## Carrier density dependence of the nonlinear absorption of intense THz radiation in GaAs |

Optics Express, Vol. 20, Issue 16, pp. 18016-18024 (2012)

http://dx.doi.org/10.1364/OE.20.018016

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### Abstract

We study the carrier density dependence of nonlinear terahertz (THz) absorption due to field-induced intervalley scattering in photoexcited GaAs using the optical-pump/THz-probe technique. The intervalley scattering in GaAs is strongly dependent on the photo-carrier density. As the carrier density is increased from 1 × 10^{17} to 4.7 × 10^{17} cm^{−3}, the nonlinear absorption bleaching increases. However, if the carrier density is increased further above 4.7 × 10^{17} cm^{−3}, the trend reverses and the bleaching is reduced. The initial increase in absorption bleaching is because, unlike low THz field, high THz field experiences intervalley scattering and nonparabolicity of the conduction band. On the other hand, a simple electron transport model shows that the reduction in intervalley scattering is mainly due to the increase in the electron-hole scattering rate with the increase in the carrier density. This increase in the electron-hole scattering rate limits the maximum kinetic energy attainable by the electrons and thus reduces the observed nonlinear absorption.

© 2012 OSA

## 1. Introduction

1. A. G. Markelz, N. G. Asmar, B. Brar, and E. G. Gwinn, “Interband impact ionization by terahertz illumination of InAs heterostructures,” Appl. Phys. Lett. **69**(26), 3975–3977 (1996). [CrossRef]

6. B. E. Cole, J. B. Williams, B. T. King, M. S. Sherwin, and C. R. Stanley, “Coherent manipulation of semiconductor quantum bits with terahertz radiation,” Nature **410**(6824), 60–63 (2001). [CrossRef] [PubMed]

7. P. Gaal, K. Reimann, M. Woerner, T. Elsaesser, R. Hey, and K. H. Ploog, “Nonlinear terahertz response of n-type GaAs,” Phys. Rev. Lett. **96**(18), 187402 (2006). [CrossRef] [PubMed]

13. A. G. Stepanov, L. Bonacina, S. V. Chekalin, and J.-P. Wolf, “Generation of 30 μJ single-cycle terahertz pulses at 100 Hz repetition rate by optical rectification,” Opt. Lett. **33**(21), 2497–2499 (2008). [CrossRef] [PubMed]

14. P. Gaal, W. Kuehn, K. Reimann, M. Woerner, T. Elsaesser, R. Hey, J. S. Lee, and U. Schade, “Carrier-wave Rabi flopping on radiatively coupled shallow donor transitions in n - type GaAs,” Phys. Rev. B **77**(23), 235204 (2008). [CrossRef]

7. P. Gaal, K. Reimann, M. Woerner, T. Elsaesser, R. Hey, and K. H. Ploog, “Nonlinear terahertz response of n-type GaAs,” Phys. Rev. Lett. **96**(18), 187402 (2006). [CrossRef] [PubMed]

7. P. Gaal, K. Reimann, M. Woerner, T. Elsaesser, R. Hey, and K. H. Ploog, “Nonlinear terahertz response of n-type GaAs,” Phys. Rev. Lett. **96**(18), 187402 (2006). [CrossRef] [PubMed]

14. P. Gaal, W. Kuehn, K. Reimann, M. Woerner, T. Elsaesser, R. Hey, J. S. Lee, and U. Schade, “Carrier-wave Rabi flopping on radiatively coupled shallow donor transitions in n - type GaAs,” Phys. Rev. B **77**(23), 235204 (2008). [CrossRef]

15. H. Wen, M. Wiczer, and A. M. Lindenberg, “Ultrafast electron cascades in semiconductors driven by intense femtosecond terahertz pulses,” Phys. Rev. B **78**(12), 125203 (2008). [CrossRef]

16. M. C. Hoffmann, J. Hebling, H. Y. Hwang, K.-L. Yeh, and K. A. Nelson, “Impact ionization in InSb probed by terahertz pump-terahertz probe spectroscopy,” Phys. Rev. B **79**(16), 161201 (2009). [CrossRef]

17. F. Blanchard, D. Golde, F. H. Su, L. Razzari, G. Sharma, R. Morandotti, T. Ozaki, M. Reid, M. Kira, S. W. Koch, and F. A. Hegmann, “Effective mass anisotropy of hot electrons in nonparabolic conduction bands of n-doped InGaAs films using ultrafast terahertz pump-probe techniques,” Phys. Rev. Lett. **107**(10), 107401 (2011). [CrossRef] [PubMed]

18. L. Razzari, F. H. Su, G. Sharma, F. Blanchard, A. Ayesheshim, H. C. Bandulet, R. Morandotti, J. C. Kieffer, T. Ozaki, M. Reid, and F. A. Hegmann, “Nonlinear ultrafast modulation of the optical absorption of intense few-cycle terahertz pulses in n - doped semiconductors,” Phys. Rev. B **79**(19), 193204 (2009). [CrossRef]

21. M. C. Hoffmann and D. Turchinovich, “Semiconductor saturable absorbers for ultrafast terahertz signals,” Appl. Phys. Lett. **96**(15), 151110 (2010). [CrossRef]

19. F. H. Su, F. Blanchard, G. Sharma, L. Razzari, A. Ayesheshim, T. L. Cocker, L. V. Titova, T. Ozaki, J. C. Kieffer, R. Morandotti, M. Reid, and F. A. Hegmann, “Terahertz pulse induced intervalley scattering in photoexcited GaAs,” Opt. Express **17**(12), 9620–9629 (2009). [CrossRef] [PubMed]

22. G. Sharma, L. Razzari, F. H. Su, F. Blanchard, A. Ayesheshim, T. L. Cocker, L. V. Titova, H. C. Bandulet, T. Ozaki, J. C. Kieffer, R. Morandotti, M. Reid, and F. A. Hegmann, “Time-resolved terahertz spectroscopy of free carrier nonlinear dynamics in semiconductors,” IEEE Photon. J. **2**(4), 578–592 (2010). [CrossRef]

## 2. Experimental set-up

10. F. Blanchard, L. Razzari, H. C. Bandulet, G. Sharma, R. Morandotti, J. C. Kieffer, T. Ozaki, M. Reid, H. F. Tiedje, H. K. Haugen, and F. A. Hegmann, “Generation of 1.5 µJ single-cycle terahertz pulses by optical rectification from a large aperture ZnTe crystal,” Opt. Express **15**(20), 13212–13220 (2007). [CrossRef] [PubMed]

*i.e.*by placing two silicon wafers after the sample to reduce the THz electric field impinging onto the detection crystal. Wire-grid polarizers are used to vary the intensity of the THz pulse at the sample position. To evaluate the THz electric field, we have used the same method reported in Ref [24

24. H. G. Roskos, M. D. Thomson, M. Kreß, and T. Löffler, “Broadband THz emission from gas plasmas induced by femtosecond optical pulses: From fundamentals to applications,” Laser Photon. Rev. **1**(4), 349–368 (2007). [CrossRef]

*E*is the THz peak electric field,

_{0}*η*is the free-space impedance (377 Ω),

_{0}*W*is the THz energy,

*ω*is the intensity beam waist, and

_{I}*g*(

*t*) is the temporal shape of the THz electric field (with a peak value normalized to 1), which can be easily retrieved from the EO sampling measurements. The THz intensity beam waist was measured to be 0.6 mm by imaging the THz beam at the focus using a pyroelectric IR camera (ElectroPhysics, model PV320). The peak THz energy (

*W*) is measured to be 0.53 μJ, using a Microtech Instrument pyroelectric detector. Substituting all the parameters into Eq. (1), the maximum THz electric field is evaluated to be 133 kV/cm. In our OPTP experiments, we have used two conditions for the THz probe beam, one with “high” THz fields at 133 kV/cm, and the other with “low” THz fields at 9 kV/cm.

^{−2}to 40 μJ.cm

^{−2}. From this parameter, the carrier density is estimated to range between 1 × 10

^{17}and 10.6 × 10

^{17}cm

^{−3}[25

25. F. A. Hegmann and K. P. Lui, “Optical pump-terahertz probe investigation of carrier relaxation in radiation-damaged silicon-on-sapphire,” Proc. SPIE **4643**, 31–41 (2002). [CrossRef]

## 3. Results and discussions

### 3.1 Effect of carrier density on nonlinear THz absorption bleaching

*Γ*valley in the conduction band in the normally insulating GaAs sample. Figure 2 shows the normalized transmission of the main peak of the THz probe pulse through optically pumped GaAs as a function of the pump-probe delay time. The nonlinear response is then determined by monitoring the peak THz electric field of the pulse. When the optical pump and the THz probe overlap in time (at a pump-probe delay of 0 ps in Fig. 2), the THz transmission through the photoexcited GaAs sample is reduced, since the sample becomes more conductive due to the increased carrier density. However, even for large pump-probe delay times (a few hundreds of picoseconds), the normalized transmission at 133 kV/cm (

*E*) is greater than that of the 9 kV/cm peak field THz pulse (

_{high}*E*). This phenomenon has been shown to be due to the intervalley scattering between the central

_{low}*Γ*and the

*L*valley of the conduction band [18

18. L. Razzari, F. H. Su, G. Sharma, F. Blanchard, A. Ayesheshim, H. C. Bandulet, R. Morandotti, J. C. Kieffer, T. Ozaki, M. Reid, and F. A. Hegmann, “Nonlinear ultrafast modulation of the optical absorption of intense few-cycle terahertz pulses in n - doped semiconductors,” Phys. Rev. B **79**(19), 193204 (2009). [CrossRef]

20. J. Hebling, M. C. Hoffmann, H. Y. Hwang, K.-L. Yeh, and K. A. Nelson, “Observation of nonequilibrium carrier distribution in Ge, Si, and GaAs by terahertz-pump terahertz probe measurements,” Phys. Rev. B **81**(3), 035201 (2010). [CrossRef]

*L*valley compared to the

*Γ*valley, and thus reduces the mobility and conductivity of these carriers. The result is a bleaching of the pump-induced absorption at high THz probe field strengths, observed previously by F. Su

*et al.*[19

19. F. H. Su, F. Blanchard, G. Sharma, L. Razzari, A. Ayesheshim, T. L. Cocker, L. V. Titova, T. Ozaki, J. C. Kieffer, R. Morandotti, M. Reid, and F. A. Hegmann, “Terahertz pulse induced intervalley scattering in photoexcited GaAs,” Opt. Express **17**(12), 9620–9629 (2009). [CrossRef] [PubMed]

*E*and

_{pump}*E*are the transmitted THz electric field through the photoexcited (

_{ref}*pump*) and unexcited (

*ref*) sample, and

*T*and

_{high}*T*are the normalized transmission at high (133 kV/cm) and low (9 kV/cm) THz fields, respectively. Figure 3 shows the THz induced nonlinearity for a carrier density ranging from 1.0 × 10

_{low}^{17}to 10.6 × 10

^{17}cm

^{−3}. It can be clearly seen from Fig. 3 that the absorption bleaching varies significantly with carrier density.

^{17}cm

^{−3}), the THz-induced nonlinearity increases with increasing carrier density from 1 × 10

^{17}cm

^{−3}onward. For the range of carrier densities we have investigated, absorption bleaching reaches a maximum at a carrier density of 4.7 × 10

^{17}cm

^{−3}; (ii) for carrier densities from 4.7 × 10

^{17}cm

^{−3}to 10.6 × 10

^{17}cm

^{−3}(the maximum carrier density under investigation), absorption bleaching decreases monotonically. For the maximum value of the carrier density (10.6 × 10

^{17}cm

^{−3}), absorption bleaching almost vanishes completely.

### 3.2 Theoretical interpretation

*Γ*valley to the

*L*valley [18

18. L. Razzari, F. H. Su, G. Sharma, F. Blanchard, A. Ayesheshim, H. C. Bandulet, R. Morandotti, J. C. Kieffer, T. Ozaki, M. Reid, and F. A. Hegmann, “Nonlinear ultrafast modulation of the optical absorption of intense few-cycle terahertz pulses in n - doped semiconductors,” Phys. Rev. B **79**(19), 193204 (2009). [CrossRef]

19. F. H. Su, F. Blanchard, G. Sharma, L. Razzari, A. Ayesheshim, T. L. Cocker, L. V. Titova, T. Ozaki, J. C. Kieffer, R. Morandotti, M. Reid, and F. A. Hegmann, “Terahertz pulse induced intervalley scattering in photoexcited GaAs,” Opt. Express **17**(12), 9620–9629 (2009). [CrossRef] [PubMed]

*L*valley is over 10 times less than in the central

*Γ−*valley. Therefore, a high-field THz pulse that is able to induce intervalley scattering of electrons into the

*L*valley will effectively reduce the conductivity in the photoexcited GaAs, thus increasing the transmitted electric field when compared with the low-field case. This results in THz free carrier absorption bleaching for sufficiently high THz electric fields. The change in the bleaching from an increasing to decreasing dependence beyond 4.7 × 10

^{17}cm

^{−3}suggests the onset of a competing scattering process governed by the carrier density.

26. P. N. Saeta, J. F. Federici, B. I. Greene, and D. R. Dykaar, “Intervalley scattering in GaAs and InP probed by pulsed far-infrared transmission spectroscopy,” Appl. Phys. Lett. **60**(12), 1477–1479 (1992). [CrossRef]

26. P. N. Saeta, J. F. Federici, B. I. Greene, and D. R. Dykaar, “Intervalley scattering in GaAs and InP probed by pulsed far-infrared transmission spectroscopy,” Appl. Phys. Lett. **60**(12), 1477–1479 (1992). [CrossRef]

28. W. Walukiewicz, L. Lagowski, L. Jastrzebski, M. Lichtensteiger, and H. C. Gatos, “Electron mobility and free-carrier absorption in GaAs: Determination of the compensation ratio,” J. Appl. Phys. **50**(2), 899–908 (1979). [CrossRef]

*τ*(

_{Γ}*low*) and hence the electron mobility [26

26. P. N. Saeta, J. F. Federici, B. I. Greene, and D. R. Dykaar, “Intervalley scattering in GaAs and InP probed by pulsed far-infrared transmission spectroscopy,” Appl. Phys. Lett. **60**(12), 1477–1479 (1992). [CrossRef]

**17**(12), 9620–9629 (2009). [CrossRef] [PubMed]

*Γ− L*intervalley scattering is used to describe the effect of carrier density on absorption bleaching in GaAs. THz transmission through the sample can be idealized as a thin conducting sheet with thickness

*d*on an insulating substrate with index

*N*, and can be expressed as follows [23]:Here,

*E*and

_{trans}*E*are the transmitted and incident THz fields,

_{inc}*Y*and

_{0}= 1/377Ω^{−1}*Y*are the free-space and substrate admittances, respectively, and

_{s}= NY_{0}*J = (n*is the current density in the film. Furthermore,

_{Γ}eν_{Γ})*e*is the electronic charge,

*n*is the electron density and

*ν*is the electron drift velocity. The electron drift velocity (

*ν*) in the

_{Γ}*Γ*valley driven by the transmitted THz field

*E*can be described by the dynamic equation:In the above equation, the Drude scattering time,

_{trans}*τ*, is dependent on the carrier density in the

_{Γ}*Γ*valley. As the carrier density increases, e-h scattering reduces

*τ*for the low field case. This in turn limits the maximum drift velocity achieved by electrons at higher carrier densities. In our simulation,

_{Γ}*τ*is a fit parameter and is obtained from the fits to the nonlinear experimental data.

_{Γ}*Γ*and

*L*valleys is determined by the intervalley scattering rates. The

*L−Γ*transfer rate (

*τ*)

_{LΓ}^{−1}is kept constant [29

29. K. Blotekjaer, “Transport equations for electrons in two-valley semiconductors,” IEEE Trans. Electron. Dev. **17**(1), 38–47 (1970). [CrossRef]

*Γ− L*scattering rate (

*τ*)

_{Γ L}^{−1}is a function of the average kinetic energy (

*ε*) [30

_{Γ}30. A. M. Anile and S. D. Hern, “Two-valley hydrodynamical models for electron transport in gallium arsenide: Simulation of Gunn oscillations,” VLSI Des. **15**(4), 681–693 (2002). [CrossRef]

*ε*= [

_{Γ}*m*

_{Γ}***(

*ν*)

_{Γ}*+*

^{2}*3k*]

_{B}T_{L}*/2*is the average kinetic energy of the electrons in the

*Γ*valley. The scattering time

*τ*is zero at low energies but starts to increase rapidly at a threshold value

_{Γ L}*ε*to a maximum value

_{th}*τ*at high energies. The nonparabolicity of the conduction band is taken into account by changing the effective mass for high THz fields, given by the following equation:Here,

_{Γ L0}*α*= 0.61 is the nonparabolicity factor for the

_{Γ}*Γ*valley in GaAs and

*m** = 0.067

_{Γ0}*m*is the effective mass at the bottom of the conduction band [27, 31]. For the simulations, the “high” and “low” THz fields used are again 133 kV/cm and 9 kV/cm, respectively. We have used this model to simulate the experimental observation, as shown in Fig. 3 (red solid line). The simulated results match well with the experimental finding. The parameters used in these simulations are shown in Table 1 .

_{e}*J*in Eq. (3), and hence modifies the transmitted field

*E*. Now the current density depends on the drift velocity, and the drift velocity depends not only on the transmitted THz electric field but also on the electron and hole population. As the population increases, the momentum component parallel to the THz E-field vector is randomized due to e-h scattering, limiting the maximum achievable velocity and hence the kinetic energy of the electrons. As the carrier density increases further, the e-h scattering rate increases approximately linearly with a logarithmic correction [32, 33

_{trans}33. J. F. Young, P. J. Kelly, N. L. Henry, and M. W. C. Dharma-Wardana, “Carrier density dependence of hot-electron scattering rates in quasi-equilibrium electron-hole plasmas,” Solid State Commun. **78**(5), 343–346 (1991). [CrossRef]

*τ*fit parameters, for the low THz field strengths (9 kV/cm) used in our simulations. Black squares in Fig. 4 shows the extracted mobility as a function of the carrier density. As previously discussed, it can be seen in Fig. 4 that as we increase the carrier density, the overall mobility of the conduction band decreases. The mobility as a function of carrier density is empirically fitted using the Caughey-Thomas relation, shown as a solid black line in Fig. 4 and given by the following expression [34

_{Γ}34. M. C. Beard, G. M. Turner, and C. A. Schmuttenmaer, “Transient photoconductivity in GaAs as measured by time-resolved terahertz spectroscopy,” Phys. Rev. B **62**(23), 15764–15777 (2000). [CrossRef]

35. D. M. Caughey and R. E. Thomas, “Carrier mobilities in silicon empirically related to doping and field,” Proc. IEEE **55**(12), 2192–2193 (1967). [CrossRef]

*μ*is the mobility and

*N*is the carrier density. As expected, the calculated mobility is well described by the relation above [34

_{e}34. M. C. Beard, G. M. Turner, and C. A. Schmuttenmaer, “Transient photoconductivity in GaAs as measured by time-resolved terahertz spectroscopy,” Phys. Rev. B **62**(23), 15764–15777 (2000). [CrossRef]

*μ*and

_{max}*μ*are floating parameters chosen via the best fit of a straight line to the plot of

_{min}*log[(μ*versus

_{max}−μ)/(μ− μ_{min})]*log*(

*N*). The values of

_{e}*α*and

*N*(

_{e}*ref*) are then obtained from the slope and unity intercept of the straight line. The fitting values obtained for

*μ*and

_{max}*μ*are 7900 cm

_{min}^{2}/Vs and 1000 cm

^{2}/Vs, in good agreement with previous work [34

34. M. C. Beard, G. M. Turner, and C. A. Schmuttenmaer, “Transient photoconductivity in GaAs as measured by time-resolved terahertz spectroscopy,” Phys. Rev. B **62**(23), 15764–15777 (2000). [CrossRef]

*α*and

*N*(

_{e}*ref*) are obtained to be 0.29 and 5.25 × 10

^{16}cm

^{−3}, respectively. When the carrier density is increased from 1 × 10

^{17}to 10.6 × 10

^{17}cm

^{−3}, the carrier mobility decreases from 4010 cm

^{2}/Vs to 3055 cm

^{2}/Vs [36

36. M. C. Nuss, D. H. Auston, and F. Capasso, “Direct subpicosecond measurement of carrier mobility of photoexcited electrons in gallium arsenide,” Phys. Rev. Lett. **58**(22), 2355–2358 (1987). [CrossRef] [PubMed]

37. D. G. Cooke, F. A. Hegmann, E. C. Young, and T. Tiedje, “Electron mobility in dilute GaAs bismide and nitride alloys measured by time-resolved terahertz spectroscopy,” Appl. Phys. Lett. **89**(12), 122103 (2006). [CrossRef]

## 4. Conclusions

*Γ*valley to the low mobility

*L*valley. The intervalley scattering rate depends on the electron energy, which is limited by e-h scattering via the reduction of the electron mobility. Even if the intervalley scattering rate is lower (thus promoting population of the

*Γ*valley), the electron will still experience a reduction in mobility due to the nonparabolicity of the conduction band. Therefore, the THz transmission through the photoexcited GaAs layer is less attenuated for high fields as compared to low fields. This initially increases the THz induced absorption bleaching (when the carrier density rises from 1 × 10

^{17}to 4.7 × 10

^{17}cm

^{−3}). However, a further increase in the carrier density beyond 4.7 × 10

^{17}cm

^{−3}leads to a monotonically decreasing absorption bleaching as a function of density. This can be explained instead by the increased e-h scattering rate as a function of density, which in turn reduces the mobility significantly to the point where intervalley scattering becomes less likely and nonparabolicity effects (happening at the edge of the conduction band) can be ignored. Thus the high field transmission approaches the low field transmission and the absorption bleaching is reduced. Despite the simplicity of our model, we find that it explains the carrier density dependence of the nonlinear THz absorption rather well.

## Acknowledgment

## References and links

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21. | M. C. Hoffmann and D. Turchinovich, “Semiconductor saturable absorbers for ultrafast terahertz signals,” Appl. Phys. Lett. |

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29. | K. Blotekjaer, “Transport equations for electrons in two-valley semiconductors,” IEEE Trans. Electron. Dev. |

30. | A. M. Anile and S. D. Hern, “Two-valley hydrodynamical models for electron transport in gallium arsenide: Simulation of Gunn oscillations,” VLSI Des. |

31. | M. Grundmann, |

32. | D. G. Cooke, “Time-resolved terahertz spectroscopy of bulk and nanoscale semiconductors,” PhD dissertation, (Department of Physics, University of Alberta, 2007). |

33. | J. F. Young, P. J. Kelly, N. L. Henry, and M. W. C. Dharma-Wardana, “Carrier density dependence of hot-electron scattering rates in quasi-equilibrium electron-hole plasmas,” Solid State Commun. |

34. | M. C. Beard, G. M. Turner, and C. A. Schmuttenmaer, “Transient photoconductivity in GaAs as measured by time-resolved terahertz spectroscopy,” Phys. Rev. B |

35. | D. M. Caughey and R. E. Thomas, “Carrier mobilities in silicon empirically related to doping and field,” Proc. IEEE |

36. | M. C. Nuss, D. H. Auston, and F. Capasso, “Direct subpicosecond measurement of carrier mobility of photoexcited electrons in gallium arsenide,” Phys. Rev. Lett. |

37. | D. G. Cooke, F. A. Hegmann, E. C. Young, and T. Tiedje, “Electron mobility in dilute GaAs bismide and nitride alloys measured by time-resolved terahertz spectroscopy,” Appl. Phys. Lett. |

**OCIS Codes**

(300.6500) Spectroscopy : Spectroscopy, time-resolved

(320.7110) Ultrafast optics : Ultrafast nonlinear optics

(320.7130) Ultrafast optics : Ultrafast processes in condensed matter, including semiconductors

(300.6495) Spectroscopy : Spectroscopy, teraherz

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: April 27, 2012

Revised Manuscript: July 5, 2012

Manuscript Accepted: July 6, 2012

Published: July 23, 2012

**Citation**

G. Sharma, I. Al-Naib, H. Hafez, R. Morandotti, D. G. Cooke, and T. Ozaki, "Carrier density dependence of the nonlinear absorption of intense THz radiation in GaAs," Opt. Express **20**, 18016-18024 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-16-18016

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### References

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