Theory of light-enhanced phonon-mediated superconductivity

We investigate the dynamics of a phonon-mediated superconductor driven out of equilibrium. The electronic hopping amplitude is ramped down in time, resulting in an increased electronic density of states. The dynamics of the coupled electron-phonon model is investigated by solving Migdal-Eliashberg equations for the double-time Keldysh Green's functions. The increase of the density of states near the Fermi level leads to an enhancement of superconductivity when the system thermalizes to the new state at the same temperature. We provide a time- and momentum-resolved view on this thermalization process and show that it involves fast processes associated with single-particle scattering and much slower dynamics associated with the superconducting order parameter. The importance of electron-phonon coupling for the rapid enhancement and the efficient thermalization of superconductivity is demonstrated, and the results are compared to a BCS time-dependent mean-field approximation.

Figure 1

THz pump enhances superconductivity. (a) Through a lattice distortion, the electronic hopping amplitude decreases from $J_0$ to $J_f$ (red shaded area). As a consequence, the superconducting order parameter ${\mathrm{\Delta }}_0$ is boosted to a larger value $\mathrm{\Delta }\left(t\right)>{\mathrm{\Delta }}_0$. At longer time scales (blue shaded area) the order parameter approaches its thermal value ${\mathrm{\Delta }}_f$ corresponding to $J_f$ in the presence of efficient electron-phonon (el-ph) coupling. (b) Sketch of equilibrium order parameters ${\mathrm{\Delta }}_0$ and ${\mathrm{\Delta }}_f$ corresponding to $J_0$ and $J_f<J_0$, respectively, leading to a larger critical temperature $T_{c,f}>T_{c,0}$.

Figure 2

Light-enhanced superconductivity. Dynamics during and after $\tau =100$ fs (dark colors) and $\tau =3$ fs (light colors) ramps for different initial equilibrium temperatures ($T_{c,0}\approx 135$ K). Solid (dashed) lines show the results of the el-ph (BCS) model and arrows indicate the final thermal equilibrium values ${\mathrm{\Delta }}_f$. Dark (light) gray shaded rectangles indicate the ramp durations.

Figure 3

Initial state and ramp duration dependence. (a) Order-parameter change during the 100-fs ramp, ${\mathrm{\Delta }}_{\text{ramp}}-{\mathrm{\Delta }}_0$, relative to ${\mathrm{\Delta }}_f-{\mathrm{\Delta }}_0$ (symbols). Here ${\mathrm{\Delta }}_{\text{ramp}}$ denotes the order parameter at the end of the ramp, i.e., at the time $t=\tau =100\mathrm{fs}$. The change scales almost linearly with the initial value ${\mathrm{\Delta }}_0$ (dashed line). (b) Dependence of final steady-state value on ramp duration $\tau$ within BCS theory for different temperatures. Arrows show the data points for 100-fs ramps corresponding to Fig. 2. The dashed line indicates $\tau {\mathrm{\Delta }}_0=\hslash$.

Figure 4

Thermalization at long times via el-ph coupling. (a) Deviations of the order parameter from respective final thermal values for various temperatures and full el-ph coupling parameter set (i) on a logarithmic scale. The approach to the final thermal value is well described by exponential decays (red lines). (b) Same as in (a) but for reduced el-ph coupling parameter set (ii). The slope in the exponential decays is indeed smaller by a factor of 0.8 as expected from the ratios of the electron-phonon couplings. Hence thermalization takes longer for smaller el-ph coupling.

Figure 5

Time- and momentum-resolved dynamics. Snapshots of the momentum-resolved deviations from final thermal equilibrium values of the normal and anomalous occupations at times as indicated above, shown in 1/8 Brillouin zone each (separated by black lines) for a case of small ${\mathrm{\Delta }}_0$ (a–c) and intermediate ${\mathrm{\Delta }}_0$ (g–i). Effective Bogoliubov dispersions $E_{\mathit{k}}\left(t\right)$ at the corresponding times along a $k_x=k_y$ momentum cut, compared to final thermal dispersions $E_{\mathit{k},f}$ (dashed curves) for the small (d–f) and large (j–l) ${\mathrm{\Delta }}_0$.

Figure 6

Cuts of momentum distributions. Momentum distributions $n_k$ and $f_k$ along a diagonal cut $k_x=k_y$ for the same data as in Fig. 5 for $T=133$ K [panels (a) and (b)] and $T=130$ K [(c) and (d)], at times as indicated above. The black curves are instantaneous nonequilibrium data; the red lines are final thermal distributions. Blue lines show thermal reference data for a lower temperature of 46 K.

Figure 7

Importance of acoustic phonons for thermalization. (a) Time evolution of the order parameter in the presence of a narrow optic phonon mode (red curve) and additionally a broad acoustic phonon branch (blue curve). Inset: Eliashberg functions for the respective cases shifted for clarity. (b) Momentum-resolved deviations from final thermal values of normal and anomalous occupations at 392 fs in the presence of the narrow optic phonon mode (without acoustic phonons), shown in 1/8 of the Brillouin zone each separated by the black line. The relaxation processes are suppressed in a momentum window around the Fermi surface, which is determined by an energy window $E_{\mathit{k}}\left(\text{392}\text{fs}\right)<{\mathrm{\Omega }}_{\text{opt}}/2$ (dashed curves).

Figure 8

Intermediate time behavior. (a) Time evolution of the order parameter for ramp duration $\tau =100$ fs (gray shaded area). An approximately linear behavior is found for intermediate times (red shaded area). Here the temperature is varied as indicated; data are for the “$1.0g^2$” el-ph coupling parameters. (b) Rates of change $\alpha$, obtained from fits to $\mathrm{\Delta }\left(t\right)$ at intermediate times, scaled by remaining deviation ${\mathrm{\Delta }}_f-\mathrm{\Delta }\left(\text{161}\text{fs}\right)$, vs instantaneous order parameter. Colored arrows point to the parameter sets at $T=133$ K (magenta) and 128 K (orange). The black line indicates the equality between the scaled rate and instantaneous order parameter.