Quantitative analysis of electrically detected Ramsey fringes in P-doped Si

This work describes detection of the laser preparation and subsequent coherent manipulation of the quantum states of orbital levels of donors in doped Si, by measuring the voltage drop across an irradiated Si sample. This electrical signal, which arises from thermal ionization of excited orbital states, and which is detected on a millisecond time scale by a voltmeter, leads to much more sensitive detection than can be had using optical methods, but has not before been quantitatively described from first principles. We present here a unified theory which relates the voltage drop across the sample to the wave function of the excited donors, and compare its predictions to experiments in which pairs of picosecond pulses from the Dutch free-electron laser FELIX are used to resonantly and coherently excite P donors in Si. Although the voltage drop varies on a millisecond time scale we are able to measure Ramsey oscillation of the excitation on a picosecond time scale, thus confirming that the donor wave function, and not just its excited state population, is crucial in determining the electrical signal. We are also able to extract the recombination rate coefficient to the ground state of the donor as well as the photoionization cross section of the excited state and phonon induced thermal ionization rate from the excited state. These quantities, which were previously of limited interest, are here shown to be important in the description of electrical detection, which, in our unoptimized configuration, is sensitive enough to enable us to detect the excitation of $\sim {10}^7$ donors.

Figure 1

A schematic of the experiment. The FELIX beam enters at the top right and by a series of mirrors (M) and beam splitters (BS) is divided into two, sent through two delay lines (DS2) which control the pulse arrival times $\left(t_d\right)$ and recombined, before being focused onto the sample with a parabolic mirror (PM). The FELIX polarization is perpendicular to the plane of the optical table. Attenuator $A_1$ is used to (approximately) equalize the beam intensities and $A_2$ controls the total intensity arriving at the target. The four-point current-biased configuration with current $I$ and voltage drop $V\left(t_{\mathrm{elec}}\right)$ and circuit capacitance $C$ is sketched. The sample I–V characteristic at 10 K is shown as an inset. The upper pair of lines and points are the total voltage drop [upper (red) data is forward bias, lower (green) data is reverse bias]. The lower pair of curves are the barrier voltage drop [upper (blue) curve is forward bias, lower (black) curve is revese bias]. The total measured voltage drop is shown as points; the solid lines show the I–V characteristics for a barrier voltage of $V_B=1.3$ V, the value used in the analysis of the experiments. (See Ref. [12], Chap. 3). The $I-V$ measurements were made at the London Centre for Nanotechnology on the same sample, and under the same conditions as the experiments described here.

Figure 2

The excitation process: A FELIX pulse nominally resonant with the 1–2 transition, but with detuning $\mathrm{\Delta }$ excites population into the upper level with instantaneous Rabi frequency ${\mathrm{\Omega }}_{21}$. After this pulse is over the donor is left in a linear combination of states 1 and 2 determined by the pulse area. A second pulse (not shown) further modifies the upper state population. The upper state can be thermally ionized into the conduction band at rate $G$, or photoionized at a rate determined by ${\sigma }_2$ the photoionization cross section from the upper level. Decay via phonon emission back to the ground state at the rate ${\gamma }_{21}$ is a competing process. Dephasing processes at rate $\mathrm{\Gamma }$ are also included in the model.

Figure 3

The effective multipulse ionization density $n_M$ as a function of initial single pulse ionization density $n_1$, for $M=100$ micropulses separated by 40 ns. We assume $n_{\mathrm{eq}}=3\times {10}^{16}{\mathrm{m}}^{-3}, n_{\mathrm{c}}=5\times {10}^{16}{\mathrm{m}}^{-3}$, and $P=5.3\times {10}^{-13}{\mathrm{m}}^3{\mathrm{s}}^{-1}$, values typical of these experiments.

Figure 4

Four fits of the voltage drop chosen at random. The red line is the best fit voltage, the blue points are the measured values. Each graph is labeled by $\{t_{\mathrm{d}},A_2\}$, where $t_{\mathrm{d}}$ is the interpulse delay in picoseconds and $A_2$ is the total attenuation. An example of the corresponding conduction band density is shown in Fig. 5.

Figure 5

The conduction band density as a function of interpulse delay $t_{\mathrm{d}}$ for experiment 5. The actual experimental values are shown as a thin (blue) line. The upper border (shown in red) and the lower border (shown in green) are used in the fitting procedure described below. Also shown, as an inset, is the experimental power spectrum, normalized to unit height, as a function of frequency in THz (blue points), and in black, a Gaussian fit to it, showing that the rapid oscillations are indeed at the $1s-2p_{\pm }$ transition frequency of 9.49 THz.

Figure 6

The fits to the experimental Ramsey fringe profiles for experiment 5, assuming a Lorentzian inhomogeneous profile (top) or a Gaussian inhomogeneous profile (bottom). As in Fig. 5 the upper and lower borders of the experimental continuum electron density as a function of interpulse delay are shown. The smooth black lines are the corresponding theoretical result. Fits to experiments 2, 4 and 6 are comperable; the fit to experiment 3 is slightly poorer. See text, and Table 3 for further details.

Figure 7

The analog of Fig. 4 for experiment 1. Here the voltage drop is typically much larger than in the previous cases.

Figure 8

$h$, the ratio of the relative change in continuum electron density to the relative change in the corresponding maximum voltage drop $\mathrm{\Delta }V$, plotted as a function of $\mathrm{\Delta }V$ for the parameters appropriate to experiment 1 (solid black line) and experiments 2 to 6 (dashed black line). Also shown are the experimental probability distributions for $\mathrm{\Delta }V$ for the six experiments, each labeled with its experimental index. They have been scaled so that the vertical axis is correct for both $h$ and the probability distributions.

Figure 9

The analog of Fig. 6 for experiment 1. Notice that the agreement between theory and experiment is poorest at large $n_M$, where this is least well determined by the measured voltage drop (see Fig. 8).