Dynamics of Cooper pair formation in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8+\delta }$

We report ultrafast measurements of the transient terahertz conductivity in the high-$T_C$ superconductor ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8+\delta }$. After depletion of the superconducting condensate by optical excitation, the ensuing dynamics is tracked with high sensitivity at various excitation densities and temperatures. We measure low-energy spectra and shapes of decay that demonstrate a bimolecular kinetics of condensate formation on a picosecond time scale and yield a measure of quasiparticle interactions hidden in linear spectroscopy.

Figure 1

(Color online) Equilibrium THz conductivity of Bi-2212, showing (a) the real part ${\sigma }_1\left(\omega \right)$ and (b) the imaginary part ${\sigma }_2\left(\omega \right)$ of conductivity at the temperatures indicated. Solid lines are fits using a two-fluid model (see text). In the normal state $(T=95\mathrm{K})$, this yields a spectral weight $({\hslash }^2∕{ϵ}_0){\rho }_{QP}=0.75{\mathrm{eV}}^2$ and a Drude scattering rate $1∕\tau =3\mathrm{THz}$ similar to previous work. (Ref. 28). At $T=6\mathrm{K}$, the condensate spectral weight is $({\hslash }^2∕{ϵ}_0){\rho }_S=0.5{\mathrm{eV}}^2$. A residual quasiparticle weight of ${\rho }_{QP}\left(6\mathrm{K}\right)∕{\rho }_{QP}\left(95\mathrm{K}\right)\approx 34\%$ remains with a reduced scattering rate ${\tau }^{-1}=1.20\mathrm{THz}$. Dashed line: the imaginary part corresponding to the residual $6\text{−}\mathrm{K}$ quasiparticle spectral weight. (c) The temperature dependence of superfluid density ${\rho }_S$ (open circles) and initial photoinduced spectral weight change $\Delta {\rho }_S$ (solid dots) obtained from two-fluid fits to equilibrium and transient THz spectra, respectively.

Figure 2

(Color online) Induced THz conductivity changes, for $T=6\mathrm{K}$ and $I_0\equiv 0.7\mu \mathrm{J}∕{\mathrm{cm}}^2$ excitation fluence. (a),(b) Measured changes in the real part of conductivity $\Delta {\sigma }_1$ and the imaginary part $\Delta {\sigma }_2$ (circles) at indicated time delays $t$ after excitation. Solid lines: a two-fluid model, corresponding to momentary depletion of 16% $\left(1.3\mathrm{ps}\right)$, 8% $\left(10\mathrm{ps}\right)$, 3.2% $\left(35\mathrm{ps}\right)$, and 2.7% $\left(50\mathrm{ps}\right)$ of the low-temperature condensate spectral weight and Drude widths, respectively, of $1∕\tau =1.35$, 1.26, 1.21, and $1.20\mathrm{THz}$. (c) The transient conductivity changes at center probe energy $5.5\mathrm{meV}$ (averaged over a $1.6\text{−}\mathrm{meV}$ interval) vs pump-probe time delay.

Figure 3

(Color online) Bimolecular temporal kinetics of condensate recovery at $T=6\mathrm{K}$. The identical transient conductivity $\Delta {\sigma }_2\left(t\right)$ is plotted (a) on a logarithmic scale and (b) as the reciprocal $1∕\Delta {\sigma }_2$. Data are averaged over a $1.6\text{−}\mathrm{meV}$ interval at center probe energy $5.5\mathrm{meV}$, for excitation fluence $I=I_0$. Equivalent dynamics is observed in the real part $\Delta {\sigma }_1$ (not shown). The solid lines are guides to eye; open and closed circles denote scans taken on different days.

Figure 4

(Color online) Density and temperature dependence of conductivity dynamics, displayed as $1∕\Delta {\sigma }_2\left(t\right)$ for the $5.5\text{−}\mathrm{meV}$ probe energy. (a) Decays for different excitation intensities $I$ as indicated, at $T=6\mathrm{K}$ and with $I_0$ defined as in Fig. 2. Solid line: a guide to the eye indicating bimolecular decay. (b) Kinetics for different temperatures as indicated, at $I=I_0$. Solid lines are fits using the model described in the text, with a constant bimolecular coefficient and $2R{n}_T=0,0.02,0.06,0.12,0.26\mathrm{THz}$ (lowest to highest temperature). For comparison, single-exponential kinetics is shown as the least square fits to the data over its entire range at $25\mathrm{K}$ (long-dashed line, decay time $\tau =7.9\mathrm{ps}$) and $55\mathrm{K}$ (short-dashed line, $\tau =3.1\mathrm{ps}$).