Electronic and phononic properties of a two-dimensional electron gas coupled to dipolar phonons via small-momentum-transfer scattering

We study the electron and phonon spectral functions as well as superconductivity for a two-dimensional electron gas coupled to dipolar phonons via small-momentum-transfer scattering. The results reported here are obtained through sign-problem-free quantum Monte Carlo simulation. Hence, aside from modeling there are no further approximations. Our results can be relevant to the high-temperature superconductor at the interface between a unit-cell-thick FeSe and ${\mathrm{SrTiO}}_3$. Our results give support to the interpretation that the observed replica band in unit-cell-thick FeSe on ${\mathrm{SrTiO}}_3$ is due to the small-momentum transfer interaction between the FeSe electrons and a branch of polar STO phonons. Moreover, we conclude that while the small-momentum electron-phonon interaction can not induce strong Cooper pairing by itself, it can enhance the Cooper pairing triggered by other pairing interactions.

Figure 1

The electron spectral function calculated for (a) $\lambda =0.3$ and (b) $\lambda =0.5$. The upper (lower) bands are the main (replica) bands, respectively. In order to obtain continuous color images we have fit the DQMC data, including the band dispersion, the linewidth, and peak intensity as a function of momentum with analytic functions. The details of the fitting are given in the Supplemental Material.

Figure 2

(a) The replica to main band intensity ratio $I_{0,R}$ as a function of $\lambda$. For a given momentum the intensity is defined as the peak value of the spectral function. The red curve is a polynomial, $a+b\lambda +c{\lambda }^2$, fit to the data. Here $a=0.003\pm 0.023,b=1.031\pm 0.152,c=0.236\pm 0.175$, which indicates that the linear term is dominant. (b) The ratio between the energy splitting between the replica and the main bands and the weighted average of the phonon energy as a function of $\lambda$. The red curve is a polynomial, $a+b\lambda +c{\lambda }^2$, fit to the data. Here $a=0.992\pm 0.042,b=0.141\pm 0.102,c=0.4671\pm 0.095$. The band intensity and energy splitting are averaged over all the momenta along the cut $k_y=0$.

Figure 3

The phonon spectral functions calculated for (a) $\lambda =0.3$ and (b) $\lambda =0.5$. Like Fig. 1, in order to obtain continuous color images we have fit the DQMC data with analytic functions. The details of the fitting are given in the Supplemental Material. Phonon linewidth and the energy renormalization (redshift) as a function of momentum for (c) $\lambda =0.3$ and (d) $\lambda =0.5$. The black solid lines represent the bare dispersion and the red dashed lines represent the renormalized dispersion. The “error bars” around the renormalized dispersion are the phonon linewidth. The results in this figure are obtained at a nonzero $k_{B}T=4\text{meV}$.

Figure 4

(a) The square of the $s$-wave and $d$-wave superconducting order parameter as a function $\lambda$. In the absence of other interactions the superconducting order parameters are small and extrapolate to zero in the thermodynamic limit (see the Supplemental Material). (b) The square of the $s$- and $d$-wave superconducting order parameters as the function $\lambda$. Here a small $s$-wave pairing field $\mathrm{\Delta }{\sum }_i(c_{i\uparrow }{c}_{i\downarrow }+\text{H.c.})$ is added to mimic the pairing induced by other electronic mechanisms. The above results are obtained in a lattice with linear dimension $L=12$.