Competition between homogeneous and inhomogeneous broadening of orbital transitions in Si:Bi

We present results for the lifetime of the orbital transitions of Bi donors in Si, measured using both frequency-domain and time-domain techniques, allowing us to distinguish between homogeneous and inhomogeneous processes. The proximity of the energy of the optically allowed transitions to the optical phonon energy means that there is an unusually wide variation in the lifetimes and broadening mechanisms for this impurity, from fully homogeneous lifetime-broadened transitions to fully inhomogeneously broadened lines. The relaxation lifetime ($T_1$) of the states ranges from the low 10's to 100's of ps, and we find that there is little extra dephasing (so that $T_1$ is of the order of $T_2/2$) in each case.

Figure 1

Energy level diagram of Bi donors in Si, values from Refs. [14, 15, 16, 20]. The left-hand side shows the phonon density of states (DOS) from Ref. [21] and indicates the position of the transverse optical (TO), longitudinal optical (LO), transverse acoustic (TA), and longitudinal acoustic (LA) phonons. The scale on the right-hand side is the energy difference between the ground state and the excited orbital states.

Figure 2

Pump-probe data measured from the (a) $2p_{\pm }$, (b) $3p_{\pm }$, and (c) $4p_{\pm }$ transitions of the $2\times {10}^{14}{\mathrm{cm}}^{-3}$ Si:Bi sample, measured at 1.6 K with pump energy of 400 nJ, 360 nJ, and 100 nJ per micropulse, respectively. The laser bandwidth used was 0.3% of the central frequency. The red line is the fit to the data using Eq. (10), with the extracted lifetimes shown in the plot.

Figure 3

Fits to the FTIR data. From the top left the lines are $2p_{\pm },3p_0,4p_0,3p_{\pm },5p_0/4p_{\pm },4f_0,5p_{\pm }$, and $6p_{\pm }$. See text for details of the fitting. The experimental absorbance vs frequency is shown with the blue points and the combined Voigt fit with the red lines. The orange and gray lines are the principle, and the purple and green lines, the secondary components to the total fit. The dashed lines are used to separate the $5p_0$ from the $4p_{\pm }$. FTIR data from Ref. [13].

Figure 4

Comparison of time-domain and frequency-domain results. Black circles: $T_2=2/\gamma$ ps from fits to the absorption spectra described in the text; red triangles: the $T_2^{*}=2/\alpha$ from Eq. (9) using data from the fits to the absorption spectrum; blue squares: $2T_1$ from the pump-probe measurements; purple diamonds: $2T_1$ from the theory of Ref. [22]. The $T_2$ value for the $6p_{\pm }$ is off the scale, and this line may be considered completely inhomogeneously broadened. Inset: A log-log plot showing a comparison between the $T_2^{*}$ as a function of $T_2$ calculated either exactly, or approximately using Eq. (5) or Eq. (9), with a constant $T_{\mathrm{inhom}}$. The results from Eq. (9) and the exact inverse FWHM of the Voigt profile are indistinguishable on the scale shown in the figure.