Semiconductor devices have become indispensable for generating electromagnetic radiation in every day applications. Visible and infrared diode lasers are at the core of information technology, and at the other end of the spectrum, microwave and radio frequency emitters enable wireless communications. But the ultrafast electromagnetic waves, whose frequency locates in terahertz (THz) region (0.3 – 30 THz; 1 THz = 1012 Hz), has remained largely underdeveloped, despite the identification of various possible applications. One of the major applications of THz spectroscopy systems is in material characterization, particularly of lightweight molecules and semiconductors  . Furthermore, THz imaging systems may find important niche applications in security screening and manufacturing quality control  - . An important goal is the development of three dimensional (3-D) tomographic T-ray imaging systems. THz systems also have broad applicability in a biomedical context, such as the T-ray biosensor . A simple biosensor has been demonstrated for detecting the glycoprotein avidin after binding with vitamin H (biotin) .
However, progresses in these areas have been hampered by the lack of efficient ultrafast electromagnetic wave / THz wave sources. As shown in Fig. 1, transistors and other electronic devices based on electron transport are limited to about ~ 300 GHz (~ 50 GHz being the rough practical limit; devices much above that are extremely inefficient) . On the other hand, the wavelength of semiconductor lasers can be extended down to only ~ 10 μm (about ~ 30 THz) . Between two technologies, lie the so called terahertz gap, where no semiconductor technology can efficiently convert electrical power into electromagnetic radiation. The lack of a high power, low cost, portable room temperature THz source is the most significant limitation of modern THz systems. A number of different mechanisms have been exploited to generate THz radiation, such as photocarrier acceleration in photoconducting antennas, second order nonlinear effects in electro-optical (EO) crystals and quantum cascade laser. Currently, conversion efficiencies in all of these sources are very low, and consequently, average THz beam powers tend to be in the nanowatt to microwatt range, whereas the average power of the femtosecond optical source is in the region of ~ 1 W. There is still a long way to go before commercial devices based on this principle can be mass-produced.
Photoconduction is one of the most common approaches for generating broadband pulsed ultrafast electromagnetic beams/THz beams. The photoconductive approach uses high speed photoconductors as transient current sources for radiating antennas . Typical photoconductors include high resistivity GaAs, InP and radiation damaged silicon wafers. Metallic electrodes are used to bias the photoconductive gap and form an antenna. The physical mechanism for THz beam generation in photoconductive antennas begins with an ultrafast laser pulse (with a photon energy larger than the bandgap of the material
Figure 2 shows the principle of generating THz radiation from antenna by photoconduction method. Several material parameters affect the intensity and the bandwidth of the resultant THz radiation. For efficient THz radiation, it is desirable to have rapid photocurrent rise and decay time. Thus, semiconductors with small effective electron masses such as InAs and InP are attractive. The maximum drift velocity is also an important material parameter, but it is generally limited by the intraband scattering rate or by intervalley scattering in direct semiconductors such as GaAs  . Because the radiating energy mainly comes from stored surface energy in the form of the static bias field, the THz radiation energy scales up with the bias and optical fluency. The breakdown field of the material is another important parameter because this determines the maximum bias that may be applied. Photoconductive emitters are capable of relatively large average THz powers in excess of ~ 40μW  and bandwidths as high as ~ 4 THz .
As described above, impulsive currents in semiconductors excited by femtosecond optical pulses radiate ultrafast electromagnetic waves in the THz frequency range, serving, for example, as an ultrafast electromagnetic wave source for THz spectroscopy. As described in Eq. (1), the ultrafast electromagnetic wave in the far field is proportional to the time derivative of the current, ∂
The novel experimental technique of the time domain THz spectroscopy for observing ultrafast carrier motion in the femtosecond time regime together with electromagnetic wave radiation that is proportional to d
2. Time domain terahertz spectroscopy by electro-optic sampling method
Time domain terahertz (THz) spectroscopy affords a powerful technique for the research of semiconductors of current industrial interest ; the time domain studies of the emitted ultrafast wave / THz wave from various semiconductor structures provide a unique way of looking directly into the temporal as well as spatial evolution of the excited carriers on the sub-picosecond time scale. In this section, time domain THz electro-optic (EO) sampling via the Pockels effect is discussed in detail, including the EO effect, the response spectrum of the EO crystal, and the experimental setup for EO sampling.
Recently, the technique called time domain THz spectroscopy has been attracting wide interest. Time domain THz spectroscopy uses short pulses of broadband THz radiation, which are typically generated using ultrafast laser pulses. This technique has been developed from a work in 1980’s at AT&T Bell Labs and IBM T. J. Watson Research Center  . Historically, freely propagating THz pulses were measured by means of either photoconductive antennas   or far infrared interferometic techniques using incoherent detectors such as bolometers  . Although the photoconductive antennas have excellent sensitivity, their frequency response is limited by resonant behavior of the Hertzian dipole structure. For the interferometric techniques, the sensitivity is far worse than that of the photoconductive antennas, because the measurement is coherent and its sensitivity is ultimately limited by the thermal background. In addition, bolometers usually require liquid helium cooling.
On the other hand, ultrafast EO sampling has been widely used in the measurement of local transient electric fields in materials  . There existed a need to extend the local electric fields measurement to free space. In 1995, three groups reported their first results using free space EO sampling independently, nearly at the same time -. Although the preliminary results were very poor, rapid progress has been made in the intervening years. It turns out that free space EO sampling is a powerful tool for THz pulse measurement, providing many advantages, such as high sensitivity, ultrabroad frequency response, ease of use, and parallel measurement capability -.
Ultrafast carrier motion in the femtosecond time regime accompanies electromagnetic wave radiation, the electric field component of which is proportional to d
2.1. Electro-optic effect
EO sampling is a phase sensitive detection technique of electromagnetic radiation which measures a birefringence in an EO crystal induced by the incident electromagnetic radiation. This birefringence in the crystal is probed as a phase shift of an optical probe beam. In the time resolved detection of THz pulse, the transient birefringence is probed with the optical pulses at different time delays.
The principle of EO sampling can be explained as follows. Suppose that the probe beam is propagating in the z-direction and
where Δ = Γ0+Γ is the phase difference between the
The static phase term Γ0 (or called the optical bias) is often set equal to
In the two beams, the signals have the same magnitudes but opposite signs. For balanced detection, the difference between
This signal is proportional to the THz induced phase change, Γ, and, Γ, in turn, is proportional to the electric field of the THz pulse, ETHz. For a (110)-oriented ZnTe crystal, the following relation holds:
2.2. The response spectrum of the electro-optic crystal
For a transient THz pulse, phase matching should be considered. When the probe pulse has a group velocity different from that of a THz pulse (so-called GVM), the probe does not always sample the same position on the THz pulse. Instead, it scans across the THz pulse as the two propagating through the crystal, leading to broadening of the measured THz waveform. Furthermore, the bandwidth of EO crystal is limited by reststrahlen band of the crystal. To estimate frequency response and upper limit of the EO sensor, we performed a calculation of the theoretical response function of EO crystals, which shows that the importance of properly accounting for the dispersion of the EO coefficient and the GVM.
The complete frequency response function R(
After a probe pulse and THz wave co-propagate through a sensor of thickness d, the accumulated GVM time is
Then, the amplitude and its phase of the EO modulation of the optical probe pulse induced by the THz wave is proportional to
The group refractive index
The transverse optical (TO) phonon and longitudinal optical (LO) phonon energies, and the lattice damping are denoted by
The Faust-Henry coefficient
The full complex response function
Figure 5 displays the calculated spectra of |
2.3. Time domain terahertz spectroscopy system
Figure 6 shows the setup for free space THz EO sampling measurements. The ulrafast laser pulse is split by a beam splitter into two beams: a pump beam (strong) and a probe beam (weak). The pump beam illuminates the sample and generates the THz radiation. The generated THz radiation is a short electromagnetic pulse with a duration on the order of one picosecond, so the frequency is in the THz range. The THz beam is focused by a pair of parabolic mirrors onto an EO crystal. The beam transiently modifies the index ellipsoid of the EO crystal via the Pockels effect, as discussed in detail in section 2.1. The linearly polarized probe beam co-propagates inside the crystal with THz beam and its phase is modulated by the refractive index change induced by the THz electric field, ETHz. This phase change is converted to an intensity change by a polarization analyzer (Wollaston prism). A pair of balanced detectors is used to suppress the common laser noise. This also doubles the measured signal. A mechanical delay line is used to change the time delay between the THz pulse and the probe pulse. The waveform of ETHz can be obtained by scanning this time delay and performing a repetitive sampling measurement. The principle of THz signal sampled by the femtosecond pulse is shown in Fig. 7. By this sampling method, we do not need the instruments for high speed measurement because the THz signals are converted to the electrical signals after the sampling by the femtosecond probe pulse. To increase the sensitivity, the pump beam is modulated by a mechanical chopper and ETHz induced modulation of the probe beam extracted by a lock-in amplifier.
3. Femtosecond acceleration of carriers in bulk GaAs
Ultrafast nonequilibrium transport of carriers in semiconductors biased at high electric fields is of fundamental interest in semiconductor physics. From a fundamental point of view, the detailed understanding of the femtosecond dynamics of carriers in an electric field is a key issue in many body physics. As an example, one may ask whether the well-developed semiclassical picture of electronic conduction breaks down on ultrafast time scales and at very high electric fields . Furthermore, clarifying carrier dynamics under extremely nonequilibrium conditions is also strongly motivated by the need to obtain information relevant for the design of ultrahigh speed devices. Indeed, transit times even less than 1 ps have been reported for compound semiconductor field effect transistors . In such ultrashort channel transistors, carriers experience very few scattering events in the channel and drift in a very nonstationary manner as schematically illustrated in the Fig. 8. Consequently, the performance of such ultrafast transistors is not mainly determined by the steady-state properties, such as saturation velocities and nobilities, but is governed by the nonstationary carrier transport subjected to high electric fields. It is, therefore, essential to understand nonequilibrium transport of carriers subjected to high electric fields in such devices. However, it has been difficult to characterize such very fast phenomena by using conventional electronics, such as sampling oscilloscopes, because of their limited bandwidth. Consequently, Monte Carlo calculations have been the only tool for discussing transient carrier transport.
So far, only a few experimental studies have been done on the femtosecond carrier acceleration under nonequilibrium in compound semiconductor   . Leitenstorfer, et al. presented the first experimental results on the ultrafast electromagnetic wave / terahertz wave emission from electrons and ions in GaAs and InP accelerated by very high electric fields and showed firm experimental evidence of velocity overshoot of electrons in the femtosecond time range  . They separated the contributions of electrons and lattices, and determine the transient acceleration and velocity of carriers with a time resolution in the order of 10 fs. The maximum velocity overshoots and traveling distances during the nonequilibrium regime have been determined for the first time.
In this section, we have investigated nonequilibrium acceleration of carriers in bulk GaAs subjected to very high electric fields by time domain THz spectroscopy . It is found that THz emission waveforms have a bipolar feature; i.e., an initial positive peak and a subsequent negative dip. This feature arises from the velocity overshoot. The initial positive peak has been interpreted as electron acceleration in the bottom of Γ the valley in GaAs, where electrons have a light effective mass, while the subsequent negative dip has been attributed to intervalley transfer from the Γ valley to the X and L valleys.
m-i-ndiode & short channel GaAs m-i-ndiode
Undoped bulk GaAs sample grown by molecular beam epitaxy was used. Sample #1 had an
3.2. Femtosecond acceleration of carriers in GaAs
m-i-ndiode & short channel GaAs m-i-ndiode
Femtosecond laser pulses from a mode locked Ti: sapphire laser operated at a repetition rate of 76 MHz was used for time domain THz spectroscopy. The full width at half maximum (FWHM) of spectral bandwidth of the femtosecond laser pulses was approximately 20 meV. The central photon energies of the light pulses were set to 1.422 eV at 300 K and 1.515 eV at 10 K, in such a way that electrons were created near the bottom of the conduction band, as well as holes near the top of the valence band. The free space EO sampling technique was used to record temporal waveforms of THz electric fields emitted from the samples  , as described in chapter 2. The EO sensor used in this experiment was a 100 μm-thick (110)-oriented ZnTe crystal. The spectral bandwidth of this sensor was approximately 4 THz  , as shown in Fig. 5 in section 2.2. The corresponding resolution is ~ 250 fs.
Figures 9 show temporal waveforms of the THz electric fields emitted from samples #1 (an
From the Maxwell equations,
THz electric field,
As shown in Fig. 9, the leading edge of the
Taking advantage of the novel experimental method, invaluable information on nonequilibrium carrier transport in the femtosecond time range, which has previously been discussed only by numerical simulations, has been obtained. The present insights on the nonstationary carrier transport contribute to better understanding of device physics in existing high speed electron devices and, furthermore, to new design of novel ultrafast electromagnetic wave oscillators.
4. Power dissipation spectrum under step electric field in Bulk GaAs in terahertz region
It is well known that negative differential conductivities (NDCs) due to intervalley transfer appear in many of the compound semiconductors, such as GaAs, under high electric fields. NDCs are of practical importance, notably for its exploitation in microwave and ultrafast electromagnetic wave / THz wave oscillators . However, since such NDC gain has a finite bandwidth, it gives intrinsic upper frequency limit to ultrafast electromagnetic wave oscillators . For this reason, a number of works have been done to investigate the mechanism, which limits the bandwidth of the gain, and found that it is mainly controlled by the energy relaxation time -. Figure 11 shows the real and imaginary parts of the differential mobility spectra, Re[
The time domain THz spectroscopy has provided us with a unique opportunity to observe the motion of electron wave packet in the sub-picosecond time range and inherently measuring the response of the electron system to the applied bias electric field . The Fourier spectra of the THz emission give us the power dissipation spectra under step electric field in THz range, from which we can find the gain region of material in THz range.
In this section, we present the measured THz radiation from the intrinsic bulk GaAs under strong biased electric field. The power dissipation spectra under step electric field in THz range have been obtained by using the Fourier transformation of the time domain THz traces  . From the power dissipation spectra, the cutoff frequency for negative power dissipation of the bulk GaAs has also been found. It is found that the cutoff frequency for the gain gradually increases with increasing electric fields up to 50 kV/cm and saturates at around 1 THz at 300 K / 750 GHz at 10 K. We also investigated the temperature dependence of cutoff frequency for negative power dissipation, from which we find that this cutoff frequency is governed by the energy relaxation process of electrons from the L valley to the Γ valley via successive optical phonon emission.
4.1. Power dissipation spectrum under step electric field in GaAs
Firstly, we recall the fact that the time domain THz emission experiments inherently measure the step response of the electron system to the applied electric field, as described in more detail in (Shimad et al., 2003). In the THz emission experiment, we first set a DC electric field,
By noting this important implication, the power dissipation spectra under step-function-like electric fields in THz range can be obtained from the thought experimental scheme, as shown in the following;
In time domain, the power density is defined as, i.e., power density equals the product of the current density,
in the linear response regime. However, this formulation which use small signal conductivity,
Mathematically, the power dissipation,
From simple mathematics,
As mentioned in section 4.3, the creation of step-function-like carrier density by femtosecond laser pulses in the actual experiment can be replaced with the application of step-function-like electric field in the thought experiment scheme, then, the
Then, we can obtain an important message from Eq. (21) that the real and imaginary part of the Fourier spectra of
From the discussion above, we know from the measured temporal waveforms of THz electric fields emitted from GaAs samples shown in Figs. 9, the power dissipation spectra in intrinsic bulk GaAs under step-function-like electric fields can be determined by using Fourier transformation of
In the case of collisions with phonons in the Γ valley, each collision redirects the momentum and restrains the drift velocity. Furthermore, intervalley scattering from the Γ valley to the L valley requires phonon with a large wave vector, and group theoretical selection rules show that the longitudinal optical (LO) phonon is the only one allowed because the L valley minimum lies on the edge of the Brillioun zone. Because the probability of LO phonon emission is proportional to <NLO+1> (NLO is the thermal equilibrium phonon population per unit volume;), which is temperature dependent, we expect the scattering time is shorter at higher temperatures. The calculation result by Ruch and Fawcett’s also showed different magnitude of drift velocity at low and high temperatures .
Furthermore, to investigate the power dissipation spectra under extremely large electric fields, we measured the time domain THz traces emitted from sample #2, an
4.2. The cutoff frequency for negative power dissipation in GaAs
4.2.1. Electric field dependence of the cutoff frequency
From the real part of the power dissipation spectra, we can find the cutoff frequencies for the gain region,
4.2.2. Temperature dependence of the cutoff frequency
To clarify the mechanism of the cutoff frequency,
4.3. Mechanism for the cutoff frequency
When a small step-function electric field is added on a large DC electric field, the electron velocity is predominantly redirected, which causes a general increase of the kinetic energy, or a heating, of the electron system until the mean collision rate with the lattice has risen sufficiently to balance the increasing supply of energy to the electrons from the electric field. The characteristic time for this thermal readjustment is known as the energy relaxation time, which is the limiting process for the upper frequency limit of transferred electron devices . However, the mechanism for the cutoff frequency for gain,
4.3.1. Energy relaxation and momentum relaxation times govern
Monte Carlo simulation has been used as a tool to estimate
4.3.2. Relaxation process of electrons in the L valley governs
Another model is that the energy relaxation time is governed by relaxation process of electrons in the L valley . The electrons start to be accelerated at the bottom of the Γ valley, scattered into the L valley, relax and dwell there, and, then, scattered back to the Γ valley. In this model,
4.3.3. Relaxation process of electrons in the Γ valley governs
The temperature dependence of
For this order estimation, we assume electrons are ballistically accelerated by the electric field
The energy relaxation process in the Γ valley is dominated by the LO phonon scattering. Since the LO phonon energy is 36.5 meV, 8 LO phonons must be successively emitted in the process for electrons to relax from the bottom of the L valley to the bottom of Γ valley (
By using Eq. (23), the energy relaxation time in the Γ valley is estimated to be 1.46 ps at 300 K and 1.95 ps at 10 K, by summing up emission times of 8 LO phonons. The temperature dependence of the energy relaxation time is due to the temperature dependence of the phonon emission process, <NLO+1>. The estimated cutoff frequency,
As shown in Fig. 15, the overall trend of this estimation of
Under very high electric fields, Wannier Stark (WS) ladder may be formed, although it has never been observed experimentally. The cyclic motion of electrons in the k space can be expressed in the real space as Fig. 18 (b). Electrons are created in the WS ladder states immediately after short pulsed photoexcitation under DC electric field
Based on our experimental data, we conclude that the cutoff frequency is governed by the energy relaxation process of electrons in the Γ valley via successive optical phonon emission. In our estimation, we neglect scattering while electrons are accelerated by the electric field in the band #1. This scattering in the acceleration process makes the sharp kink in the cutoff frequency,
where the momentum relaxation time of electrons in band #1 (Γ valley) is about
At 50 kV/cm, l is calculated to be ~ 60 nm, almost the same as the mean free path of electrons. Therefore, we can conclude that when
The other possible explanation is that the strong band mixing gives the sharp kink in
In summary, the power dissipation spectra under step electric field in bulk GaAs in THz range have been investigated by Fourier transformation of the ultrafast electromagnetic waveforms/THz waveforms emitted from intrinsic bulk GaAs photoexcited by femtosecond laser pulses under strong bias electric fields. The cutoff frequency for the gain region, that is of practical importance, notably for its exploitation in ultrafast electromagnetic wave generators, has been also found. The cutoff frequency gradually increases with increasing electric field within 50 kV/cm, saturates and reaches 1 THz (300 K) above 50 kV/cm. Furthermore, illustrated by the electric and temperature dependence of the cutoff frequency, we also find out that this cutoff frequency is mainly dominated by the energy relaxation process of electrons from the L to the Γ valley via successive optical phonon emission.
Ultrafast electromagnetic waves emitted from semiconductors under high electric fields, which are closely related with ultrafast nonequilibrium transport of carriers in semiconductor, are of fundamental interest in semiconductor physics. Furthermore, clarifying carrier dynamics under extreme nonequilibrium conditions is also strongly motivated by the need to obtain information relevant for the design of ultrahigh speed devices and ultrafast electromagnetic wave emitters. It is, therefore, essential to understand nonequilibrium transport of carriers subjected to high electric fields in such devices. The time domain terahertz (THz) spectroscopy gives us a unique opportunity of observing motions of electron wave packets in the sub-ps time range and measuring the response of electron systems to applied bias electric fields. By understanding the nonequilibrium transport of carriers in bulk GaAs, the intrinsic property of the negative differential conductivity (NDC) in the THz region, which means the gain, can be clarified.
In section 2, we describe the experimental technique of the time domain THz spectroscopy used in this work. Ultrafast carrier motion in the femtosecond time regime accompanies electromagnetic wave radiation that is proportional to d
In section 3, the femtosecond acceleration of carriers in bulk GaAs under very high electric fields,
In section 4, we have investigated the gain due to intervalley transfer under high electric fields, which is of practical importance for its exploitation in ultrafast electromagnetic wave oscillators. The power dissipation spectra under step-function-like electric fields in THz range have been obtained by using the Fourier transformation of the time domain THz traces. From the power dissipation spectra, the cutoff frequencies for negative power dissipation (i.e., gain) in bulk GaAs have been determined. The cutoff frequency gradually increases with increasing electric field,
Finally, section 5 summarizes this chapter. As demonstrated in this chapter, taking advantage of the novel experimental method, invaluable information on nonequilibrium carrier transport in the femtosecond time range, which has previously been discussed only by numerical simulations, has been obtained. The present insights on the nonstationary carrier transport contribute to better understanding of device physics in existing high speed electron devices. Furthermore, the gain in GaAs due to electrons intervalley transfer under high electric fields obtained from the Fourier transformation of the time domain THz traces is also discussed, which is of practical importance for its exploitation in ultrafast electromagnetic wave oscillators.
The authors thank Mr. JunAn Zhu, a student from the University of Shanghai for Science and Technology, for editing the manuscript of this chapter.