Phonons Do Not Always Help Superconductivity: When Retardation Fails — Epoche B2
Re-evaluating Electron-Phonon Coupling: When Retardation Fails Every introductory account of the Bardeen-Cooper-Schrieffer (BCS [1] ) theory tells the same reassuring story [2] . An electron races through the crystal, dents the lattice, and the sluggish positive ions drift towards the dent; the resulting puddle of excess positive charge then draws in a second electron. The lattice has, in effect, glued two electrons into a Cooper pair , and it is these pairs that carry the resistanceless supercurrent. The moral usually drawn is that phonons — the quantised lattice vibrations — are unambiguously the friends of superconductivity: crank up the electron-phonon coupling and the critical temperature $T_c$ should climb. This essay re-examines that moral. The phonon-mediated interaction is not universally attractive , and there are concrete, physically realisable conditions under which the electrons' bare Coulomb repulsion wins, so that lattice coupling produces no pairing at all. To see why, we must build up four things the tidy story skips: what pairing really demands, why the two competing interactions live on different timescales, how that difference is compressed into a single number, and what ultimately fixes the sign of the net interaction. What pairing actually requires In 1956 Cooper showed that a filled Fermi sea — electrons stacked one per state up to the Fermi energy $E_F$ — is unstable to pairing as soon as the electrons near that surface feel any net attraction, however feeble. The size of the effect is set by two quantities: $N(0)$, the electronic density of states per spin at the Fermi level, and the net interaction $V_{\text{net}}$ between two such electrons. In weak coupling the transition temperature is $$ k_B T_c \simeq 1.13\,\hbar\omega_D\,\exp\!\left(-\frac{1}{g}\right), \qquad g \equiv N(0)\,V_{\text{net}}. $$ Here $k_B$ is Boltzmann's constant, $\hbar\omega_D$ is the Debye energy (the highest phonon energy, which also sets the thin shell around $E_F$ in which pairs form), and $g$ is the dimensionless net coupling. The physics lives entirely in that exponent. Because $g$ sits under a $-1/g$, $T_c$ is not governed by the raw magnitude of the attraction but by whether $g$ is positive at all and, if so, how large. A net interaction that is even marginally repulsive ($V_{\text{net}}\lt 0$ in the convention where attractive is positive) admits no pairing solution: $T_c=0$. The entire question of superconductivity therefore collapses to a single sign — is $g\gt 0$? — and everything below is about what determines it. Two interactions on two timescales Two contributions make up $V_{\text{net}}$: the bare Coulomb repulsion between the electrons, and the phonon-mediated attraction. The naive story treats them as if they act at the same instant and simply adds them, $V_{\text{eff}} = V_{\text{Coulomb}} + V_{\text{phonon}}$, concluding that the pair binds whenever the attraction outweighs the repulsion. That bookkeeping is wrong, because the two interactions act on wildly different timescales. The Coulomb repulsion is fast and strong: two electrons feel each other's (screened) charge over the electronic timescale $\hbar/E_F \sim 10^{-16}\,\text{s}$, and it operates across the whole occupied band, an energy range of order $E_F \sim$ a few eV. The phonon attraction is slow. The ions are thousands of times heavier than the electrons ($M/m \sim 10^{4}$), so the lattice can only respond over the phonon timescale $1/\omega_D \approx 3\times10^{-14}\,\text{s}$ for $\hbar\omega_D=25\,\text{meV}$ — a full lattice oscillation, $2\pi/\omega_D$, taking some $10^{-13}\,\text{s}$. That is two orders of magnitude slower than the electrons. The retardation is the whole point: in the time the ions take to gather, an electron travelling at $v_F\sim2\times10^{6}\,\text{m}\,\text{s}^{-1}$ has moved $v_F/\omega_D\sim5\times10^{-8}\,\text{m}$, several hundred ångström, so the second electron is drawn to a spot the first has long left. The screened Coulomb repulsion between them has not been switched off — it never is — but at that separation it is feeble, while the lattice's attraction is still being delivered. Attraction and repulsion are separated in time , and therefore in energy. In frequency space this shows up in the interaction mediated by exchanging a phonon of frequency $\omega_q$, $$ V_{\text{ph}}(\omega) = |g_q|^{2}\,\frac{2\,\omega_q}{\omega_q^{2}-\omega^{2}}, $$ where $\omega$ is the energy transferred between the two electrons and $g_q$ is the electron-phonon matrix element. For slow processes, $|\omega|\lt\omega_q$, the denominator is positive and $V_{\text{ph}}\gt 0$: attractive. For fast processes, $|\omega|\gt\omega_q$, it flips sign and becomes repulsive — it does not merely switch off. So the lattice glue helps only within a window of width $\sim\hbar\omega_D$ about the Fermi surface; outside that window it adds to the repulsion rather than opposing it. The two-square-well model and the Coulomb pseudopotential Morel and Anderson captured this in 1962 with a deliberately crude but honest caricature [3] . Give each interaction a dimensionless strength: $\lambda \approx N(0)V_{\text{ph}}$ for the phonon attraction inside the window, and $\mu \approx N(0)V_c$ for the Coulomb repulsion, which acts everywhere. The pairing kernel, as a function of the single-electron energies $\xi,\xi'$ measured from $E_F$, is then two nested square wells: $$ N(0)\,V = \lambda - \mu \quad (\,|\xi|,\,|\xi'| \lt \hbar\omega_D\,), \qquad N(0)\,V = -\mu \quad (\,\hbar\omega_D \lt |\xi|,\,|\xi'| \lt E_F\,). $$ Inside the narrow phonon window the electrons feel both the phonon attraction and Coulomb repulsion, for a net interaction $\lambda-\mu$; over the vast remaining band, up to $E_F$, only the repulsion $-\mu$ survives (Fig. 1). Now comes the decisive step. When the gap equation is solved with this kernel, the wide-band repulsion does not enter with its bare strength $\mu$. Because a pair can virtuall