Controlling competing orders via nonequilibrium acoustic phonons: Emergence of anisotropic effective electronic temperature

Ultrafast perturbations offer a unique tool to manipulate correlated systems due to their ability to promote transient behaviors with no equilibrium counterpart. A widely employed strategy is the excitation of coherent optical phonons, as they can cause significant changes in the electronic structure and interactions on short time scales. One of the issues, however, is the inevitable heating that accompanies these resonant excitations. Here, we explore a promising alternative route: the nonequilibrium excitation of acoustic phonons, which, due to their low excitation energies, generally lead to less heating. We demonstrate that driving acoustic phonons leads to the remarkable phenomenon of a momentum-dependent effective temperature, by which electronic states at different regions of the Fermi surface are subject to distinct local temperatures. Such an anisotropic effective electronic temperature can have a profound effect on the delicate balance between competing ordered states in unconventional superconductors, opening a so far unexplored avenue to control correlated phases.

Figure 1

Momentum-dependent effective electronic temperature generated by nonequilibrium acoustic phonons. (a) The lower panel illustrates the instantaneous distortions on a square lattice caused by the excitation of an acoustic phonon mode. The solid (semitransparent) lines and symbols refer to the undistorted (distorted) lattice. Scattering by nonequilibrium acoustic phonons promotes a momentum-dependent redistribution of electronic quasiparticles, which is translated as a steady-state effective temperature profile that varies along the Fermi surface, depicted in the upper panel (blue and red represent local temperatures that are colder and hotter than the average, respectively). (b) Anisotropic sound velocity $v_{s,\lambda }$ of the two in-plane acoustic phonon modes as a function of the propagation direction $\phi$. The polarization directions along three high-symmetry directions are also shown. (c) Electron-phonon matrix element $M_{q,-}$ for the low-energy phonon mode as a function of the propagation direction $\phi$. In this and in the next figures, we set the ratio $(C_{11}-C_{12})/\left(2C_{66}\right)=1/4$, such that $v_{s,-}(\pi /4)/v_{s,-}\left(0\right)=1/2$.

Figure 2

Anisotropic redistribution of electronic state and nonuniform heating of the Fermi surface states. (a) Due to the geometric constraint imposed by momentum conservation, Eq. (7), the electronic states capable of absorbing acoustic phonons are not uniformly distributed around the Fermi surface (black dashed line). States near the red thick (blue thin) regions have a larger (smaller) phase space for phonon scattering. The inset shows the density of electronic states $N\left({\theta }_F\right)$ available for the scattering of an electron at a Fermi surface angle ${\theta }_F$ by a phonon of energy ${\omega }_0$, for different values of the dimensionless parameter $\kappa \equiv \left(\frac{{\omega }_0}{2{\varepsilon }_F}\right)/\left(\frac{v_s\left(0\right)}{v_F}\right)$. (b) Momentum-dependent temperature profile $\delta T\left({\theta }_F\right)$, normalized by the average heating $〈\delta T〉$, for different values of $\kappa$. The phonon nonequilibrium distribution $n_B$ is depicted in the inset. We set the ratio ${\mathrm{\Omega }}_0\equiv {\omega }_0/2T=5.8$. Due to the scattering of electrons by nonequilibrium phonons, certain regions of the Fermi surface are locally hotter (red shade) or colder (blue shade) than the average. The typical momenta associated with the hot/cold regions change as a function of $\kappa$. (c) Color plot of the anisotropy of the nonuniform temperature, as defined by the logarithm of the ratio $\delta T(\pi /4)/\delta T\left(0\right)$, as a function of the two dimensionless parameters $\kappa$ and ${\mathrm{\Omega }}_0$. Blue (yellow) denotes dominant heating near ${\theta }_F=0,\pi /2$ (${\theta }_F=\pi /4$).

Figure 3

Nonequilibrium steady-state phase diagram for competing superconductivity (SC) and spin-density wave (SDW). The steady-state nonequilibrium phase diagram for the two-band model of the iron pnictides is shown in (a), and the highlighted regions are zoomed in (b). It contains phases with SDW (cyan), SC (yellow), and coexistent SDW+SC (orange) order. The inset shows the superimposed circular holelike and elliptical electronlike Fermi pockets, with the latter displaced by the SDW ordering vector $\mathbf{Q}$. The crossing points are the hot spots ${\mathbf{k}}_{\text{hs}}$, which govern the SDW instability; their positions are controlled by the parameter ${\delta }_{\mu }$, which also determines the carrier concentration. The nonequilibrium phase diagram was obtained for the anisotropic effective temperature profile $\delta T\left({\theta }_F\right)$ shown in Fig. 2 (here we set $\kappa =0.27$) and illustrated in the inset (red and blue denote regions hotter and colder than the average heating $〈\delta T〉$, respectively). For comparison, we also show the equilibrium phase diagram (solid lines) shifted by the average heating $〈\delta T〉/T_M\approx 0.1$. Here, we set ${\delta }_m/T_M=0.9$. All quantities are expressed in terms of the equilibrium SDW transition temperature $T_M$ for ${\delta }_{\mu }=0$.

Figure 4

The scattering density as a function of the phonon direction $\phi$, as given by Eq. (B4).

Figure 5

Illustration of the contour integration that allows one to extract the relations used to determine the out of equilibrium coefficients in Eq. (C5a). The gray area contains the features of the unknown function $f$ and the turquoise area corresponds to the degenerate pole, which might only be resolved by causality. As a consequence, the difference between retarded and advanced components (as typically appearing from the Keldysh Green's function) is given by the residue of single poles or multiple poles.