Nonlinear transient absorption bleaching of intense few-cycle terahertz (THz) pulses is observed in photoexcited GaAs using optical-pump – THz-probe techniques. A simple model of the electron transport dynamics shows that the observed nonlinear response is due to TH-zelectric-field-induced intervalley scattering over sub-picosecond time scales as well as an increase in the intravalley scattering rate attributed to carrier heating. Furthermore, the nonlinear nature of the THz pulse transmission at high peak fields leads to a measured terahertz conductivity in the photoexcited GaAs that deviates significantly from the Drude behavior observed at low THz fields, emphasizing the need to explore nonlinear THz pulse interactions with materials in the time domain.
©2009 Optical Society of America
The far-infrared nonlinear optical properties of semiconductors have been the focus of much research over the past twenty-five years using primarily nanosecond to microsecond duration intense terahertz pulse sources and incoherent (i.e., total energy) detection methods [1-6]. For example, third-harmonic generation at terahertz (THz) frequencies in bulk semiconductors  and semiconductor nanostructures  has been previously reported, as well as terahertz-electric-field-induced impact ionization in InAs heterostructures  and dynamical Franz-Keldysh effects in excitonic absorption features in InGaAs quantum wells . Ultrafast pulse-slicer techniques have also been used to study Rabi oscillations in donor states in GaAs over picosecond time scales .
More recently, however, the development of intense, coherently-detected, few-cycle THz pulse sources [7-13] has led to a new and powerful tool for studying the nonlinear optical response of materials at terahertz frequencies over picosecond and sub-picosecond time scales [7,8,12,14-19]. For instance, coherent THz emission and Rabi oscillations in n-type GaAs [7,16], cross-phase modulation in electro-optic crystals [8,14], lattice anharmonicity and self-phase modulation in LiNbO3 , and THz-electric-field-induced impact ionization in InSb [17, 18] as well as intervalley scattering in n-doped InGaAs films using z-scan techniques  have been recently studied using intense, few-cycle THz pulses. In this paper, we use optical-pump – terahertz-probe (OPTP) techniques, which have been used extensively over the past 12 years to probe ultrafast carrier dynamics and transport in semiconductors [20-26], to study transient absorption bleaching of intense THz probe pulses in photoexcited GaAs.
In our OPTP experiments, a large aperture ZnTe optical rectification source  is used to generate high-power, few-cycle, THz probe pulses with energies of 0.6 µJ and a bandwidth extending from 0.1 to 3 THz. A 0.5 mm-thick SI-GaAs wafer is placed at the focus of the THz probe beam and is photoexcited with a collinear 800 nm, 50 fs, pump beam at an excitation fluence of 8 µJ/cm2. The spot size diameters on the sample for the THz probe beam and the optical pump beam are 1.6 mm and 12 mm, respectively. Free-space electro-optic sampling in a second ZnTe crystal is used to detect the THz pulse transmitted through the GaAs sample. Wire-grid polarizers were used in the THz detection scheme to keep the THz electric field within the linear detection regime at the detector crystal. Other wire-grid polarizers as well as Si wafers were used to adjust the amplitude of the THz probe pulse at the sample corresponding to “high THz field” and “low THz field” conditions studied here with peak electric fields of 173 kV/cm and 4 kV/cm, respectively, as estimated from the THz pulse energy and spot size at the sample. All the experiments were performed under a dry nitrogen purge at room temperature.
3. Results and discussion
3.1 Absorption bleaching
Figure 1 shows the normalized transmission of the main peak of the THz probe pulse as a function of pump-probe delay time, which is a common method for probing ultrafast carrier dynamics in semiconductors in the OPTP technique [24, 25]. The inset shows the long-term dynamics. The 800 nm pump pulse photoexcites electrons and holes in the normally insulating GaAs sample, with the electrons being injected into the higher mobility central Γ valley in the conduction band. At low THz probe fields, the transmission of the THz pulse through the photoexcited GaAs sample is approximately 60% of the transmission through the unexcited sample. However, at the higher THz probe field the relative change in transmission is greatly reduced compared to that observed in the low-field case, thus suggesting a bleaching of the THz pulse absorption at high THz fields.
Figure 2 shows the transmitted THz waveforms measured at a pump-probe delay time of 10 ps and normalized to the peak of the THz waveform transmitted through the unexcited sample at negative delay times. The absorption bleaching can be clearly seen in the relative amplitude of the transmitted THz waveform at a high THz field compared to that at low THz field. In particular, there is almost no change in electric-field amplitude or phase shift in the trailing portion of the THz waveform at high THz probe fields. We note that the observed absorption bleaching is inconsistent with an impact ionization mechanism, as it would produce an enhancement in THz pulse absorption [17, 18].
3.2 Intervalley scattering mechanism
We propose that the absorption bleaching observed at high THz probe fields is due in part to THz-pulse-induced intervalley scattering of electrons between the Γ and L valleys in the GaAs conduction band [27-29], as illustrated in Fig. 3 below. Since the mobility of electrons in the L valley is much less than that in the central Γ valley, a high-field THz probe pulse that is able to induce intervalley scattering will effectively see less conductivity in the photoexcited GaAs sample as compared to the low-field case where no intervalley scattering occurs. This results in an increase in transmission of the THz probe pulse at high THz fields, especially in the trailing part of the waveform after the main peak. Self-induced nonlinear absorption of intense THz pulses due to intervalley scattering has also been observed recently in time-resolved THz z-scan measurements in n-doped InGaAs . We note that optically-induced intervalley scattering is well-known in ultrafast pump-probe and OPTP experiments performed using higher pump photon energies (e.g., 400 nm pump pulses) that can excite carriers directly into the satellite valleys via optical phonon scattering [20-23,25,30]. In addition, the observation of electric-field-induced intervalley scattering has recently been reported in OPTP experiments on DC-biased GaAs wafers . Here we show intervalley scattering induced by the THz probe pulse itself in an OPTP experiment.
3.3 Dynamic intervalley-electron-transfer model
A simple model incorporating Γ-L intervalley scattering is used to describe the temporal dynamics of the observed nonlinear THz transmission in the photoexcited GaAs sample . The THz electric field, Et, transmitted through the thin conducting photoexcited layer of conductivity σ and thickness d can be expressed in terms of the incident field, Ei, and current density J as :
where Y0=(377 Ω)-1 and Ys=NY0 are the free-space and sample admittances, respectively, and N=3.6 is the index of refraction of GaAs at THz frequencies. (The optical penetration depth at 800 nm of 1 µm is used for the thickness d of the conducting layer.) Neglecting contributions to the conductivity from the lower mobility holes and L-valley electrons [27,28], the current density J=en ΓvΓ can be determined, where e is the electronic charge, n Γ is the Γ-valley electron density, and vΓ is the drift velocity. The electron motion driven by the transmitted THz field, Et, and the population of electrons in the Γ-valley can be described by the dynamic equations:
Where Γ τ and m*Γ denote the relaxation time and effective mass of the Γ-valley electrons, respectively, n0 is the total electron density, and τ -1 ΓL and τ -1 LΓ are the intervalley scattering rates from one valley to the other. The L-Γ valley transfer rate, τ -1 LΓ, is kept constant . The Γ-L scattering rate, τ -1 ΓL, is considered as a simplified function of the average kinetic energy, , associated to the electrons in the Γ valley :
where τ -1 ΓL0 represents the maximum Γ-L intervalley scattering rate and εth is a threshold energy with a range given by Δεth. (The lattice temperature T is set to room temperature in the calculations. Heating of the lattice by the optical pump pulse is less than 0.05 K and is therefore negligible.) Since Γ-L intervalley scattering takes place via the emission or absorption of optical phonons, the energy-dependent function of the Γ-L valley transfer rate, τ -1 ΓL(ε Γ), is tentatively made “smooth” in the energy range of 2Δεth=0.08eV, which is slightly larger than twice the optical phonon energy in GaAs (i.e., 2ħωLO=0.07eV ). The smooth function is inserted via a seventh-order polynomial section that is continuous up to the third derivative . The intervalley scattering rate as a function of the electron energy is shown in Fig. 4. Nonparabolic effects are also taken into account by assuming that the effective mass varies as m*Γ(ε Γ)=m*Γ0(1+a Γ ε Γ), where αΓ=0.61 is the nonparabolic parameter for the Γ valley in GaAs and m*Γ 0=0.067me is the effective mass of the electrons at the bottom of the conduction band [27,28].
The incident electric field, Ei, is determined from the THz pulse waveform transmitted through both interfaces of the unexcited GaAs sample and scaled to the appropriate electric field amplitude according to the measured THz pulse energy and spot size at the sample. The THz waveform, Et, transmitted through the first air-GaAs interface is then calculated numerically according to Eqs. (1)-(4). (For example, Eq. (1) gives Et=2/(1+N) Ei in unexcited GaAs.) The good agreement between the calculated and experimental transmitted THz waveforms at 10 ps after the photoexcitation induced by the pump pulse can be seen in Fig. 2, where the relative RMS deviation in both the low and high field cases is 7% . The total carrier density and intravalley relaxation time obtained from the fit to the low-field data in Fig. 2(a) are n0=(1.7±0.1)×1017 cm-3 and τ Γ=0.16±0.02 ps, respectively. This corresponds to a carrier mobility of 4200 cm2/Vs, which is consistent with other OPTP experiments in GaAs . This is also consistent with a total carrier density of about 1.9×1017 cm-3 estimated from the measured pump fluence incident on the sample and assuming a pump beam reflectance of about 40%. At high THz fields, the same value for n0 is used as in the low-field case, but the best fit to the observed THz transmission in Fig. 2(b) gives τ Γ=0.051±0.005 ps and requires intervalley scattering with τ ΓL0=0.022±0.015 ps and τLΓ=3.0±1.0 ps. If intervalley scattering is removed from the model calculation, the best fit RMS deviation in the high-field case increases from 7% to 11%. However, as shown in Fig. 2(c), the absolute deviation (or difference) between the experimental and calculated waveforms increases significantly immediately after the main THz peak if intervalley scattering is not included in the calculation. As shown in Fig. 2(b), the fraction of electrons left in the Γ valley is reduced significantly when the electric field of the THz probe pulse becomes high enough to induce intervalley scattering. The initial Γ-L population transfer occurs over a time scale of 77 fs (10%-90%), which is comparable to Γ-L intervalley scattering times of about 80 fs reported elsewhere . The transfer time back to the Γ valley of τ LΓ=3.0±1.0 ps also agrees well with the approximately 2 ps transfer times reported in other OPTP experiments [21,30]. The extracted threshold energy from the fitting is εth=0.16±0.01 eV, which is lower than the actual Γ-L valley separation of 0.29 eV. This obvious discrepancy in threshold energy may be attributed to thermal smearing effects related to the hot electron distribution in the Γ valley [23,27], which are not accounted for in our simple model. (We will discuss this point further in Section 3.4.) Table 1 below summarizes the best-fit parameters used in the dynamic intervalley-electron-transfer model.
Figure 5 shows the incident and transmitted THz waveforms, the corresponding drift velocities, and the total electron energies obtained from the model for both low and high THz probe fields. Note that the driving field, Et, in the low-field case in Fig. 5(a) has a peak value around 1 kV/cm, and is therefore insufficient to drive intervalley scattering, which typically requires fields in excess of 4 kV/cm in the steady-state [27-29]. In the high-field case presented in Fig. 5(d), however, the electrons are driven by the larger THz field to a maximum energy of 0.14 eV, which exceeds the minimum threshold for intervalley scattering in our model, as shown in Fig. 5(f). It is interesting to note that the intravalley relaxation time of τ Γ=0.051±0.005 ps extracted from the fit at high THz probe field is much shorter than the corresponding low-field value of 0.16±0.02 ps, which is likely due to heating of the electron distribution by the intense THz pulse and an increase in scattering from polar optical phonons . It should be mentioned that the high-field THz-pulse-induced increase of the Γ-valley intravalley scattering rate also contributes to the observed absorption bleaching. However, the dynamic Γ-L intervalley transfer dominates in the transmission of the tail of the THz pulse waveform, as shown in Fig. 2 (c) where the deviation from the experimental data exhibits a significant increase in the trailing portion of the waveform when intervalley scattering is switched off.
3.4 Comparison to velocity overshoot in GaAs
It is well known that velocity overshoot of electrons can take place under the application of a large DC electric field in GaAs, which is then followed by strong intervalley scattering over sub-picosecond time scales [27-29]. For example, the maximum velocity acquired by an electron in GaAs after the application of an electric field of 40 kV/cm is about 7×107 cm/s [28, 29]. This velocity is attained in a time of about 150 fs, after which intervalley scattering to the lower mobility L valley sets in and the average carrier velocity is greatly reduced. Using the same parameters we used in the dynamic intervalley-electron-transfer model to fit our THz data, a calculation of the average electron velocity <vΓ>=(n Γ/n0) vΓ upon application of a step-like 60kV/cm electric field revealed a pronounced velocity overshoot in a time less than 200 fs, as shown in Fig. 6, consistent with typical velocity overshoot phenomena in GaAs. This provides some justification for the low threshold value for intervalley scattering of εth=0.16 eV obtained from our fits, despite the fact that the Γ-L energy separation in GaAs is 0.29 eV. In fact, the energy associated with an electron with a velocity of 7 x 107 cm/s is only 0.1 eV, and yet this is where intervalley scattering is known to occur in GaAs. We emphasize again that this represents the average electron velocity, and thermal smearing due to carrier heating effects must be taken into account in a more detailed analysis. Nevertheless, our simple model seems to capture the essential physics of the transient absorption bleaching that we observe here when we illuminate photoexcited GaAs with intense THz pulses. We note that recent THz-pump-THz-probe measurements have also observed transient absorption bleaching in GaAs due to intervalley scattering .
3.5 The complex conductivity
where Epump(ω) and Eref(ω) are the Fourier transforms of the time-domain THz waveforms transmitted through the pumped and non-pumped GaAs, respectively. Figure 7 shows the complex conductivities extracted from the THz waveforms shown in Fig. 2. At low THz probe field, the observed conductivity spectrum is consistent with a Drude response [23-25], as shown in Fig. 7(a), with the same fit parameters for n0 and τ Γ given in Table 1. However, the high-field conductivity spectrum shown in Fig. 7(b) differs significantly from the Drude behavior observed at low fields, and in fact cannot be fitted to any conductivity model. This is simply due to the fact that the conductivity of the photoexcited layer in the GaAs sample is a nonlinear function of the THz probe field at high fields, and so the conductivity cannot be extracted by the simple Fourier analysis used in most OPTP experiments since it requires the conductivity to be independent from the amplitude of the electric field. This emphasizes the need to look at the temporal evolution of THz waveform rather than the frequency spectrum when probing materials at high peak THz fields. Figure 7(c) shows how the model fit becomes significantly less accurate if intervalley scattering is ignored.
In conclusion, we have observed absorption bleaching of intense THz pulses in photoexcited GaAs due to THz-field-induced intervalley scattering and intravalley carrier heating. This work also demonstrates the need to study nonlinear THz pulse interactions in the time domain rather than in the frequency domain.
We wish to acknowledge financial support from NSERC, NSERC Strategic Projects, and INRS. L. R. wishes to acknowledge a Marie Curie Outgoing International Fellowship (contract n. 040514). L. V. T. thanks the Avadh Bhatia Fellowship for financial support.
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