the ionic lattice; a second carrier can lower its <energy> in that distortion. Integrating out a <phonon> gives a frequency-dependent interaction with an attractive low-frequency region. A schematic Matsubara-frequency contribution is $V_{\rm ph}(\mathbf q,i\omega_n)\propto-|g_{\mathbf q}|^2\omega_{\mathbf q}/(\omega_n^2+\omega_{\mathbf q}^2)$. It must be combined with the screened Coulomb repulsion, not treated as a statement that the bare electron-electron <force> is attractive.
Retardation separates the electronic and <phonon> <energy> scales and reduces the repulsive contribution in the pairing channel to a Coulomb pseudopotential. In the simplest weak-coupling approximation, a positive net dimensionless pairing strength gives
$$
k_BT_c\sim1.14\hbar\omega_D\exp[-1/(\lambda_{\rm ep}-\mu^*)].
$$
The formula assumes a <phonon> cutoff and simplified coupling; strong-coupling calculations need the spectral interaction rather than this one-number replacement. The <phonon> <frequency>'s dependence on ionic <mass> motivates an <isotope effect>, whose actual exponent can be modified by other physics.
An interaction need not be attractive in every <momentum> channel to support pairing. Spin fluctuations or other electronic correlations can favor a sign-changing gap: in the gap equation, repulsive scattering between two regions can contribute constructively when their gap signs differ. Such mechanisms are candidates for <unconventional superconductivity>. Observing a d-wave gap identifies symmetry information but does not, by itself, establish a unique microscopic pairing mechanism in a cuprate.
Back to article page
