Theory of PLE Spectroscopy in Semiconductors
Photoluminescence Excitation (PLE)...
Theory of photoluminescence excitation spectroscopy in semiconductors K Hannewald,1,2 S Glutsch,1 F Bechstedt1 1 Institut für Festkörpertheorie und Theoretische Optik, Friedrich-SchillerUniversität Jena, Max-Wien-Platz 1, 07743 Jena, Germany 2 Present address: Department of Applied Physics, Eindhoven University of Technology, P.O.Box 513, 5600 MB Eindhoven, The Netherlands Abstract. We develop a microscopic theory of photoluminescence excitation spectroscopy (PLE) in semiconductors that is not restricted to thermalized carrier distributions. The phenomenological assumption of equivalence between PLE and absorption spectra is confirmed only for thermal equilibrium. For nonequilibrium, i.e, small time intervals between pulse excitation and luminescence detection, we observe significant differences which can be attributed to incomplete relaxation and bottleneck effects, and should be observable in experiment.
1. Introduction Besides absorption and photoluminescence (PL) experiments, photoluminescence excitation (PLE) measurements are a widely used spectroscopic tool for the characterization of optical transitions in semiconductors. In contrast to the successful development of microscopic theories for absorption [1-3] and PL [4,5] in recent years, an analogous first-principles description of PLE experiments is still missing. Here, we present the first microscopic theory of PLE spectroscopy in semiconductors. A typical luminescence experiment in semiconductors can be divided into three stages, as indicated in Fig. 1. First, the sample is excited out of the ground state which is described by a completely filled valence band and an empty conduction band. Here, we consider optical band-to-band excitation using a femtosecond laser pulse with a mean photon energy of «&pump. The laser pulse creates electron-hole pairs due to a transfer of electrons from the valence into the conduction band (Fig. 1a). Second, the nonequilibrium electron and hole distributions tend to relax back into the ground state. The initial intraband relaxation is caused by energy transfer to the crystal lattice, i.e., a step-by-step excitation of lattice vibrations (Fig. 1b), which are at low temperatures primarily longitudinal optical (LO) phonons in polar semiconductors such as Gallium Arsenide (GaAs). Finally, the electron-hole pairs recombine under emission of light which is the photoluminescence process. Due to the attractive Coulomb interaction between the charge carriers, the emission spectrum does not only contain contributions from states at or above the fundamental energy gap Egap but also sharp discrete lines just
Figure 1. Sketch of the basic processes involved in a typical luminescence experiment in optically excited semiconductors. below Egap which originate from bound excitonic states. In PLE spectroscopy, the spontaneous emission from the sample is detected at a fixed photon energy, typically at the lowest excitonic resonance Eexc in high-purity materials (Fig. 1c). The intensity of this signal is then recorded as a fuQFWL WK SXP IUHTXHQF &pump. It is usually assumed that the PLE spectrum obtained in this way is roughly equivalent to the linear absorption spectrum. A qualitative phenomenological discussion of the relation between PLE and absorption spectra can be found in several textbooks [6,7]. The supposed equivalence between PLE and absorption signals in semiconductors strongly relies on the assumption that the recombination times are much larger than the intraband relaxation times, i.e., if the laser-excited electrons and holes have enough time to relax completely into quasi-equilibrium before radiative recombination. In this case, the emission intensity at the lowest exciton becomes independent of the relaxation rate, and the PLE spectrum can be argued phenomenologically to mimic the absorption signal . However, this assumption has never been justified by microscopic studies. Furthermore, another intriguing question has remained completely unaddressed so far in WK OLWHUDWXUH :KD KDSSHQ L WK WLP ûW between pulse excitation and PL detection is kept shorter than the intraband thermalization times? Then, the one-to-one correspondence between PLE and absorption signals should break down completely, and no simple arguments can be given how the PLE signal looks like. In order to answer these questions, we develop here the first nonequilibrium description of PLE spectroscopy.
2. Theory and modelling The PLE theory is developed by means of the following 3-step strategy which incorporates the ultrafast nonequilibrium aspects (pulse excitation and intraband relaxation) as well as Coulomb interaction (excitonic effects). First, we determine the time evolution of the electron and hole distributions within the conduction and valence band, respectively, during pulse excitation and relaxation. This is done by means of electron-LO-phonon quantum-kinetic equations  which have previously been employed to describe successfully ultrafast four-wave mixing [9,10] and absorption signals [1-3] from coherently excited semiconductors. Details of the numerical implementation are given in Ref. . Second, the luminescence intensity at the exciton energy Eexc is determined by converting the charge carrier distributions into spontaneous emission spectra applying a recently developed Green’s function approach to PL in semiconductors . This theory relates the PL spectrum to the semiconductor polarization function which is then obtained from an explicit solution of the BetheSalpeter equation in ladder approximation. The key result can be expressed as an Elliottlike formula  for luminescence that, in contrast to previous results , allows to treat arbitrary nonthermal situations. Moreover, compared to the alternative method of photon-assisted density matrices [13,14], our approach overcomes several shortcomings such as negative PL and overestimated excitonic signals, as demonstrated in Ref. . Finally, in the third step, the PLE signals are obtained if the above procedure is repeated YDU\LQ WK SXP IUHTXHQF &pump while keeping the other parameters (pump intensity, pump pulse length, and lattice temperature) fixed. For the explicit calculations, we use bulk GaAs parameters, Egap = 1.52 eV and «&LO = 36 meV. The exciton bindining energy is 4.7 meV and the homogeneous line broadening is 0.94 meV. The lattice temperature is fixed at T = 0 K. The pump pulse is assumed to be Gaussian shaped with a peak intensity of 0.1 MW/cm2, and the spectral resolution of the luminescence signals is optimized by choosing the pump pulse length to be 320 fs which is significantly larger than the LO phonon period of 115 fs in GaAs. 3. Results and discussion In Fig. 2, we present the results for the time evolution of the PLE spectra. It follows that
OV [FLWDWL the PLE spectrum significantly depends WK WLP LQWHUYD û EHWZHH and PL detection. For short intervals up to a few picoseconds, the PLE spectra are not similar at all to the linear absorption but strongly dominated by the signal at the exciton which is about three orders of magnitude larger than the PLE intensity above the band gap. This is due to the fact that for high pump frequencies the excited electrons and holes have not yet completely relaxed towards their band minima. As a consequence, the emission is not solely from the exciton, but by definition only the excitonic emission contributes to the PLE signal. Furthermore, there are pronounced satellite peaks in the continuum region which are completely absent in the linear absorption spectrum. They clearly reflect bottleneck effects that may occur in the charge-carrier relaxation process. Once the electrons (or holes) have reached kinetic energies below the threshold for LOphonon emission, the intraband relaxation process drastically slows down. The strength of this effect depends mainly on the ratio between «&pump − Egap and «&LO, resulting either in a weak or a strong bottleneck. This, in turn, gives rise to the peculiar satellite features seen in the PLE signals at early times. It is worthwhile to note that for the more
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energy (eV) Figure 2. PLE intensity in bulk GaAs vs. pump energy «&pump for times ût = 0.7, …, 28.0 ps after pulse excitation (from top to bottom). The pump intensity is 0.1 MW/cm2, the pump pulse length is 320 fs, and the lattice temperature is T = 0 K. polar II-VI semiconductors such as Zinc Selenide similar LO-phonon cascades have been observed in stationary PLE experiments and were interpreted as hot-carrier effects [15,16]. In the long-time limit, the picture changes dramatically and the PLE spectrum resembles much more the absorption spectrum. Since the bottleneck effects are slowly removed due to quantum-kinetic effects, the dips in the satellite structures become less D û
WK UHSOLFD DU and less prRQRXQFHG D OPRV RWDOO VPHDUH out. Furthermore, the ratio between the PLE signal at the exciton and the continuum has become almost the same as in the absorption signal because the PLE signal from the continuum L RQ RUGH R PDJQLWXG ODUJH WKD IR û ! ZKHUHD ! WK" 3/# VLJQD$ from the exciton has decreased strongly by two orders of magnitude. While the first effect can again be attributed to the removal of the bottleneck effect on long time scales, the behavior for «&pump ≈ Eexc is quite unexpected. An analysis of the time evolution of 4
the charge carrier distributions under these pump conditions (not shown) reveals the important role played by quantum-kinetic effects such as scattering processes without energy conservation. For «&pump ≈ Eexc, the pump-induced electrons and holes occupy initially only a very narrow region close to the band extrema. In the semiclassical limit, these charge carriers would not experience any intraband relaxation at all at zero temperature. However, this is no longer true in quantum mechanics where the timeenergy uncertainty principle renders a possibility for upward scattering even at T = 0 K. As a consequence, the electrons and holes are gradually redistributed to higher kinetic energies which, in turn, explains the significant decrease in the excitonic PLE signal seen in Fig. 2. The resemblance between the absorption spectrum and the PLE signal in the long WLP
assumed equivalence between PLE and absorption. Nevertheless, if both spectra are plotted on a linear scale , we observe some slight deviations: the PLE spectrum exhibits a broader exciton line and small but noticeable modulations in the continuum signal. As explained above, the latter effect is related to bottleneck effects and we expect it to vanish on nanosecond time scales, especially when the scattering with acoustic phonons (not considered here) becomes important, too. The other effect, the line broadening of the exciton in PLE, is mainly determined by the spectral width of the pump pulse, in contrast to the excitonic linewidth in absorption which is only subject to dephasing, i.e, homogeneous line broadening. 4. Summary In conclusion, we have developed a novel description of photoluminescence excitation spectroscopy in semiconductors. The theory is not restricted to thermal equilibrium but also allows the treatment of highly nonequilibrium systems such as optically excited electron-hole pairs created by ultrafast femtosecond laser pulses. The phenomenological assumption of equivalence between PLE and absorption spectra is confirmed in the longtime limit, i.e., approaching thermal equilibrium. For small time intervals between excitation and luminescence detection, we observe significant differences which can be attributed to incomplete relaxation and bottleneck effects, and should be easily observable in experiment since the setup can be customized to require only picosecond time resolution. Thus, our findings offer a new possibility for time-resolved studies of hot-carrier phenomena by means of emission spectroscopy. References        
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