## 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 Optical Society of America

## 1. Introduction

Investigations of high-field charge transport in semiconductors using far-infrared radiation have been of significant interest for more than two decades, primarily because of the relevance to device physics [1–6]. With the recent development of laser-based intense, few-cycle terahertz (THz) sources and coherent detection techniques [7–13], it is now possible to study the nonlinear optical response of semiconductors at THz frequencies on picosecond (and even sub-picosecond) timescales. For example, long-lived coherent THz emission centered around 2 THz and carrier-wave Rabi oscillations [14] have been excited using intense THz radiation [7], due to the strong THz coupling to the impurity levels of n-type GaAs [7, 14]. THz electric-field-induced impact ionization in InSb [15, 16], and mapping of the effective mass anisotropy in the non-parabolic conduction band of an InGaAs thin film [17] have been reported using an intense THz pulse. Moreover, these sources have allowed the observation of nonlinear THz absorption bleaching due to intervalley scattering in InGaAs, GaAs, Si, and Ge using THz-pump/THz-probe (TPTP) or optical-pump/THz-probe (OPTP) based experiments [18–21].

Carrier dynamics in semiconductors can be monitored in the THz regime by either the OPTP or TPTP techniques. In the TPTP technique, one typically uses samples with fixed doping levels, and hence the free carrier density is also fixed. In OPTP, one can vary the free carrier density of the semiconductor sample by changing the fluence of the optical pump. This allows one to study the influence of carrier density on charge carrier dynamics, which is important for understanding and improving various optoelectronic devices.

Recently, nonlinear absorption bleaching of intense THz pulses has been observed in photoexcited GaAs using the OPTP technique [19]. To date, however, the effect of carrier density on this nonlinear THz effect remains unexplored. In this paper, we report the carrier-density dependent nonlinear THz response of a photoexcited GaAs sample illuminated with different pump fluences. As in previous work [22], we attribute the absorption bleaching to THz-field induced intervalley scattering and compare our experimental results with simulation results obtained using an intervalley-scattering-based Drude model.

## 2. Experimental set-up

The THz source used in this study is described in Ref [10]. Briefly, a 75 mm diameter large-aperture <110> ZnTe crystal is used as the nonlinear medium to generate high-power, few-cycle intense THz pulses via optical rectification, with a bandwidth extending from 0.1 to 2.8 THz. The temporal shape of the THz pulse is shown in Fig. 1 , with the corresponding spectrum presented in the inset.

In the current work, a standard OPTP technique [23] based on the use of the above intense THz pulse is employed to study the effect of carrier density on the THz-induced nonlinearities in GaAs. A 0.5 mm thick, undoped GaAs wafer is placed at the focus of the THz beam, and is photoexcited by an 800 nm, 50 fs pump beam with a diameter of 8 mm FWHM. Free-space electro-optic (EO) sampling in a second 0.5 mm thick <110> ZnTe crystal is used to detect the THz pulse transmitted through the GaAs sample. Detection linearity is maintained by keeping the maximum probe beam modulation measured at the two photodiodes well below polarization over-rotation, *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], based on the following relation:

*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.

A half-wave plate and a polarizer were used to vary the 800 nm optical pump fluence, ranging from ~2 μJ.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], assuming an absorption depth of 1 μm. In order to investigate the effect of carrier density on the THz-induced nonlinearity, we measured the nonlinear THz absorption bleaching of the optically pumped GaAs at both “high” and “low” THz field strengths.

## 3. Results and discussions

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

The 800 nm optical pump beam excites the electrons from the valence band to the central *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 _{high}*) is greater than that of the 9 kV/cm peak field THz pulse (

*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 [18–20], which increases the effective mass of a significant population of carriers in the

*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]

However, the effect of the carrier density on intervalley scattering has not been studied yet. Such effect can be quantified by defining the “THz induced absorption bleaching”, given as:

*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.

The absorption bleaching data in Fig. 3 can be divided into two carrier-density regions: (i) at relatively low carrier densities (< 4.7 × 10^{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

The difference between THz transmission at low and high THz fields has been explained in previous works and has been attributed to the intervalley scattering of electrons from the *Γ* valley to the *L* valley [18, 19]. In short, the mobility of electrons in the *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.

Previous linear THz spectroscopic studies have demonstrated that the density of the photo-injected electrons and holes strongly influences the mobility of carriers in GaAs [26]. Analogous to impurity scattering, electron-hole (e-h) scattering limits the electron mobility [26–28]. As the effective mass of a hole is much larger than that of an electron, e-h scattering is insufficient to transfer energy from an electron to a hole. However, e-h scattering is effective for momentum relaxation. This reduces the Drude scattering time at low THz fields *τ _{Γ}* (

*low*) and hence the electron mobility [26].

A simple Drude-based model [19], incorporating *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]:

*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:

_{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}We also note that the change in electron populations in the *Γ* and *L* valleys is determined by the intervalley scattering rates. The *L−Γ* transfer rate (*τ _{LΓ}*)

^{−1}is kept constant [29], while the

*Γ− L*scattering rate (

*τ*)

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

*ε*) [30], where

_{Γ}*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:

_{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}During the absorption bleaching process, the incident field accelerates the electrons in the conducting layer of the sample and induces a population transfer between the different valleys of the conduction band. This in turn affects the current density *J* in Eq. (3), and hence modifies the transmitted field *E _{trans}*. 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]. This has been taken into account in our model by decreasing the intravalley scattering rate in Eq. (4). The intervalley scattering rate depends on the energy of the electrons. If the kinetic energy of the electrons is reduced due to an increase in the e-h scattering, the intervalley scattering rate will also be reduced. This discussion suggests that absorption bleaching strongly depends on the charge carrier density.

As a self-consistency check in our simulation of the THz nonlinearity, we calculated the carrier mobility from the *τ _{Γ}* 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, 35].

*μ*is the mobility and

*N*is the carrier density. As expected, the calculated mobility is well described by the relation above [34], showing self-consistency in our modeling of the THz nonlinearity. The values of

_{e}*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]. The value of

*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, 37].

## 4. Conclusions

In conclusion, we have investigated the effect of carrier density on the nonlinear absorption bleaching of intense THz pulses transmitted through photoexcited GaAs. The increase in absorption bleaching with the increase in carrier density follows from the initial increase in conductivity of the GaAs sample. For high THz fields, however, the transmission is affected less by an increase in carrier density, as compared to the low field case. The main reason for the relatively lower decrease in transmission for high fields is a reduction in electron mobility, due to two reasons: nonparabolicity of the conduction band and intervalley scattering from the high mobility *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

We would like to acknowledge the financial support from Fonds de recherche du Québec - Nature et technology (FQRNT) and Natural Science and Engineering Research Council of Canada (NSERC).

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