Linear and Nonlinear Excitonic Absorption and Photoluminescence Spectra in Cu 2 O: Line Shape Analysis and Exciton Drift

Transcription

1 A. Jolk and C. F. Klingshirn: Excitonic Absorption and PL Spectra in Cu 2 O 841 phys. stat. sol. (b) 206, 841 (1998) Subject classification: Cc; Hx; S10.1 Linear and Nonlinear Excitonic Absorption and Photoluminescence Spectra in Cu 2 O: Line Shape Analysis and Exciton Drift A. Jolk and C. F. Klingshirn Institut fur Angewandte Physik, Universitat Karlsruhe, Kaiserstr. 12, D Karlsruhe, Germany (Received October 29, 1997) A complete analysis of the line shape of the excitonic absorption and photoluminescence lines of the yellow series of Cu 2 O is given. A detailed fit to the absorption lines up to the n ˆ 5 exciton according to Toyozawa's theory gives precise values for the excitonic resonance energies, allowing to calculate the excitonic Rydberg. The fit to the photoluminescence lines shows very good agreement with experimental data at different temperatures. At high excitation, the fit to the excitonic absorption lines reveals the fundamental mechanism for bleaching of exciton absorption lines in Cu 2 O. Based on these findings, a new all-optical method for the observation of exciton transport in Cu 2 Ois proposed. 1. Introduction The yellow excitonic series in cuprous oxide (Cu 2 O) has received considerable interest in the past, both as an excellent example of basic optical properties of direct semiconductors and as one of the most prominent candidates for the observation of high-density effects. Recent work by Benson et al. [1,2] suggested that Bose-Einstein condensation of excitons can be deduced from a characteristic change of the exciton propagation. Kopelevich et al. [3] ascribe this effect to a phonon-wind driven transport. Scattering-rate calculations by Ivanov et al. [4] seem to indicate that Bose-Einstein condensation of excitons in Cu 2 O may not be possible within the exciton lifetime due to the limited relaxation rate of 1s excitons near k ˆ 0. Still the observation of the propagation of dense exciton clouds gives important information on the understanding of high density excitonic effects in Cu 2 O. In this work, we first provide a thorough analysis of the linear optical properties of our Cu 2 O samples by analyzing the line shape of the yellow excitonic absorption series and of the photoluminescence (PL) spectrum. The detailed understanding of the absorption line shape according to Toyozawa's theory [5] allows us in a second part to investigate the effect of a high density of excitons on the shape of the absorption lines. Based on these results, we propose a new all-optical method for the observation of exciton transport and present a first realization of this experimental technique. Cu 2 O is a direct semiconductor with a band gap of ev at 6 K, having a cubic lattice with inversion symmetry. Therefore, parity is a good quantum number for all transitions in Cu 2 O. Since the highest valence and the lowest conduction bands both have positive symmetry, the formation of excitons by a first-class optical transition is forbidden in the dipole approximation. Excitons can only be produced by dipole transi-

2 842 A. Jolk and C. F. Klingshirn tion if the excitonic envelope function is of negative parity (second-class transition). This is most noticeably the case with the np excitons. Due to the electron±hole exchange interaction the 1s exciton splits into an orthoexciton (at ev at 4 K) and a paraexciton 12 mev lower in energy. The orthoexciton is weakly visible with thick samples in linear absorption spectra due to quadrupole transition. Despite the very weak coupling to the radiation field, a polariton effect has been observed [6]. On the other hand, 1s and higher ns excitons can be directly created by two-photon absorption [7]. This behavior is well-known and has been observed before [8,9]. By a different path, excitons of positive parity can be generated if an optical phonon participates in the process. Since in Cu 2 O the energy of the lowest LO phonon is considerably lower than the exciton binding energy, 1s excitons can be produced by absorption of one photon under simultaneous emission of an LO phonon. By this process, an exciton with almost arbitrary wavenumber can be formed. The absorption coefficient is thus proportional to the density of final exciton states, which is in itself proportional to the square root of the excess energy. This process manifests itself as a distinct absorption onset that follows the expected square root dependence starting one LO phonon energy above the 1s exciton resonance energy. For higher temperature, even an absorption onset one LO phonon energy below the exciton resonance has been observed [10]. High-quality samples of Cu 2 O exhibit a beautiful hydrogenic series of absorption peaks up to the 12p exciton [9]. The resonance energies of the np excitons follow for n 2 the hydrogen series E n ˆ E g Ry*=n 2 ; 1 where E n is the resonance energy of the np exciton, E g is the band-gap energy, and Ry* is the excitonic Rydberg, or exciton binding energy. A fit to this series for n 2 allows to calculate E g and Ry* very precisely. The n ˆ 1 para- and orthoexcitons do not coincide with the series as is normal for Ry* > hw LO [11]. In a previous work [9], this fit has been evaluated using the peak position apparent from the absorption spectra as exciton resonance energy. A detailed analysis of the line shape of the absorption lines reveals that this is not exactly true. Toyozawa [5] and Ueno [12] give an expression based on a model that includes offdiagonal transition matrix elements for the phonon-assisted generation of an exciton. If the absorption lines are sufficiently narrow, each absorption line is of an asymmetric Lorentzian shape that can be fitted to the absorption spectra in order to measure the exact resonance energies of the np excitons. Under optical excitation with a low-power continuous wave (cw) laser tuned above the excitonic band edge, Cu 2 O exhibits several photoluminescence lines that can be ascribed to direct optical recombination of 1s (ortho-)excitons as well as to a phonon-assisted recombination. Analogous to the indirect absorption process observed in the linear absorption measurements, an exciton can both emit or absorb an optical phonon and recombine, emitting a photon that is shifted from the 1s exciton resonance energy by the momentum conserving LO phonon energy hw LO and the kinetic energy of the exciton. The line shape of the latter indirect luminescence lines fits very well to the product of the Boltzmann distribution function and the excitonic density of states. The resulting parameters such as effective exciton temperature and LO phonon energy are in very good agreement with the literature.

3 Linear and Nonlinear Excitonic Absorption and PL Spectra in Cu 2 O 843 Photoluminescence from the np exciton states can not be observed because the exciton relaxation into the lowest 1s exciton state is a much faster process than the recombination of np excitons. Using a standard pump±probe set-up, we measured the absorption spectra of Cu 2 O under simultaneous excitation by a strong pulsed dye laser. A fit to the absorption line shape reveals that for moderate excitation, the predominant effect is the broadening of the exciton lines due to increased scattering or excitation induced dephasing. When the density of excitons increases further, the oscillator strength decreases. With increasing excitation intensity the excitons with large n are affected first. This is due to the fact that the smaller excitons, i.e. those with smaller main quantum number n have a much smaller scattering and screening probability than the larger ones. 2. Experimental Set-Up 2.1 Linear absorption measurements The measurements of the linear absorption were performed using a 60 cm grating monochromator with a 1800/mm grating. The transmitted light from the spontaneous emission of an optically pumped dye cuvette was detected using a 1024 channel optical multichannel analyzer (OMA). The sample consisted of a 50 mm thick Cu 2 O platelet on a 1 mm sapphire or a selfsupported 300 mm thick platelet. It was cooled in a liquid helium bath cryostat. In all cases the samples have been cut and polished from natural single crystals from Congo. The orientation of the sample surfaces was 111Š. The transmission spectra both of the sample and of a reference absorber were recorded and processed to yield the absorption spectrum. Scattering losses and reflection at the sample surfaces were neglected. Since the excitonic oscillator strength in Cu 2 Ois relatively weak, the reflectivity does not noticeably change within the considered energy interval. Thus, correcting for the reflected light from the sample surface does not influence the shape of the absorption spectra. 2.2 Photoluminescence measurements For the measurements of the photoluminescence, we used a 300 mm Cu 2 O sample held at temperatures between 6 and 40 K. Optical excitation was done with the 514 nm (2.41 ev) line of an Ar cw laser at an excitation density not exceeding 100 W/cm 2. The emitted light in backscattering geometry was chopped and detected using a 2 0:27 m double grating monochromator with 1200/mm gratings and a photomultiplier connected to a lock-in amplifier. 2.3 Pump and probe measurements We also carried out differential transmission spectroscopy (DTS) measurements, typically known as `pump and probe' measurements. We measured the transmission of the 50 mm sample held at 5 K, both in the dark and under simultaneous high excitation with a so called `pump laser' pulse of about 5 ns duration. We used an optically pumped dye laser tuned to wavelengths in the Cu 2 O yellow excitonic series or just above the band edge (570 nm at 6 K). The maximum excitation energy flux density of the pump pulse was 1 MW/cm 2. 56a physica (b) 206/2

4 844 A. Jolk and C. F. Klingshirn The transmitted light was detected using a 60 cm grating monochromator with a 1800/mm grating and a 1024 channel optical multichannel analyzer (OMA) as for the linear absorption measurements. The repetition rate of the laser system was 20 Hz. The recorded transmission spectra and reference spectra were combined to yield the spectrum of differential absorption. 3. Results 3.1 Linear absorption measurements The yellow excitonic series in Cu 2 O shows several distinct regions of interest. Fig. 1 shows an overview of the absorption spectrum. The spectrum was recorded with a sample of 300 mm thickness at a temperature of 6 K. One remarks a clear absorption onset at about 2.05 ev that corresponds to the phonon assisted production of 1s excitons. The dotted line is a fit to a square root dependence. The direct absorption into a 1s exciton state is not observable with this sample due to the small thickness. The absorption line corresponding to the 1s orthoexciton is found at 2.03 ev in thicker samples (above 1 mm). In the region between 2.14 ev and the band edge at 2.17 ev, the np exciton absorption lines are clearly resolved. With the low spectral resolution in this figure, only the n ˆ 2 and 3 lines are distinguishable. The baseline of the absorption seems to deviate from the fitted background even below the np exciton peaks. The origin of this deviation is not known, eventually this is an additional square-root dependence involving a phonon with higher energy. In order to fit the absorption spectra to an analytical formula, one has to assume a reasonable background function. According to Toyozawa's analysis [5], the absorption peaks follow an ªasymmetric Lorentzianº. A single absorption peak can thus be described by the expression a w ˆa 0 G 2A w w 0 w w 0 2 G 2 ; 2 where a 0 is proportional to the oscillator strength, w 0 is the resonance frequency, and G is the damping constant (the inverse of the phase relaxation time). The resulting line Fig. 1. Linear absorption of Cu 2 O in the spectral region of the yellow exciton series, measured at T ˆ 6 K. The sample used in this measurement was 300 mm thick, and a 0.27 m grating monochromator with 600/mm gratings was used. Due to the limited resolution of this set-up, only the n ˆ 2 and n ˆ 3 lines at 2.15 ev and 2.16 ev are resolved. Above 2.17 ev, the transmitted intensity was too low to be reliably detected

5 Linear and Nonlinear Excitonic Absorption and PL Spectra in Cu 2 O 845 Fig. 2. Calculated asymmetric Lorentzian according to Toyozawa. The parameters chosen to calculate this function were a 0 ˆ G ˆ 0:3 mev, A ˆ 1 and hw 0 ˆ 2:15 ev. Note that the maximum of the curve does not coincide with the resonance energy unless A ˆ 0 (symmetrical Lorentzian) shape for typical Cu 2 O parameters is sketched in Fig. 2. The asymmetry parameter A is found to be negative and close to unity for all the np peaks in Cu 2 O. A complete excitonic absorption series can be described as the sum of several asymmetric Lorentzians. Fig. 3 shows a nonlinear least-squares fit to the first four peaks of a typical absorption spectrum. The background absorption has been described by a linear background corresponding to a fit to the phonon-assisted absorption into the 1s state over a small energy interval. Superimposed to this is an absorption onset one LO phonon energy above the 2p resonance to account for the phonon-assisted absorption into the 2p state. The fitted function follows the experimental data very closely. The resonance energy of the absorption peaks is one of the fit parameters and is usually found with very small error bars (typically below 0.1 %). A fit of the hydrogen series (eq. (1)) for n 2 to the exciton resonance energies gives values for the band edge E g ˆ 2:172 ev and the excitonic Rydberg Ry* ˆ 93:26 mev. The n ˆ 1 orthoexciton has a binding energy of 140 mev and is situated at ev [9]. 3.2 Photoluminescence measurements The most prominent PL lines observable at low temperature are the direct 1s orthoexciton recombination line at hw ˆ 2:032 ev and the one LO phonon replica 13.2 mev lower in energy. Fig. 3. Fit of the asymmetric Lorentzian to experimental data. Crosses ( ) correspond to experimental data, the solid line is the best fit. The dotted line is the background function used for the fit. The background absorption onset at ev has been introduced phenomenologically and corresponds to a phonon-assisted absorption into a 2p exciton state. On the upper x-axis the resonance energies of the np excitons are marked 56a*

6 846 A. Jolk and C. F. Klingshirn Fig. 4. Fit of the broadened PL function to experimental data. Experimental data were taken at T ˆ 6K The theoretically expected line shape of the one LO phonon replica is the product of the Boltzmann distribution function with the exciton density of states which corresponds to a square root dependence of the excess energy. This holds as long as the exciton density is not too high so that the boson character of the excitons can be neglected. The weak direct recombination line due to a quadrupolar recombination of a 1s exciton is expected to be of symmetrical Lorentzian shape. The PL lines are normally broadened because of both the limited resolution of the detection system as well as the intrinsic line broadening. We accounted for the former effect by deconvolving the measured spectra with a detection broadening function that was essentially Gaussian. Fig. 4 shows a fit of a broadened PL function to experimental data. The fit parameters were hw ˆ 2: ev for the 1s exciton resonance energy, hw ˆ 13:2 mev for the LO phonon energy, and T ˆ 8:6 K for the effective exciton temperature. This agrees well with the measured temperature T ˆ 6 K of the cryostat and thus of the crystal lattice. Fig. 5 shows a fit of the (non-broadened) PL function including several phonon replicas to experimental data taken at a lattice temperature T ˆ 37 K. The fit function agrees very well with the experimental data over two orders of magnitude. The fit gave an effective exciton temperature of T ˆ 38:6 K which is in reasonable agreement with the measured lattice temperature. Fig. 5. Fit of the non-broadened PL function to experimental data at T ˆ 37 K. Note that the y-axis is logarithmic

7 Linear and Nonlinear Excitonic Absorption and PL Spectra in Cu 2 O 847 At this high temperature, another LO phonon replica is observable one phonon energy above the direct 1s recombination line at ev. It corresponds to the ªanti-Stokes processº, where an exciton recombines under simultaneous absorption of an LO phonon. The relative heights of the Stokes and the anti-stokes line are given by I S =I A ˆ hw S hw A 4 w 4 S 1 f Bose =f Bose ˆ exp hw LO=kT ; 3 w A where f Bose is the Bose distribution function and hw LO is the LO phonon energy [13]. Given the LO phonon energy hw ˆ 13:2 mev, the effective exciton temperature can be calculated from the fitted heights of the phonon replicas giving T ˆ 34:9 K. All three values for lattice and exciton temperatures agree within experimental error. 3.3 Pump and probe measurements ªPump and probeº type measurements on Cu 2 O have been published before [14] but a precise understanding of the bleaching mechanism is still lacking. Fig. 6 shows the absorption spectrum of Cu 2 O under simultaneous optical excitation in the continuum with different excitation intensities. Because of the limited absorption length of the excitation pulse we corrected for the non-excited part of the sample using a simple two layer model. The excitation intensity is given in the plot inset relative to about Fig. 6. Absorption spectra under simultaneous excitation with different intensities. The curves are vertically offset for clarity. The inset gives the relative pump intensity; the maximum pump density was about 1 MW/cm 2 corresponding to an exciton density of the order of cm 3. On the upper x-axis the position of the exciton absorption lines at zero pump intensity is marked. Data were taken at T ˆ 5:5 K, excitation wavelength was l exc ˆ 560 nm (2.21 ev) well above the band edge

8 848 A. Jolk and C. F. Klingshirn 1 MW/cm 2 which corresponds to a 1s exciton density of roughly cm 3. This density is still well below the critical Mott density because of the small exciton Bohr radius in Cu 2 O. It should be noted that with increasing excitation intensity, the higher exciton lines are first affected, while the lowest 2p line is nearly unaffected by the pumping up to the highest values. When free carriers are created in Cu 2 O by optical excitation above the yellow band edge, they recombine within a few 100 fs to excitons and relax very quickly (on a 30 ps time scale) into the 1s state [14]. The 5 ns optical pump pulse we used for the pump and probe type measurements is much longer than these time scales. The bleaching we observe can only be caused by 1s excitons since they have an even longer lifetime of several 100 ns. Thus we expect the differential absorption to be reasonably independent of the exact wavelength used for pumping. We verified this by pumping in the continuum above 2.17 ev as well as in resonance with several np exciton states. The differential absorption spectra were essentially unchanged in all of these cases. 4. Discussion and Outlook The linear absorption spectrum of Cu 2 O can be described by Toyozawa's expression eq. (2) which fits very well to experimental data. This allows to calculate precisely the excitonic band edge and the Rydberg energy. The same fit can be performed for the absorption lines under high excitation. We find that with moderately increasing excitation intensity, the oscillator strength of the absorption peaks first remains essentially unchanged, while the line width increases significantly. When the excitation increases even further, the line vanishes completely. We interpret this behavior as follows: At relatively moderate excitation, exciton±exciton scattering is the most important interaction mechanism, reducing the phase relaxation time and thus increasing the absorption line width without drastic changes of the oscillator strength. Similarly, a collision-assisted absorption process as described by Mysyrowicz et al. [15] explains the observed broadening but does not account for the exact line shape of the absorption spectrum. The observation that the higher exciton lines broaden before the lower ones agrees well with both of these pictures. At higher excitation, the absorption into larger exciton states is completely blocked. This can be ascribed to the renormalization of the band gap due to the large density of excitons and/or to a screening of the Coulomb interaction between electrons and holes by the excitons. Screening of Coulomb interaction by electrically neutral but polarizable excitons has been observed in GaAs [16]. For a theoretical treatment, see e.g. [17,18]. The analysis of the PL shows that the line shape is very well described by the product of the Boltzmann distribution function and the exciton density of states. We conclude that the free carriers generated by optical excitation above the band gap recombine into 1s excitons and reach a thermodynamic equilibrium with the lattice. This observation is in agreement with the known time scales of the different relaxation processes in Cu 2 O. The above suggests that the bleaching observed in the pump and probe type measurements is indeed due to a finite density of 1s excitons. The detection of a bleaching signal at a certain photon energy can thus serve as a probe for the 1s exciton density, once the dependence of the differential absorption on the exciton density has been quantitatively established. We thus propose a new all-optical method for the observation of exciton propagation in Cu 2 O single crystals (see Fig. 7). A weak cw dye laser is tuned to a photon energy at

9 Linear and Nonlinear Excitonic Absorption and PL Spectra in Cu 2 O 849 Fig. 7. Schematic of the propagation measurement. The excitation spot (`pump') is at a certain distance from the probe spot. The generated excitons propagate through the crystal as indicated by the grey scale which the differential absorption spectrum under high excitation is very pronounced. This laser is focused on the Cu 2 O sample and the transmitted light is detected using a fast photomultiplier connected to an oscilloscope. At a given distance d from the probe spot, the pump pulse is focused on the sample. The generated excitons propagate through the crystal and eventually reach the probe spot where they cause an absorption change which is detected by the oscilloscope. The oscilloscope is triggered by an electrical pulse from the pump laser which has a constant delay from the light pulse. This delay can be measured by sending some scattered light from the pump pulse directly onto the photomultiplier. We performed some first experiments along these lines. Fig. 8 shows the time-resolved differential absorption in the 3p exciton resonance at different distances d from the pump spot. It is clearly seen that the excitons do propagate through the crystal, over distances up to 400 mm. The time delay between the pump pulse (at t ˆ 1:8 ms) and the maximum of the bleaching curve shifts to later times with increasing distance d. The origin of the positive part of the function at zero distance is not yet understood. We tentatively ascribe it to a local heating of the lattice because of its long decay time of the order of 1 ms. Apart from the exciton propagation, this measurement is a direct observation of a decay process of the exciton population in Cu 2 O extending over several ms. While it is generally accepted that paraexcitons in Cu 2 O have a lifetime that exceeds 100 ms, no Fig. 8. Time-resolved differential absorption at different distances d given in the plot key (in mm). The cw probe laser was tuned to the 3p exciton resonance. The pump pulse hits the sample at t ˆ 1:8 ms. T ˆ 6:5 K,l exc ˆ 555 nm (2.23 ev), I exc ˆ 200 kw=cm 2

10 850 A. Jolk and C. F. Klingshirn: Excitonic Absorption and PL Spectra in Cu 2 O direct observation has been published so far. The differential absorption is by no means proportional to the exciton density but it serves as an indicator of exciton presence. The optical method proposed here does not share some of the drawbacks of the galvanic method used by Benson et al. [1,2]. Most prominently, the distance between excitation and detection of the excitons can be freely chosen, and the exciton detection is not bound to the sample surfaces and Schottky barriers where several nonlinear effects intervene, such as screening of the field in the barrier and subsequent saturation of the detection. Further investigations are underway to establish the interdependence of the differential absorption and the exciton density. A detailed analysis of the exciton propagation has the potential to aid in the understanding of high-density effects in Cu 2 O. 5. Conclusion In the present paper, we have given a detailed analysis of the line shape of both the absorption and the photoluminescence of the yellow exciton series in Cu 2 O. Analyzing the absorption under simultaneous high excitation, we identified exciton±exciton scattering as the most important bleaching mechanism. Based on these observations, we propose a new all-optical method for the observation of exciton propagation. This method has been realized and demonstrated. An excitonic decay process of several ms duration has been observed. Acknowledgements We would like to thank Prof. Dr. A. Mysyrowicz (Palaiseau) for stimulating discussions and the Deutsche Forschungsgemeinschaft for financial support. References [1] E. Benson, E. Fortin, and A. Mysyrowicz, Solid State Commun. 101, 313 (1997). [2] E. Benson, E. Fortin, and A. Mysyrowicz, phys. stat. sol. (b) 191, 345 (1995). [3] G. A. Kopelevich, S. G. Tikhodeev, and N. A. Gippius, Soviet Phys. ± J. Exper. Theor. Phys. 82, 1180 (1996). [4] A. L. Ivanov, C. Ell, and H. Haug, Phys. Rev. E 55, 6363 (1997). [5] Y. Toyozawa, J. Phys. Chem. Solids 25, 59 (1964). [6] D. Frohlich, A. Kulik, B. Uebbing, V. Langer, H. Stolz, and W. von der Osten, phys. stat. sol. (b) 173, 31 (1992). [7] C. Uihlein, D. Frohlich, and R. Kenklies, Phys. Rev. B 23, 2731 (1981). [8] J. B. Grun and S. Nikitine, J. Physique 24, 355 (1963). [9] H. Matsumoto, M. Hasuo, S. Kono, and N. Nagasawa, Solid State Commun. 97, 125 (1996). [10] J. B. Grun, M. Sieskind, and S. Nikitine, J. Phys. Chem. Solids 19, 189 (1961). [11] C. F. Klingshirn, Semiconductor Optics, Springer-Verlag, Heidelberg [12] T. Ueno, J. Phys. Soc. Jpn. 26, 438 (1969). [13] D. W. Snoke, A. J. Shields, and M. Cardona, Phys. Rev. B 45, (1992). [14] J. P. Wolfe, J. L. Lin, and D. W. Snoke, in: Bose-Einstein Condensation, Eds. A. Griffin, D. W. Snoke, and S. Stringari, Cambridge University Press, Cambridge 1995 (Chap. 13, pp. 281 to 329). [15] A. Mysyrowicz, D. Hulin, and E. Hanamura. in: Ultrafast Phenomena VII, Eds. C. B. Harris, E. P. Ippen, G. A. Mourou, and A. H. Zewai, Vol. 53, Springer Series Chem. Phys., Springer-Verlag, Berlin/Heidelberg 1990 (pp. 244 to 246). [16] G. W. Fehrenbach, W. Schafer, J. Treusch, and R. G. Ulbrich, Phys. Rev. Lett. 49, 1281 (1982). [17] R. Zimmermann, Many-Particle Theory of Highly Excited Semiconductors, Vol. 18, Teubner Texte Phys., Teubner, Leipzig [18] H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 2nd ed., World Scientific Publ. Co., Hong Kong 1993.