Scrutinizing the Debye plasma model: Rydberg excitons unravel the properties of low-density plasmas in semiconductors

For low-density plasmas, the classical limit described by the Debye-Hückel theory is still considered as an appropriate description even though a clear experimental proof of this paradigm is lacking due to the problems in determining the plasma-induced shift of single-particle energies in atomic systems. We show that Rydberg excitons in states with a high principal quantum number are highly sensitive probes for their surrounding making it possible to unravel accurately the basic properties of low-density nondegenerate electron-hole plasmas. To this end, we accurately measure the parameters of Rydberg excitons such as energies and linewidths in absorption spectra of bulk cuprous oxide crystals in which a tailored electron-hole plasma has been generated optically. Since from the absorption spectra exciton energies, as well as the shift of the single-particle energies given by the band edge, can be directly derived, the measurements allow us to determine the plasma density and temperature independently, which has been a notoriously hard problem in semiconductor physics. Our analysis shows unambiguously that the impact of the plasma cannot be described by the classical Debye model, but requires a quantum many-body theory, not only for the semiconductor plasma investigated here, but in general. Furthermore, it reveals an exciton scattering mechanism with coupled plasmon-phonon modes becoming important even at very low plasma densities.

Figure 1

Transmission spectra of Rydberg excitons from $n=5$ onwards recorded at $T=1.35\mathrm{K}$ for simultaneous application of a pump laser at a photon energy of 2.168 883 0 eV (blue arrow) operated at different powers as shown. The full lines are the experimental results and the dotted lines show the best fit using the CTM method [49]. The upper panel shows the spectra at low pump powers, which were fitted assuming an Urbach-like tail [Eq. (6)]. The black line for 50 $\mu \mathrm{W}$ is a fit without an Urbach tail. The lower panel shows the spectra for high pump powers, which were fitted assuming only a broadened continuum [Eq. (5)]. The deviations of the experimental traces from the fit between the $P$ lines are due to optical absorption into other angular-momentum states that become allowed through the effect of the impurity-induced electric fields [39]. Note the nonlinear scale of the $x$ axis proportional to the square root of energy difference of nominal band edge $E_{g0}$ and photon energy allowing displaying all lines with equal resolution. The red arrows mark the position of the apparent band edge $E_g$, where the absorption lines of the $P$ states vanish and the transmission goes over to the flat continuum. The band-edge shifts $\mathrm{\Delta }=E_g-E_{g0}$ are plotted as black squares in Fig. 4.

Figure 2

Results of the fit of the transmission spectra for various pump powers. (a), (b): Oscillator strengths for different pump powers from 0.$2\mu \mathrm{W}$ to $5\mathrm{mW}$ grouped into low and high power ranges. The pump powers are the same as in Fig. 1. The colored full lines give the dependence of oscillator strength on quantum number according to exciton theory as discussed in the text. One must be aware that the lines represent a physical value only at the discrete principal quantum numbers; in between they should be regarded as a guide to the eye. While in the HPR [panel (b)] the fit results (within experimental error) all fall onto this line; in the LPR [panel (a)] we see the well-known deviations due to the effect of charged impurities. Panels (c) and (d) show the dependence of linewidth of the Lorentzians [given as HWHM (half-width at half maximum) ${\mathrm{\Gamma }}_n$] on principal quantum number $n$. The different pump powers are represented by the same symbols as in (a) and (b). The dashed black lines give the expected dependence from phonon scattering, while the colored lines contain an additional contribution as discussed in the text. The inset in (d) shows the pump-power dependence of the strength factor $C_{\mathrm{add}}$. The blue full line gives a pump-power dependence $P^{0.8}$.

Figure 3

Dependence of energy shifts on the principal quantum number $n$ for selected pump powers from 20 $\mu \mathrm{W}$ to 5 mW as indicated. The full lines give the results of many-particle theory proportional to the fourth power of the principal quantum number [Eq. (9)]. The proportionality constants allow deducing the plasma densities and temperatures shown in Figs. 4 and 4 as blue triangles.

Figure 4

Panel (a) shows the shift of the band edge Δ as a function of the pump power. Black square symbols are the shifts as determined from the transmission spectra (red arrows in Fig. 1), while the red dots and the blue triangles are the shifts calculated by many-body theory [Eq. (1.4)] from the densities and temperatures obtained from fitting the linewidth broadening [Fig. 2] and the shift of the resonance energies (Fig. 3), respectively. In the calculation we neglected the contribution of the charged impurities, resulting in a too-small shift at low pump powers, as expected. Panel (b) gives the effective temperatures of the EHP obtained from the line-energy and band-edge shifts (blue triangles) and from the line broadening (red dots) compared to results of the relaxation and cooling model (see Appendix pp2) for the phonon-assisted Auger scenario (black line). Panel (c) shows the dependence of the electron-hole pair concentration on pump power obtained from the $P$ lines energy shift and the band-edge shift (blue triangles), from the linewidth broadening (red dots) and from the rate and cooling model (black line).

Figure 5

Breakdown of the Debye plasma model. Panel (a) compares the variations of the line-energy shifts on principal quantum number $n$ for selected pump powers from 20 to $5000\mu \mathrm{W}$ as indicated (same data as in Fig. 3) with the predictions of the Debye plasma model [Eq. (10), full lines]. The dotted lines give the results of the many-particle theory [Eq. (9)] as taken from Fig. 3. The ratios of density and temperature obtained from the proportionality constants are plotted as blue dots in (b), while the red triangles show the same quantity obtained from the band-edge shift [Eq. (11)]. The error bars in (b) are obtained from the fits in case of energy shifts and from the accuracy in the determination of the band edge of about $20\mu \mathrm{eV}$.

Figure 6

Comparison of the measured transmission spectrum (red) with the fit according to CTM theory (blue). For low [panel (a)] and high [panel (b)] pump power as indicated. The difference between both transmission spectra is shown as the green line. While one clearly sees residual absorption lines into $F$ states (from $n=5\mathrm{to}7$) [59], the deviations at higher principal quantum numbers ($n>10$) are due to states that become allowed by the electric field of residual charged impurities [39] and to interference effects of the setup [49]. The magenta line gives the corresponding theoretical reflection spectrum (shifted by 0.15). Pump laser: photon energy 2.168 883 eV.

Figure 7

Overview of the asymmetry parameters from the fits of the transmission spectra. The pump powers are given at the left of the curves. The full lines are guides to the eye; the dotted lines gives the constant value of $A=-0.275$ at low principal quantum numbers for all measurements. For increasing pump power the lines are shifted by 0.1 from dataset to dataset.

Figure 8

Urbach tail decay parameters from Eq. (A2). $E_{ur1}$(red squares), $E_{ur2}$ (blue diamonds), and $A_{ur}$(green dots) on pump power in the LPR. The lines are guides to the eye.

Figure 9

Results of the rate model. The curves give the concentrations of the following species: primarily excited $ortho$-exciton (dashed black line), … para-excitons at $k=0$ (full blue line), Auger-created electron-hole pairs (magenta line). The dotted lines of the same color give the results of the effective rate model used to simulate the cooling behavior (see Sec. B5). The red dots and blue triangles give the results of the analysis of the experimental data (see main text).

Figure 10

Scattering rates of electrons (a) and holes (b) vs kinetic energy. Blue lines denote scattering by the LO2 mode via Fröhlich interaction, red lines that of the LO1 mode. In (a) scattering by acoustic phonons is denoted by the magenta line. In (b) scattering by the LO2 mode in the lower part of the valence band is denoted by a dashed magenta line, while scattering by acoustical phonons by a black line. The dashed black line gives the acoustical scattering in the upper part of the valence band (note the multiplication by a factor of 10).

Figure 11

Temperatures of the EHP vs excitation laser power. The blue triangles and red dots denote the values derived from the analysis of the line shifts and line broadening, respectively (see main text). The black line gives the result of the calculation for the average temperature if we assume that the phonon-assisted Auger scattering is dominant (see text). The blue dashed line gives the temperature of the electrons alone, while the dashed-dotted blue line gives the temperature of the holes.