Microscopic description of displacive coherent phonons

We develop a Hamiltonian-based microscopic description of laser pump induced displacive coherent phonons. The theory captures the feedback of the phonon excitation upon the electronic fluid, which is missing in the state-of-the-art phenomenological formulation. We show that this feedback leads to chirping at short timescales, even if the phonon motion is harmonic. At long times, this feedback appears as a finite phase in the oscillatory signal. We apply the theory to ${\mathrm{BaFe}}_2{\mathrm{As}}_2$, explain the origin of the phase in the oscillatory signal reported in recent experiments, and we predict that the system will exhibit redshifted chirping at larger fluence. Our theory also opens the possibility to extract equilibrium information from coherent phonon dynamics.

Figure 1

(a) Sketch of an $A_{1g}$ coherent phonon motion in a two-atom (red balls) unit cell. A macroscopic number of atoms oscillate with identical phase and frequency ${\omega }_0$. Green arrows indicate the instantaneous velocities at two instants. The motion preserves the point group symmetry. (b) The effect of the laser pump is idealized as a temperature quench from a measured base temperature $T_L$ to a high temperature $T_H$ over a short time set to zero, and the subsequent relaxation of temperature over a time-scale ${\tau }_e$. In the theory, $(T_H,{\tau }_e)$ are phenomenological parameters (see text). The temperature and timescales are representative.

Figure 2

Calculations for representative parameter values. Frequency ${\omega }_0/\left(2\pi \right)=5.5$ THz, $X_1/{\omega }_0=0.5,{\tau }_1=0.7$ ps, ${\tau }_2=0.6$ ps, and ${\gamma }^{-1}=5$ ps, and for different strengths of the lattice feedback term $X_2$. $X_2=0$ corresponds to the phenomenological theory [46]. (a) Coherent phonon displacement $u\left(t\right)$, see Eqs. (8) and (9) and the associated text. The inset, a blow-up of the dashed rectangle, shows signature of the finite phase $\varphi$ for different values of $X_2$. (b) The effects of the feedback at different timescales. At short times $\left(t\lesssim {\tau }_2\right)$ a finite $X_2$ leads to chirping. At long times $\left(t\gg {\tau }_2\right)$, it leads to a finite phase $\varphi$, see also inset in (a). ${\tau }_2$ is defined in Eq. (5).

Figure 3

Quantitative description of the $A_{1g}$ coherent phonon [frequency ${\omega }_0/\left(2\pi \right)=5.5$ THz] of ${\mathrm{BaFe}}_2{\mathrm{As}}_2$, and comparison with experiment [31]. (a) Calculated equilibrium expectation value of the weighted electron density ${\langle \widehat{O}\rangle }_T$ for $\lambda =0.25$. (b) Solid (black) line: the $T$ dependence in (a) is transformed into a time dependence using Eq. (2) for representative values of the phenomenological parameters $(T_H,{\tau }_e)$. Base temperature $T_L=140$ K. Dashed (red) line: fit using Eq. (5), and estimate of $(X_1,{\tau }_1)$. (c) Solid lines: temporal variation of x-ray form factor calculated using Eq. (8) at different fluences (FL in $\mathrm{mJ}/{\mathrm{cm}}^2)$. The table gives estimates of $(T_H,{\tau }_e)$ used in the calculation. The fit uses ${\gamma }^{-1}=5$ ps, which is the equilibrium lifetime [56]. Symbols represent data points extracted from Ref. [31].