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Optics Express

Optics Express

  • Editor: J. H. Eberly
  • Vol. 1, Iss. 12 — Dec. 8, 1997
  • pp: 364–369
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Polarization dependent quantum beats of homogeneously broadened excitons

Takao Aoki, Georg Mohs, Takeshi Ogasawara, Ryo Shimano, Makoto Kuwata-Gonokami, and A. Atsushi Yamaguchi  »View Author Affiliations


Optics Express, Vol. 1, Issue 12, pp. 364-369 (1997)
http://dx.doi.org/10.1364/OE.1.000364


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Abstract

The polarization dependence of quantum beats from the A-exciton and the B-exciton in a GaN sample of exceptional quality is studied with four-wave mixing experiments. When changing the incident polarizations from collinear to crossed linear, a π-phase shift of the beats is observed while the decay rate remains unchanged. This confirms previous theoretical predictions.

© Optical Society of America

1. Introduction

In general, four-wave mixing (FWM) offers much information about coherent oscillations in semiconductors and the intrinsic dephasing properties of excitons. Particularly if two transitions with different energies are excited simultaneously, the FWM signal vs. time delay shows quantum beats with a period corresponding to the energy difference. The observation of quantum beats in semiconductors such as GaAs quantum wells[1–3

1. S. Schmitt-Rink, D. Bennhardt, V. Heuckeroth, P. Thomas, P. Haring, G. Maidorn, H. Bakker, K. Leo, D. Kim, J. Shah, and K. Köhler, “Polarization dependence of heavy- and light-hole quantum beats” Phys. Rev. B 46, 10460–10463 (1992). [CrossRef]

], ZnSe[4

4. T. Saiki, M. Kuwata-Gonokami, K. Ohkawa, and T. Mitsuyu, “Role of biexciton state in excitonic resonant nonlinearity in homoepitaxial ZnSe” Solid State Commun. 95, 679–683 (1995). [CrossRef]

], CdS[5

5. D. Weber, W. Petri, U. Woggon, C.F. Klingshirn, S. Shevel, and E.O. Göbel, “Quantum beats of polaritons in hexagonal CdS crystals” Phys. Rev. B 55, 12848–12851 (1997). [CrossRef]

], GaN[6

6. R. Zimmermann, M.R. Hofmann, A. Euteneuer, J. Möbius, D. Weber, W.W. Rühle, E.O. Göbel, B.K. Meyer, H. Amano, and I. Akasaki, “Quantum beat spectroscopy on excitons in GaN” ICAM’97/E-MRS’97, Strasbourg(1997), L-XII.3.

,7

7. A.J. Fischer, W. Shan, G.H. Park, J.J. Song, D.S. Kim, D.S. Yee, R. Horning, and B. Goldenberg, “Femtosecond four-wave-mixing studies of nearly homogeneously broadened excitons in GaN” Phys. Rev. B 56, 1077–1080 (1997). [CrossRef]

] and many more have already been reported. In those experiments, it has been observed that the phase of the quantum beats shifts by π if the incident polarization is changed from collinear to crossed linear. To explain this phase shift, Schmitt-Rink et al. proposed a six-level model based on the band structure of the semiconductor, and applied the optical Bloch equations (OBE) to this model[1

1. S. Schmitt-Rink, D. Bennhardt, V. Heuckeroth, P. Thomas, P. Haring, G. Maidorn, H. Bakker, K. Leo, D. Kim, J. Shah, and K. Köhler, “Polarization dependence of heavy- and light-hole quantum beats” Phys. Rev. B 46, 10460–10463 (1992). [CrossRef]

]. The calculated FWM signal intensity as a function of time delay τ in the two polarization configurations showed the π-phase shift. However, in their experimental results, they observed not only a phase shifts, but also a decay rate and signal intensity change which they could not reproduce in their calculations. Saiki et al. calculated the FWM signal intensity using a third-order perturbational approach in which they took exciton-exciton interactions into account [2

2. T. Saiki, M. Kuwata-Gonokami, T. Matsusue, and H. Sakaki, “Photon echo induced by two-exciton coherence in a GaAs quantum well” Phys. Rev. B 49, 7817–7820 (1994). [CrossRef]

]. The model predicts different decay rates for the two polarization configurations if inhomogeneous transitions are assumed. This was the case for the sample used by Schmitt-Rink et al. and fully explains their experimental results. However, if the transitions are homogeneously broadened, the theory by Saiki et al. predicts the same decay rate for the two polarization configurations. Quantum beats of homogeneously broadened excitons have already been observed by A.J. Fischer et al.[7

7. A.J. Fischer, W. Shan, G.H. Park, J.J. Song, D.S. Kim, D.S. Yee, R. Horning, and B. Goldenberg, “Femtosecond four-wave-mixing studies of nearly homogeneously broadened excitons in GaN” Phys. Rev. B 56, 1077–1080 (1997). [CrossRef]

], but, even though they also present polarization dependent FWM experiments, their data still leaves ambiguities with respect to the decay times for both polarization cases. This makes it impossible to judge the validity of suggested model by Saiki et al. In this paper, we focus on the polarization dependence of the quantum beats in a high quality GaN epitaxial layer with homogeneously broadened exciton transitions. We analize our data qualitatively on the basis of the model by Saiki et al. and obtain good agreement.

2. Experiments

Wurtzite GaN epitaxial layers with a thickness of 100μm are grown by vaper-phase epitaxy on a sapphire substrate[8

8. A. Usui, H. Sunakawa, A. Sakai, and A.A. Yamaguchi, “Thick GaN Epitaxial Growth with Low Dislocation Density by Hydride Vapor Phase Epitaxy” Jpn. J. Appl. Phys. 36, L899–L902 (1997). [CrossRef]

] and the substrate is removed after growth. Fig. 1(a) shows the reflection spectrum near the bandgap taken at 5K. The resonances of the A-exciton n=1 state at 356.1nm and B-exciton n=1 state at 355.5nm are clearly observed. The oscillator strength of the A-exciton transition and the B-exciton transition are almost equal as can be judged from the modulation depth. These resonances are sharp and deep, which suggests that the quality of the sample is high. At shorter wavelengths the spectrum shows three additional resonances. We attribute the resonances at 354.1nm to the A-exciton n=2 state, at 353.5nm to the B-exciton n=2 state, and at 353.3nm to the C-exciton n=1 state, respectively. Assuming hydrogen like energy states for the exciton we find an exciton binding energy of 26 meV in this sample. The sample is mounted in a closed-cycle helium cryostat and kept at 14K in all the following measurements.

We use a Kerr-lens mode-locked Ti-sapphire laser with 200fs pulse width and 76MHz repetition rate. The second harmonic of the laser is generated with an LBO crystal yielding pulses with a center wavelength of 355.7nm and a full width at half maximum of 2nm and an average power of 6mW. The laser spectrum plotted in Fig.1(b) shows that both the A-exciton and B-exciton are excited simultaneously.

Fig. 1. (a) The reflection spectrum of our GaN sample. Each label for a resonance indicates the corresponding transition. (b) The laser spectrum as used for excitation in the FWM experiments.

In order to generate the FWM signal, we use the two-pulse self-diffraction geometry[9

9. T. Yajima and Y. Taira, “Spatial optical parametric coupling of picosecond light pulses and transverse relaxation effect in resonant media” J. Phys. Soc. Jpn. 47, 1620–1623 (1979). [CrossRef]

] in reflection. Two pulses with wavevectors k 1 and k 2 reach the sample at times 0 and τ, respectively. The FWM signal in the direction 2k 1 - k 2 is measured as a function of time delay τ between the two pulses with positive time delay being defined as pulse k 2 arriving first. The signal is detected using a photo-multiplier tube and measured with lock-in technique. Double chopping is used to reference the lock-in amplifier.

The zero position of τ is determined by monitoring both signal directions, 2k 1-k 2 and 2k 2-k 1, which are equal yet time reversed with respect to τ. The symmetry point of the two traces is defined as zero. By examining the excitation-power dependence of the signal intensity we confirm that all measurements are done strictly in the third-order regime.

3. Results and discussion

Fig.2 shows the FWM signal in the direction 2k 1 - k 2 for (a) collinear and (b) crossed linear polarization. Clear beats are observed in both polarization configurations. The dynamic range well exceeds two orders of magnitude and the modulation depth is almost constant with time delay except in regions where the signal is comparable to the noise. In the collinear polarization configuration, the beats start with a maximum at τ = 0. On the other hand, in the crossed linear polarization configuration, the beats start with a minimum. The signal in these two polarization configurations is π out of phase. We measure a decay time of 0.67ps in the collinear polarization configuration and 0.60ps in the crossed linear polarization configuration. These two values lie within 11% of each others and should be regarded as equal.

Fig. 2. The FWM signals in the direction 2k 1 - k 2 as a function of time delay τ between the pulse k 1 and pulse k 2 for collinear (a) and crossed linear (b) incident polarizations. Positive time delay is defined as pulse k 2 arriving first.

This polarization dependent phase shift of the quantum beats has previously been observed in GaAs quantum wells. Schmitt-Rink et al. proposed a six-level model describing the J=12 electrons and J=32 holes, and applied the optical Bloch equations to this model[1

1. S. Schmitt-Rink, D. Bennhardt, V. Heuckeroth, P. Thomas, P. Haring, G. Maidorn, H. Bakker, K. Leo, D. Kim, J. Shah, and K. Köhler, “Polarization dependence of heavy- and light-hole quantum beats” Phys. Rev. B 46, 10460–10463 (1992). [CrossRef]

]. The obtained beats in the FWM signal change their phase by π when changing the incident polarizations. In their experimental results, the phase of the beats changed as predicted in their calculations similar to our measurements. A.J. Fischer et al. also observed a π phase shift of the quantum beats in GaN [7

7. A.J. Fischer, W. Shan, G.H. Park, J.J. Song, D.S. Kim, D.S. Yee, R. Horning, and B. Goldenberg, “Femtosecond four-wave-mixing studies of nearly homogeneously broadened excitons in GaN” Phys. Rev. B 56, 1077–1080 (1997). [CrossRef]

], however, their results show a maximum at time delay zero for the crossed polarization configuration and a minimum in the collinear case which is exactly opposite to the calculations by Schmitt-Rink et al. and our measurements. Since the electronic structure of GaN is similar to that of GaAs no deviation from the predictions for GaAs should be expected. This difference in experimental findings may be due to an arbitrary assignment of time delay zero. We carefully determine time zero by monitoring both FWM directions and assign zero to the crossing point of the two traces as mentioned earlier.

In the results of Schmitt-Rink et al. the signal intensity and decay rate did also change when changing the incident polarizations, which can not be explained with their model. As a single particle model, interactions among and between electrons and holes are not taken into account. Saiki et al. explained how the FWM signal changes its decay rate and intensity from the excitonic point of view[2

2. T. Saiki, M. Kuwata-Gonokami, T. Matsusue, and H. Sakaki, “Photon echo induced by two-exciton coherence in a GaAs quantum well” Phys. Rev. B 49, 7817–7820 (1994). [CrossRef]

]. They considered the ground state, the one-exciton state, and the two-exciton state in their model. The two-exciton state involves a stable biexciton state, its excited states, and a repulsive interaction for unbound two exciton states. Within this model the decay rate does change for the case of an inhomogeneously broadened system when changing the incident polarization configurations, but does not change for the case of a homogeneously broadened system. In our GaN sample, the exciton transition is homogeneously broadened and the same decay rate for both polarization configurations is observed different to the earlier reported cases of GaAs quantum wells[1

1. S. Schmitt-Rink, D. Bennhardt, V. Heuckeroth, P. Thomas, P. Haring, G. Maidorn, H. Bakker, K. Leo, D. Kim, J. Shah, and K. Köhler, “Polarization dependence of heavy- and light-hole quantum beats” Phys. Rev. B 46, 10460–10463 (1992). [CrossRef]

, 2

2. T. Saiki, M. Kuwata-Gonokami, T. Matsusue, and H. Sakaki, “Photon echo induced by two-exciton coherence in a GaAs quantum well” Phys. Rev. B 49, 7817–7820 (1994). [CrossRef]

].

In the collinear polarization configuration only the ground state and the one exciton state contribute dominantly to the FWM process similar to the two level model. If we neglect two exciton states, no signal is expected in the crossed linear configuration due to selection rules. However, in the excitonic model biexciton states exist and can be excited from the one exciton state. The interaction among the excitons results in the additional nonlinearity that again allows for FWM signal. This is different from the model by Schmitt-Rink et al. Out of four two-particle states they exclude two (same helicity states) in their calculations which corresponds to an artificial interaction between the included states (opposite helicity states) and therefore to a high nonlinearity. This results in FWM signal for the crossed linear polarization configuration, which is, however, more or less a coincidence of the model arising from the truncation of the possible two-particle states. Because the origin of the FWM signal is not correctly accounted for, the model by Schmitt-Rink et al. cannot reproduce the different signal intensities in both polarization configurations. In the model by Saiki et al. on the other hand, the processes that are responsible for the generation of the FWM signal are fundamentally different for both polarization configurations and may result in different signal amplitudes. However, both cases are equivalent up to the point when the second pulse arrives at the sample. Only the second pulse carries the information of the utilized polarization configuration since it may be either parallel or perpendicular in polarization with respect to the first pulse. Furthermore, it is well known that in the case of a homogeneously broadened transition the signal is emitted in form of a free induction decay right after the second pulse arrives at the sample whereas the signal pulse in the inhomogeneous case is emitted in form of a photon echo at a time τ after the second pulse. This means, that since in the homogeneous case the signal is emitted right after the second pulse has reached the sample only the dephasing of the polarization generated by the first pulse contributes to the decay of the FWM signal. Since this is equivalent for both polarization configurations the same decay rate should be expected. In the inhomogeneous case, however, the system is in two fundamentally different states from the arrival of the second pulse at time τ up to the emission of the signal pulse at time 2τ for the two polarization configurations as pointed out earlier. During this time, in the collinear configuration only the dephasing of the one exciton state contributes to the decay of the FWM signal whereas in the crossed linear configuration also the dephasing of the two exciton states contributes. Since this dephasing is generally faster than for the one exciton state a faster decay for the crossed linear polarization configuration as compared to the collinear configuration has to be expected. This explanation describes the experimental observations by Schmitt-Rink et al. and our own experimental results well. The observed decay time of 0.67ps corresponds to a homogeneous width of 1meV which is much narrower than the previously reported value.

4. Conclusions

In summary, we have studied the polarization dependence of quantum beats in GaN for homogeneously broadened excitons in the third-order regime. We observe a π-phase shift and the same decay rate when changing the incident polarizations from collinear to crossed linear. Our experimental results confirm previously reported theoretical predictions.

Acknowledgments

We are grateful to H. Sunakawa and A. Usui of NEC Corporation for providing the sample. This work was supported by the CREST (Core Research for Evolutional Science and Technology) program of the Japan Science and Technology Corporation (JST).

Footnotes

* also with: CREST, Japan Science and Technology Corporation (JST)

References

1.

S. Schmitt-Rink, D. Bennhardt, V. Heuckeroth, P. Thomas, P. Haring, G. Maidorn, H. Bakker, K. Leo, D. Kim, J. Shah, and K. Köhler, “Polarization dependence of heavy- and light-hole quantum beats” Phys. Rev. B 46, 10460–10463 (1992). [CrossRef]

2.

T. Saiki, M. Kuwata-Gonokami, T. Matsusue, and H. Sakaki, “Photon echo induced by two-exciton coherence in a GaAs quantum well” Phys. Rev. B 49, 7817–7820 (1994). [CrossRef]

3.

T.F. Albrecht, K. Bott, A. Schulze, M. Koch, S.T. Cundiff, J. Feldmann, W. Stolz, P. Thomas, S. W. Koch, and E. O. Göbel, “Disorder mediated biexcitonic beats in semiconductor quantum wells” Phys. Rev. B 54, 4436–4439 (1996). [CrossRef]

4.

T. Saiki, M. Kuwata-Gonokami, K. Ohkawa, and T. Mitsuyu, “Role of biexciton state in excitonic resonant nonlinearity in homoepitaxial ZnSe” Solid State Commun. 95, 679–683 (1995). [CrossRef]

5.

D. Weber, W. Petri, U. Woggon, C.F. Klingshirn, S. Shevel, and E.O. Göbel, “Quantum beats of polaritons in hexagonal CdS crystals” Phys. Rev. B 55, 12848–12851 (1997). [CrossRef]

6.

R. Zimmermann, M.R. Hofmann, A. Euteneuer, J. Möbius, D. Weber, W.W. Rühle, E.O. Göbel, B.K. Meyer, H. Amano, and I. Akasaki, “Quantum beat spectroscopy on excitons in GaN” ICAM’97/E-MRS’97, Strasbourg(1997), L-XII.3.

7.

A.J. Fischer, W. Shan, G.H. Park, J.J. Song, D.S. Kim, D.S. Yee, R. Horning, and B. Goldenberg, “Femtosecond four-wave-mixing studies of nearly homogeneously broadened excitons in GaN” Phys. Rev. B 56, 1077–1080 (1997). [CrossRef]

8.

A. Usui, H. Sunakawa, A. Sakai, and A.A. Yamaguchi, “Thick GaN Epitaxial Growth with Low Dislocation Density by Hydride Vapor Phase Epitaxy” Jpn. J. Appl. Phys. 36, L899–L902 (1997). [CrossRef]

9.

T. Yajima and Y. Taira, “Spatial optical parametric coupling of picosecond light pulses and transverse relaxation effect in resonant media” J. Phys. Soc. Jpn. 47, 1620–1623 (1979). [CrossRef]

OCIS Codes
(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing
(320.7150) Ultrafast optics : Ultrafast spectroscopy

ToC Category:
Focus Issue: Coherent Phenomena in Solids

History
Original Manuscript: September 30, 1997
Published: December 8, 1997

Citation
Takao Aoki, Georg Mohs, Takeshi Ogasawara, Ryo Shimano, Makoto Kuwata-Gonokami, and A. Yamaguchi, "Polarization dependent quantum beats of homogeneously broadened excitons," Opt. Express 1, 364-369 (1997)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-1-12-364


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References

  1. S. Schmitt-Rink, D. Bennhardt, V. Heuckeroth, P. Thomas, P. Haring, G. Maidorn, H. Bakker, K. Leo, D. Kim, J. Shah, and K. Kohler, "Polarization dependence of heavy- and light-hole quantum beats" Phys. Rev. B 46, 10460-10463 (1992). [CrossRef]
  2. T. Saiki, M. Kuwata-Gonokami, T. Matsusue, and H. Sakaki, "Photon echo induced by two-exciton coherence in a GaAs quantum well" Phys. Rev. B 49, 7817-7820 (1994). [CrossRef]
  3. T.F. Albrecht, K. Bott, A. Schulze, M. Koch, S.T. Cundi, J. Feldmann, W. Stolz, P. Thomas, S. W. Koch, and E. O. Gobel, "Disorder mediated biexcitonic beats in semiconductor quantum wells" Phys. Rev. B 54, 4436-4439 (1996). [CrossRef]
  4. T. Saiki, M. Kuwata-Gonokami, K. Ohkawa, and T. Mitsuyu, "Role of biexciton state in excitonic resonant nonlinearity in homoepitaxial ZnSe" Solid State Commun. 95, 679-683 (1995). [CrossRef]
  5. D. Weber, W. Petri, U. Woggon, C.F. Klingshirn, S. Shevel, and E.O. Gobel, "Quantum beats of polaritons in hexagonal CdS crystals" Phys. Rev. B 55, 12848-12851 (1997). [CrossRef]
  6. R. Zimmermann, M.R. Hofmann, A. Euteneuer, J. Mobius, D. Weber, W.W. Ruhle, E.O. Gobel, B.K. Meyer, H. Amano, and I. Akasaki, "Quantum beat spectroscopy on excitons in GaN" ICAM'97/E-MRS'97, Strasbourg(1997), L-XII.3.
  7. A.J. Fischer, W. Shan, G.H. Park, J.J. Song, D.S. Kim, D.S. Yee, R. Horning, and B. Goldenberg, "Femtosecond four-wave-mixing studies of nearly homogeneously broadened excitons in GaN" Phys. Rev. B 56, 1077-1080 (1997). [CrossRef]
  8. A. Usui, H. Sunakawa, A. Sakai, and A.A. Yamaguchi, "Thick GaN Epitaxial Growth with Low Dislocation Density by Hydride Vapor Phase Epitaxy" Jpn. J. Appl. Phys. 36, L899-L902 (1997). [CrossRef]
  9. T. Yajima and Y. Taira, "Spatial optical parametric coupling of picosecond light pulses and transverse relaxation eect in resonant media" J.Phys. Soc. Jpn.47, 1620-1623 (1979). [CrossRef]

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