Terahertz generation with ballistic photodiodes under pulsed operation

C Müller-Landau; S Malzer; H B Weber; G H Döhler; S Winnerl; P Burke; A C Gossard; S Preu

doi:10.1088/1361-6641/aae5e4

1. Introduction

High performance terahertz (THz) systems nowadays are used in many scientific and industrial applications. THz radiation allows wireless transmission with data rates of 100 Gbit s^–1 [1], spectroscopy and imaging of biological [2] and chemical [3] substances. It can be used for monitoring and quality control in food industry [4]. Broadband THz generation with femtosecond laser pulses in photoconductive semiconductors is a widely used method for these applications. Such systems have already been reported for a spectral range from 0.1 to 6.5 THz [5, 6]. However, typical time domain THz spectrometers (THz-TDS) only achieve a spectral resolution in the range of 1 to a few GHz [7, 8].

Electronic continuous-wave (CW) systems based on frequency-multiplied microwave sources [9] or quantum cascade lasers [10] offer excellent spectral resolution but only within a narrow frequency window. CW photomixing systems are the most promising approach for high spectral resolution with a wide tunability in the THz-frequency range [11–13]. The first photomixers have been made from low-temperature gallium arsenide (LT-GaAs) photoconductors operated at 800nm [14] with μW-level powers under CW operation. Nowadays, specially designed pin diodes for operation at telecom wavelengths like uni-travelling carrier (UTC) diodes [15], use electron transport only and achieve much higher power up to mW level [16, 17] in the sub-THz range (up to 300 GHz), owing to the optimized trade-off between RC-time and transit-time roll-off. At the same time, the n-i-pn-i-p superlattice concept was established [16]. In this approach, the transit time and the RC time can be optimized separately by using a stack of transit time optimized pin diodes. The resulting capacitance, responsible for the RC-time, can be controlled by the number of stacked pin-diodes, leading to an ideal trade-off frequency beyond 1 THz [11, 16, 17]. The key parameter for a short transit time (high 3 dB roll-off frequency) is a sufficiently short intrinsic layer allowing for ballistic transport. Ballistic (or quasi-ballistic) transport refers to a ballistic acceleration of electrons within the first few 100 fs to velocities much higher than the saturation velocity, before efficient scattering mechanisms (mainly side valley scattering) decelerate the electrons to the saturation velocity. This effect is also termed as 'velocity overshoot'. Holes are not expected to show any noticeable velocity overshoot. A thorough analysis of the ballistic electron transport in p-i-n diode based structures is required to approve the concept and for optimization.

Time-resolved (pulsed) THz spectroscopy has proven to be a powerful tool to study charge carrier dynamics in GaAs [18], time dependent field screening effects in small-size GaAs-devices [19] and in LT-GaAs [20, 21]. In photoconductive materials, the short life time, τ_rec < 1 ps is the reason that scattering dominates the transport and no velocity overshoot has been reported yet [12]. Contrary, in semi-insulating (SI)-GaAs or SI-InGaAs ballistic transport is evident and consequently velocity overshoot. Ballistic electron transport in GaAs THz-emitters has been studied by Monte-Carlo simulations [22] and pump and probe experiments [23]. Furthermore, it has been shown that GaAs/AlGaAs p-i-n heterostructures are efficient pulsed THz emitters, even though they were not designed for ballistic transport [24].

In this paper, we report on time-resolved ballistic transport studies which are compared to CW measurements and simulations. Pulse excitation can create a very high charge carrier density in the n-i-pn-i-p phototomixer and therefore allows to study THz power saturation behavior and suppression of ballistic transport. Although saturation occurs at fairly low optical powers, a high conversion efficiency even under pulsed operation could be demonstrated.

2. Simulation of the THz power generated by the photomixer

In order to investigate the saturation behavior of photodiodes and determine maximum current densities, we simulate the generation, motion and generated AC current of electrons and holes by a 1 D model. Subsequently, the THz power emitted by the device is calculated with the help of an equivalent circuit. The simulated device is a three-period n-i-pn-i-p superlattice photomixer [17] with an intrinsic layer comprising a 150 nm long linear Al-grading from In_0.53Ga_0.47As to In_0.53Al_0.08Ga_0.39As towards the n-contact, followed by a 50 nm In_0.53Al_0.08Ga_0.39As transport layer. The majority of the carriers is generated in the first ∼50 nm of the graded layer. At the interface of the diodes, a quasi-metallic ErAs-layer is implemented to ensure a high current density in the p–n-junction [25] and little loss of RF power. A schematic of the band structure is illustrated in figure 1(a). There are two key differences to other THz diode concepts, such as UTC diodes [15] or triple transit region diodes [26]: (I) the complete absorption region is within the intrinsic layer. All carriers are right away accelerated, no diffusion to the transport layer is necessary as in the case of UTC diodes with the absorption region composed of a p⁻-layer, decreasing the transit-time 3 dB frequency. (II) There are no abrupt band discontinuities which could give rise to excessive scattering and potentially also inter-valley transfer. Only a minor field kink in the range of 4 kV cm^–1 is inevitable at the intersection between the graded region and the region with constant Al-content as illustrated in figure 1(a).

**Figure 1.** (a) Band diagram of one period in real space. The increasing Al-content (not to scale) in the graded layer towards the n-contact limits the absorption (yellow arrow indicates the photon energy) to the beginning of the intrinsic layer. Black: ideal, unscreened band diagram. Red: screening by accumulated carriers in the structure. ErAs recombination diodes (grey) reduce the screening effect by effectively recombining electrons and holes (blue). (b) Band structure in k-space. A photon is absorbed, generating an electron–hole pair. The electron is firstly ballistically accelerated. At a kinetic energy of E_LO, LO phonon scattering becomes effective. Once the electron reaches the kinetic energy E_ΓL it scatters into one of the L-valleys.
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2.1. General setup of the simulation

The simulation considers only the 200 nm long intrinsic layer of the device, where transport takes place. It is discretized in δz = 2 nm steps. The temporal evolution of the simulation uses increments of δt = 5 fs, being a fraction of the smallest scattering times involved. First, carriers are generated (at rest) from the incident optical power, P_L(t) at each time step at the bottom of the conduction band or top of valence band for electrons and holes, respectively. For pulsed operation, we assume a temporal Gaussian distribution of the laser power with an FWHM of 100 fs as used in the experiment. For CW, the laser power is defined as ${P}_{j}={P}_{0}\cos (\omega {t}_{j})$ with a total laser power of P₀ and ω the difference frequency of the lasers, i.e. the THz frequency to be generated. The charge density at position x_i generated at time t_j then calculates to ${Q}_{{ij}}=\alpha ({x}_{i})\tfrac{{{eP}}_{L}({t}_{j})}{h\nu }\delta t\delta z$ . The absorption coefficient, α(x_i) depends on the local band gap and is estimated from experimental data of a bare In_0.53Ga_0.47As, shifted to shorter wavelengths by the respective change in band gap due to an increased Al-content of the quarternary In_0.53Al_xGa ${}_{0.47-x}$ As. The quarternary band gap as well as the empiric fit to the experimental absorption coefficient are summarized in equations (2) and (3) in the appendix. After generation, the distortion of the band structure due to charge separation is calculated. This leads to field kinks like illustrated in figure 1(a) at the blue line. For the initial time step, there is no distortion since electrons and holes are at identical positions. Next, the carriers propagate one time step in the distorted band structure according to the transport model described in detail in section 2.2. Carriers are removed from the simulation once they reach the n- or the p-contact. The total (displacement) current, j(t), is calculated by the $j(t)=\sum _{{ij}}{Q}_{{ij}}{v}_{{ij}}(t)$ . The simulation aborts after an elapsed time between 6 and 30 ps that is chosen sufficiently long for the required frequency resolution but short enough to reduce calculation time. The emitted THz power is calculated from the derived photocurrent following the formalism described in section 2.3. The simulation is then repeated for all investigated DC biases. In order to access the forward direction correctly, we estimate the diode parameters as well as the serial resistance, R_S = 20 Ω, from the diode IV-characteristics. In reverse direction, where most of the THz power is generated, a bias drop at the serial resistance is de facto irrelevant.

2.2. Carrier transport model

We simulate the carrier transport in the device by a simple ballistic 1D transport model where we only account for LO phonon scattering and abrupt inter-valley scattering for electron energies that reach the (effective) intervalley energy of ${E}_{{\rm{\Gamma }}L}^{{\rm{eff}}}=0.55\,\mathrm{eV}$ . Electrons start at zero velocity with purely ballistic transport, with an acceleration of $a={eE}/{m}^{* }$ , where E is the local electric field, e the elementary charge and m^* the effective mass. Phonon scattering has only little impact on the transport at room temperature in this time regime. Reaching the phonon energy of E_LO = 34 meV, electrons can emit LO phonons as illustrated in figure 1(b), but the scattering time for this process is in the ps-range. With higher electron energies phonon scattering becomes more pronounced. When electrons are accelerated up to the effective inter-valley energy, E_ΓL, the onset of inter-valley scattering is dominating the scattering processes. At this point, electrons find themselves in the (L-)valley at negligible kinetic energy. Here, acceleration starts again but with a substantially higher effective mass and a high rate for scattering into the equivalent side-valleys. This randomizes the direction of carrier motion and finally results in transport at the saturation velocity. For our simulation these processes are simplified. A few k_BT above the L-valley energy, E_ΓL = 0.5 eV, the scattering process is very effective, with scattering times in the range of a few 10 fs, except for extremely low accelerating fields where also only very little THz radiation is generated. Electrons are therefore assumed to decelerate instantaneously from the ballistic peak velocity (∼1.4 − 2 × 10⁸ cm s^–1), to the saturation velocity of 1.3 × 10⁷ cm s^–1 [27] once they reach a kinetic energy of ${E}_{{\rm{\Gamma }}L}^{{\rm{eff}}}=0.55\,\mathrm{eV}$ . In the L-valley, electron propagation is described by the Drude-model. We neglect carrier–carrier scattering and alloy scattering as their contributions are only secondary corrections as far as momentum relaxation times are concerned. Holes are considered by a Drude-like acceleration up to the saturation velocity of approximately 5 × 10⁶ cm s^–1 [28]. When the carriers arrive at the highly doped contacts, they experience strong scattering and thermalize on a femtosecond time scale. Therefore they are are no longer contributing to the simulation area.

As the n-i-pn-i-p superlattice diode consists of three p-i-n diodes (current sources) connected in series, we allow for a static bias U_st to built up due to charge accumulation between the diodes as the only fitting parameter. One period of the n-i-pn-i-p superlattice photomixer is illustrated in figure 1(a). A detailed description of the parameters used for the band structure, scattering times, effective masses, and velocities are summarized in the appendix.

2.3. Calculation of the emitted THz power

The AC current in the pin-diodes creates a field of ${E}_{{\rm{THz}}}(t)\sim \delta j(t)/\delta t$ [2] or in the frequency domain, ${E}_{{\rm{THz}}}(\omega )\sim {\rm{i}}\omega j(\omega )$ . The emitted THz-power is ${P}_{{\rm{THz}}}(\omega )\sim {\omega }^{2}{j}^{2}(\omega )$ . This equation holds for Hertzian dipoles only [29]. When antennas are used, it is more instructive (at least in the frequency domain) to use the concept of radiation resistance as the antenna may strongly alter the emitted spectrum. For the case of a Hertzian dipole, the radiation resistance is ${R}_{{\rm{rad}}}\sim {\omega }^{2}$ [29]. Replacing the Hertzian dipole radiation resistance by the radiation resistance of the antenna, the emitted power can be denoted as $P(\omega )=\tfrac{1}{2}{R}_{{\rm{rad}}}{j}^{2}(\omega ){A}^{2}$ , where A is the cross section of the device. In the experiment, we use a broadband, self-complementary logarithmic-periodic antenna that shows only little frequency dependence of the real part and almost negligible imaginary part, at least below 800 GHz where most of the power is measured. Above 800 GHz, the antenna performance decreases, confirmed by simulations. In contrast to the Hertzian dipole with a ${R}_{{\rm{rad}}}\sim {\omega }^{2}$ -dependence (originating from the time derivative), we consider for the logarithmic-periodic antenna a constant radiation resistance R_rad = 70 Ω where the emitted THz field is directly proportional to the photocurrent in the frequency domain. The generated THz spectra are filtered by the RC roll-off due to the antenna (R) and device capacitance (C) with an onset at ${f}_{3{\rm{dB}}}^{{RC}}=150\,\mathrm{GHz}$ . For pulsed measurements, a further high pass roll-off attributed to the electro-optic sampling (EOS) detection technique arises: the lower the THz frequency, the larger the THz spot and therefore the lower the overlap between THz and laser beam in the detection crystal. This results in a high pass behavior which is fitted to a cut-off frequency of 400 ± 100 GHz.

3. Pulsed THz emission with pin diodes

The setup for pulsed measurements, shown in figure 2(a), consists of a Toptica FFS 1550 nm mode-locked fiber laser, with a repetition rate of 78 MHz, a pulse duration of 100 fs and a maximum optical power of 100 mW. A part of the laser beam illuminates the device through a lens with a focal length of 15 cm. The device is with about 80 μm² much smaller than the expected spot size given by the numerical aperture of the lens, resulting in a homogeneous, plane-wave illumination and an approximately constant carrier concentration within its cross section. The generated current by the n-i-pn-i-p superlattice photomixer is fed into a broadband, logarithmic-periodic antenna with a bandwidth of ∼60 GHz–1.5 THz. The THz beam is imaged by a silicon lens (diameter 10 mm, hyperhemispherical offset 1 mm) followed by two off-axis parabolic mirrors on a ZnTe crystal.

The second part of the laser beam is frequency-doubled and then combined with the THz signal for electro-optic detection in the ZnTe crystal. A set of balanced photodiodes reads out the THz-induced electro-optic effect in the crystal, which is the measure for the THz field strength. We used lock-in technique to improve the signal to noise ratio. An acousto-optic modulator working at 89.05 kHz chopped the laser signal exciting the photomixer. Figure 2(b) shows the spectrum generated by the pin diode and detected with EOS at an intermediate carrier density of 6 × 10¹¹ cm^–2. There is excellent agreement between simulation and theory, however, with the exception of a slightly stronger roll-off towards high frequencies. This is partly due to the degraded antenna performance above 0.8 THz but also indicates that the simulated transit-time, shown in the inset, may be shorter than that in the experimental results.

In order to evaluate the dependence of the THz power on the accelerating field within the n-i-pn-i-p superlattice photomixer, the time domain traces at several biases were measured in order to confirm that the position of the peak does not change. Then, the time delay is adjusted for a maximum THz field strength, E_pk, in the time domain. Figure 3(a) shows the normalized peak THz power, ${P}_{{pk}}\sim {E}_{{pk}}^{2}$ versus DC bias. For low carrier density ( ${{n}}^{(2D)}\leqslant 4\,\times \,{10}^{11}$ cm^–2), the power-voltage characteristics shows a pronounced peak. This feature already proves a major signal contribution from the ballistic transport since it is strongly field-dependent [16] while transport at saturation velocity would not show a field-dependence in the region beyond a few tens of kV cm^–1. In particular, no peak signature would be visible. We note that for pulses much shorter than the transit-time, the structure of the peak field, E_pk, versus bias resembles that of the average carrier velocity, v_av^tr versus bias as ${E}_{{pk}}({U}_{{DC}})\sim j({U}_{{DC}})\sim {{\rm{env}}}_{{\rm{av}}}^{{\rm{tr}}}({U}_{{DC}})\sim {v}_{{\rm{av}}}^{{\rm{tr}}}({U}_{{DC}})$ , while the carrier concentration, $n\sim {P}_{L}$ , can be considered as a delta-shaped pulse in the time domain with all carriers generated within the same time slot. For our case, however, the pulse duration of 100 fs is already a large fraction of the transit time (≈200 fs), blurring this relation, as separation of early generated carriers starts to screen the built-in field for subsequently generated carriers. Figure 3(a) also shows normalized simulation results without any relative scaling. The only fitting parameter of the simulation is the initially screened bias due to the superlattice structure, U_scr = 0.3 V/period, in order to match theoretical and experimental bias for optimum performance at low optical powers. There is excellent agreement between experiment and theory both in terms of shape and (relative) emitted THz power versus bias for a photocurrent range of more than one order of magnitude. Only in forward direction, i.e. at very low electric fields, we see some discrepancy. This is most likely due to the simplifying assumption that carriers scatter immediately into the side-valley once reaching the inter-valley energy, ${E}_{{\rm{\Gamma }}L}^{{\rm{eff}}}$ with an instantaneous decrease of the velocity to the saturation velocity. Further, the values for the saturation velocities of both electrons and holes were extrapolated from the literature in the high field range [27, 28]. They are substantially smaller for fields below 10 kV cm^–1, i.e. when the device is biased in forward direction.

**Figure 3.** (a) THz peak power versus bias. Thick lines: experiment, thin lines: simulation. Both experimental and theoretical powers are normalized to the power at ${n}^{(2D)}=9\,\times \,{10}^{11}$ cm^–2 and U_DC = −4 V for comparison. The inset shows the saturation of the peak power versus carriers per pulse (points). Unsaturated detectors would show the depicted parabolic increase (solid line) of power with carrier density. The carrier density of 1.58 × 10¹² cm^–2 where the power is reduced by 3 dB is marked by the red cross. (b) Deformation of the band structure causing saturation of the THz output of the photodiode at high optical powers. The built-in field is inhomogeneously screened within a few 10 fs, resulting in low field regions with little carrier acceleration and, consequently, much longer transit times. The simulated results are taken for a Gaussian pulse of 100 fs duration at a time of 100 fs after the peak for an internal (accelerating) reverse bias of −0.2 V (−1.5 V external bias for the three-period device, indicated by the arrow in (a)).
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**Figure 3.** (a) THz peak power versus bias. Thick lines: experiment, thin lines: simulation. Both experimental and theoretical powers are normalized to the power at ${n}^{(2D)}=9\,\times \,{10}^{11}$ cm^–2 and U_DC = −4 V for comparison. The inset shows the saturation of the peak power versus carriers per pulse (points). Unsaturated detectors would show the depicted parabolic increase (solid line) of power with carrier density. The carrier density of 1.58 × 10¹² cm^–2 where the power is reduced by 3 dB is marked by the red cross. (b) Deformation of the band structure causing saturation of the THz output of the photodiode at high optical powers. The built-in field is inhomogeneously screened within a few 10 fs, resulting in low field regions with little carrier acceleration and, consequently, much longer transit times. The simulated results are taken for a Gaussian pulse of 100 fs duration at a time of 100 fs after the peak for an internal (accelerating) reverse bias of −0.2 V (−1.5 V external bias for the three-period device, indicated by the arrow in (a)).
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From the simulations, we determine an optimum field for pulsed operation for the lowest experimentally investigated carrier density of 2.8 × 10¹¹ cm^–2 of 47 ± 3 kV cm^–1. This optimum field strength is about 30% higher than the field strength of 35 kV cm^–1 that delivers the shortest ballistic transit-times (around 230 fs) in the simulation at vanishing carrier density (see figure 2(b), inset), indicating that the applied field is already screened to some amount. This is in agreement with the deformation of the band structure by screening of carriers as illustrated in figure 3(b) for 100 fs after the maximum optical excitation. It reveals that already these low carrier densities generate a field screening of 10–15 kV cm^–1. A higher external reverse bias is therefore necessary to restore the optimum transport condition. At higher photocarrier densities, however, the potential becomes so strongly distorted that the transport across the whole intrinsic layer is far from optimum. Ballistic overshoot may even result in a field opposing the built-in field within the intrinsic layer as shown in figure 3(b) for a simulated carrier density of 1.4 × 10¹² cm^–2. If the field is far from optimum for ballistic transport the peak height in figure 3(a) is reduced and shifted to higher negative bias. At the highest photocurrents, screening appears at time scales of a few 10 fs, reducing the average carrier acceleration and their separation for subsequent time steps. This, in turn, reduces the strength of screening and explains the less distorted band at the highest investigated current densities. Here, the electric field is very low at the beginning of the intrinsic layer, where most of the carriers are generated. The low field strongly increases the transit-time and the THz output of the device saturates. Even higher laser power does not notably increase the THz output as shown for the highest investigated carrier densities of 2.5 × 10¹² cm^–2and 4.2 × 10¹² cm^–2 as shown in the inset of figure 3(a). In CW experiments, we even found a roll-over in the high power limit, i.e. less generated THz power with increasing laser power. Such early saturation is indeed expected for pin diodes where all generated carriers contribute to the photocurrent. In photoconductors, only a small fraction (i.e. the gain, g ∼ 0.1%) contributes to the photocurrent, resulting in far less pronounced screening [12]. Therefore, photoconductors are typically used in pulse experiments, while pin diodes are much more powerful sources in CW setups where their output power benefits from the large photocurrent while screening is much less severe due to the much smaller peak photocurrent densities. Remarkably, with n-i-pn-i-p superlattice photomixers we achieved with a photocurrent as low as 4.2 μA (carrier density of 4.2 × 10¹¹ cm^–2) a peak to peak THz field strength of 0.47 V cm^–1. This field strength is comparable to results obtained with 1550 nm compatible large area photoconductive emitters [30, 31], although the absorbed laser power is about three orders of magnitude higher than in the n-i-pn-i-p superlattice photomixer.

4. Ballistic transport under CW operation

It is well known that ballistic contributions enhance the THz power of CW operated pin based photomixers [16]. However, quantitative studies for 1550 nm operated devices are missing; to the knowledge of the authors, only ballistic transport in GaAs-based devices has been directly investigated [22]. Therefore, a similar device as for the pulsed measurements is characterized in a 1550 nm CW photomixing setup that is shown in the inset of figure 4. It consists of two DFB laser diodes that are combined and subsequently amplified by an erbium doped fiber amplifier. The THz signal is collimated by a silicon lens mounted on the backside of the sample, followed by two parabolic mirrors, that image the THz signal on a Golay cell detector. One laser signal is mechanically chopped for lock-in detection. The photocurrent density is 3.75 kA cm⁻², a value where little saturation is expected as confirmed by simulations. The single frequency operation under CW conditions allows to frequency-resolve the emitted power versus bias with the (almost) frequency-independent Golay cell detector.

**Figure 4.** Measured and simulated spectra for CW operation for a photocurrent density of 3.75 kA cm⁻² under optimum transport conditions, normalized to the power at $f\to 0\,\mathrm{Hz}$ . The inset shows the experimental CW setup as described in the text.
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**Figure 4.** Measured and simulated spectra for CW operation for a photocurrent density of 3.75 kA cm⁻² under optimum transport conditions, normalized to the power at $f\to 0\,\mathrm{Hz}$ . The inset shows the experimental CW setup as described in the text.
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Figure 5(a) depicts a measurement of THz power versus bias, featuring a similar peak structure as in the pulsed measurements with an even more pronounced peak at the optimum bias. This proves again ballistic contributions to the transport, enhancing the transit-time 3 dB frequency at optimum transport conditions. At low frequencies (f_THz ≪ 0.66 THz), the peak becomes less pronounced since transport at saturation velocity is sufficient to separate electrons from holes on a time scale of $1/(2{f}_{{\rm{THz}}})$ . Interestingly, the peak also becomes less pronounced for frequencies above 0.66 THz as shown by the inset of figure 5 (a). This clearly contradicts the expectation of the classical roll-off equation,

$\begin{eqnarray}&&{\eta }_{{\rm{tr}}}^{{\rm{class}}}(f,{f}_{{\rm{tr}}}^{3{\rm{dB}}})=\displaystyle \frac{1}{1+{(f/{f}_{{\rm{tr}}}^{3{\rm{dB}}})}^{2}},\end{eqnarray} \tag{ 1 }$

with ${f}_{{\rm{tr}}}^{3{\rm{dB}}}\approx 0.5/({\tau }_{{\rm{tr}}})$ [11]. The relative peak height in figure 5(a) is calculated by normalizing the THz power at optimum bias to that at the highest reverse bias used in this experiment. The transit-time is shortest at the optimum bias with a value of ${\tau }_{{\rm{tr}}}^{{\rm{opt}}}$ and increases strongly for higher accelerating fields, i.e. higher reverse biases, confirmed by the pulsed simulations shown in the inset of figure 2(b). At frequencies (much) above the transit-time 3 dB frequency for optimum transport, equation (1) simplifies to ${\eta }_{{\rm{tr}}}^{{\rm{class}}}(f,{\tau }_{{\rm{tr}}})\approx {(2{\tau }_{{\rm{tr}}}f)}^{-2}$ . Equation (1) thus predicts a (strictly monotonous) convergence of the relative peak height to a value of ${({\tau }_{{\rm{tr}}}(-2V)/{\tau }_{{\rm{tr}}}^{{\rm{opt}}})}^{2}$ , where ${\tau }_{{\rm{tr}}}(-2V)$ is the transit-time at the highest applied reverse bias of −2 V in the experiment. This is not reproduced by the experiment that shows a pronounced maximum of the relative peak height around 0.66 THz. Actually, equation (1) is only an approximative solution for frequencies below the transit-time 3 dB frequency as shown in [11]. Above the transit-time 3 dB frequency, the performance strongly depends on the actual carrier transport (ballistic contributions versus transport at saturation velocity) and the potential shape of the absorption region. Therefore, the frequently used equation (1) describes the transit-time roll-off behavior of the diode above its 3 dB frequency inadequately and cannot be used to derive the transit-time. We note that the measurement of the power-voltage characteristics yields much higher accuracy on the optimum transport field than fits of the transit-time 3 dB frequency of the measured spectra. The latter are prone to alignment errors, distortion by the diffraction limit at low frequencies, or antenna emission efficiency and radiation resistance, resulting in a substantial error in addition to fitting errors resulting from using an inappropriate function for the roll-off. In contrast, power-voltage characteristics as shown in figure 5(a) cancel all setup-specific dependencies since only relative power values are relevant and therefore are well suited to examine ballistic performance and determine the optimum transport field.

Figure 5(b) shows simulation results for the THz-power-bias dependence of the diode. Both peak structure and general frequency-dependent behavior show excellent qualitative agreement. The simulation parameters are the same as for the pulsed simulation, except for a larger static bias, ${U}_{{\rm{st}}}=0.5\,{\rm{V}}$ . In pulsed experiments, the device can relax to its dark condition between subsequent pulses, while in CW, there is a constant generation of carriers that accumulate between the diodes, justifying a larger static screening bias. However, the peak in the simulations appears at about two times higher frequency. Further, the experimentally fitted transit-time 3 dB frequency of ∼0.55 ± 0.1 THz is about three times lower than the 3 dB frequency of 1.7 THz fitted to the simulated spectrum as shown by the spectra in figure 4. The measured spectrum, however, depends strongly on the antenna performance. From RF simulations, we determined that the antenna efficiency of the logartihmic-periodic antenna starts to decrease above ∼800 GHz because it is perturbed by the photomixer and sample geometry. Another sample, equipped with a logarithmic-periodic spiral antenna showed a fitted transit-time 3 dB frequency of 0.85 ± 0.15 THz. Still, the experimental roll-off is stronger than the simulated one. Also the simulated pulsed data in figure 2(b) overestimate slightly the THz output at frequencies above 0.8 THz. In pulsed operation, however, the spectral characteristics is dominated by the shortest time-constant within the current generation process, in this case the optical pulse duration. Similar to photoconductive sources [12, 29], we would not expect a major impact of the transit-time on the shape of the spectrum (but surely on emitted THz power) in pulsed operation for τ_tr ≫ τ_pls. We conclude that the discrepancy is mainly due to the performance limits of the logarithmic-periodic antenna. In contrast, in CW operation, the carrier transit-time directly enters the expression for the transit-time roll off, explaining the larger discrepancy of the CW simulation. Even if further scattering mechanisms are included in the simulations to describe for the deceleration of carriers, this is not sufficient to explain the discrepancy between simulation and experiment. We conclude that one or several of the following reasons are responsible: (i) the 1-dimensional description is insufficient for a quantitative description of the onset of L-valley transfer, which seems to happen at earlier times than simulated, (ii) that other, neglected scattering mechanisms, such as electron–electron scattering, are responsible for early side valley transfer, (iii) that band gap shrinkage with carrier generation deeper in the band (i.e. with excess potential energy and consequently a lower kinetic energy threshold for side-valley scattering) at higher temperatures due to heating of the device at high power levels decreases the ballistic transport time, or (iv) because we neglect excess potential energy by optical generation of carriers above the conduction band edge. Cases (iii) and (iv) effectively reduce the necessary kinetic energy before side valley scattering occurs.

The aforementioned effects may also have an impact on the optimum transport field. Despite these discrepancies, the simulation results allow for qualitative conclusions: The largest peak height usually appears at frequencies lower than the transit-time 3 dB frequency, i.e. defining a lower boundary for the transit-time 3 dB frequency. An exact relation, however, cannot be specified since it depends also on the reverse bias used for normalization. Here, the simulated peak of the power-voltage characteristics is at 1.35 THz, while the fitted 3 dB frequency of the simulated data is at 1.7 THz, i.e. 25% higher. From the measured power-voltage characteristics with a peak at 0.66 THz we conclude that the transit-time 3 dB frequency of the device is around 0.83 THz, in excellent agreement with the fitted transit-time 3 dB frequency of 0.85 THz from the logarithmic-periodic antenna.

The simulations further allow to analyze the shape of the spectrum emitted by a pin diode photomixer and deduce improved expressions for the transit-time. In most publications, the transit-time 3 dB frequency is approximated as ${f}_{{\rm{tr}}}^{3{\rm{dB}}}\approx 0.5/({\tau }_{{\rm{tr}}})$ . In [11] we already found that this approximate description does not adequately describe the frequency characteristics. With some simplifying assumptions on the transport, we obtained an analytical 3 dB frequency for ballistic transport of ${f}_{{\rm{bal}}}^{3{\rm{dB}}}\approx 0.55/({\tau }_{{\rm{tr}}})$ and ${f}_{{\rm{sat}}}^{3{\rm{dB}}}\approx 0.44/({\tau }_{{\rm{tr}}})$ for saturation transport. These 3 dB frequencies, however, only describe the frequency dependence well below the 3 dB frequency. From the simulation results obtained in this paper, a better overall agreement is obtained for ${f}_{{\rm{tr}}}^{3{\rm{dB}}}\approx 0.4/{\tau }_{{\rm{tr}}}$ , i.e. higher frequencies are substantially stronger suppressed than frequencies below the 3 dB frequency, where equation (1) is still a good approximation. Therefore, the transit-time 3 dB frequency fitted to the simulated spectra is 1.7 THz for a transit time of 0.23 fs.

5. Conclusions

We have examined the impact of ballistic transport on THz emission in n-i-pn-i-p superlattice photomixers under pulsed and CW excitation experimentally and by numerical studies for both types of excitation. The numerical studies generally underestimate ballistic transit-times, but allow for many qualitative predictions. The appearance of a peak in the THz power-voltage characteristics was well reproduced, proving ballistic transport within the diodes. The frequency dependence of the power-voltage characteristics allows for conclusions on the transit-time of the electrons within the device and further shows that the usual roll-off expression inadequately describes the THz-spectra generated by pin diodes. The experimental results show that n-i-pn-i-p photomixers are very efficient THz emitters, even under pulsed emission. If a pin diode or a n-i-pn-i-p superlattice photomixer could be implemented in a large area configuration [32], where the laser power is distributed over a large area in order to prevent local saturation by carrier screening, such devices could become very powerful pulsed THz sources. CW operation is not perfect, because the transport is mixed by ballistic and saturation effects.

Acknowledgments

We thank P Michel and the ELBE team for their dedicated support.

Appendix

This appendix lists details on the simulation parameters and assumptions. The aluminum content is defined as x in In_0.53Al_xGa_0.47−xAs. Most expressions have been taken from [27, 28, 33, 34, 35, 36].

A.1. Band structure

Direct band gap at Γ-valley versus Al-content, x [33]:
$\begin{eqnarray}&&{E}_{G}(x)=0.955{x}^{2}+x+{E}_{G,0}.\end{eqnarray} \tag{ 2 }$
The optical absorption, and hence the carrier generation rate, has been obtained by a fit to experimental data of InGaAs at 300 K around the absorption edge with the following function:
$\begin{eqnarray}&&\alpha ({E}_{{\rm{ph}}})=23\,870\,{{\rm{cm}}}^{-1}\cdot {({E}_{{\rm{ph}}}-{E}_{G}(x))}^{0.3855},\end{eqnarray} \tag{ 3 }$
for E_ph > E_G(x) and zero otherwise, where E_ph is the photon energy and E_G(x) is the band gap of the quarternary with Al-content, x. We assume that carriers are generated at rest at the bottom of the conduction band and neglect any excess potential energy. The frequency bandwidth of the laser pulse in the pulsed simulation is therefore ignored. We further assume that the shape of the absorption profile (i.e. prefactor and exponent in equation (3)) does not change considerably with $x\ne 0$ .
Distance of bottom of conduction band to bottom of L-band versus Al-content:
$\begin{eqnarray}&&{E}_{{\rm{\Gamma }}L}^{{\rm{eff}}}=(0.5+{\rm{\Delta }}-0.26x){\rm{eV}}.\end{eqnarray} \tag{ 4 }$
We assume an abrupt deceleration of the electrons once side-valley scattering takes place. Since the scattering cross section is still fairly low when the electron just reached the inter-valley energy, $E=(0.5-0.26x$ ) eV, we allow the electron to gain ${\rm{\Delta }}=2{k}_{{\rm{B}}}T\approx 0.05\,\mathrm{eV}$ higher energy before abrupt scattering sets in. Values are taken from [36] partially extrapolated from figure 6.50. The error is expected to be below k_B^T.
Valence band offset for In_0.53Al_xGa_0.47−xAs and In_0.53Ga_0.47As:
$\begin{eqnarray}&&{\rm{\Delta }}{E}_{V}=-0.35{\rm{\Delta }}{E}_{G}(x),\end{eqnarray} \tag{ 5 }$
${\rm{\Delta }}{E}_{G}(x)$ being the band gap difference between the compounds.
Conduction band offset for In_0.53Al_xGa ${}_{0.47-x}$ As and In_0.53Ga_0.47As:
$\begin{eqnarray}&&{\rm{\Delta }}{E}_{C}=0.65{\rm{\Delta }}{E}_{G}(x).\end{eqnarray} \tag{ 6 }$
Band offset L-band for In_0.53Al_xGa ${}_{0.47-x}$ As and In_0.53Ga_0.47As is assumed to be similar to that of the CB edge:
$\begin{eqnarray}&&{\rm{\Delta }}{E}_{L}=0.65{\rm{\Delta }}{E}_{G}(x).\end{eqnarray} \tag{ 7 }$

A.2. Effective masses

Data are taken from [28, 34] and from the web page of the Ioffe institute:

Electron effective mass in Γ valley versus Al-content:
$\begin{eqnarray}&&{m}_{{\rm{\Gamma }}}^{* }(x)=(0.041+0.132\,5x-0.069\,4{x}^{2}){m}_{0},\end{eqnarray} \tag{ 8 }$
with m₀ the electron rest mass.
Electron effective mass in L-valley:
$\begin{eqnarray}&&{m}_{L}^{* }=0.55{m}_{0}.\end{eqnarray} \tag{ 9 }$
Due to lack of data, the dependence of the effective mass in the L-valley on the Al-content is neglected.
Hole effective mass:
$\begin{eqnarray}&&{m}_{h}^{* }(x)=(0.45-0.208\,3x){m}_{0}.\end{eqnarray} \tag{ 10 }$
Light holes are neglected since their density of states is much smaller and the hole contribution to the THz power is fairly small in general.
As the electrons reach kinetic energies around 0.55 eV before L-valley scattering, deviations from the parabolic band shape, and hence an increase of the effective mass, are approximated using the Kane model:
$\begin{eqnarray}&&{P}^{2}=\displaystyle \frac{2{E}_{G}(x)}{{m}_{e}^{* }(x)},\end{eqnarray} \tag{ 11 }$

$\begin{eqnarray}&&{m}_{e,{Kane}}^{* }(x,v)=\displaystyle \frac{2{E}_{G}(x)}{{P}^{2}-{v}^{2}}.\end{eqnarray} \tag{ 12 }$
where v is the carrier velocity and ${m}_{e}^{* }(x)$ is the effective mass at the Γ point for an aluminum content, x.

A.3. 1D scattering rates

The simulation considers only the transport direction with effective 1 D expressions for the momentum scattering rates taken from [36] equation (2.83), justified by the assumption that the carrier velocity by the accelerating field along the transport direction is much larger than the carrier velocities in transverse directions due to diffusion and scattering. Side valley scattering is very efficient at a few k_BT above L-valley energy with a scattering time in the range of a few 10 fs. Therefore, L-valley scattering is assumed abrupt after reaching the effective L-valley scattering energy, ${E}_{{\rm{\Gamma }}L}^{{\rm{eff}}}$ .

Electron scattering rate (inverse scattering time) for emission of an LO phonon:
$\begin{eqnarray}&&\begin{array}{l}{{\rm{\Gamma }}}_{{\rm{LO}},{\rm{em}}}={\tau }_{{\rm{LO}},{\rm{em}}}^{-1}=({N}_{0}+1)\cdot \displaystyle \frac{1.607\times {10}^{18}\,{\rm{m}}\,{{\rm{s}}}^{-2}}{v}\\ \quad \cdot \,\left(\sqrt{1-r}+0.5r\mathrm{ln}\displaystyle \frac{1+\sqrt{1-r}}{1-\sqrt{1-r}}\right),\end{array}\end{eqnarray} \tag{ 13 }$
with $r={E}_{{\rm{LO}}}/{E}_{{\rm{kin}}}(x,v)$ , ${E}_{{\rm{kin}}}=\tfrac{1}{2}{m}_{e,{\rm{Kane}}}^{* }(x,v){v}^{2}$ , and N₀ = 0.34. The LO phonon energy, E_LO = 0.034 eV, has been assumed to be independent from the Al-content.
Electron scattering rate for absorption of an LO phonon:
$\begin{eqnarray}&&\begin{array}{l}{{\rm{\Gamma }}}_{{\rm{LO}},{\rm{abs}}}={\tau }_{{\rm{LO}},{\rm{abs}}}^{-1}={N}_{0}\cdot \displaystyle \frac{1.607\times {10}^{18}\,{\rm{m}}\,{{\rm{s}}}^{-2}}{v}\\ \quad \cdot \,(\sqrt{1+r}-r\mathrm{ln}[\sqrt{1/r}+\sqrt{1/r+1}]).\end{array}\end{eqnarray} \tag{ 14 }$
Electron scattering time in L-valley:
$\begin{eqnarray}&&{\tau }_{e,L}=3.1\,{\rm{ps}},\end{eqnarray} \tag{ 15 }$
calculated from an electron mobility of 10 000 cm² V^–1 s^–1 and an effective mass of 0.55 m₀ in the L-valley. This scattering time is much longer than the transport time for the investigated structure and therefore plays only a negligible role.
Hole scattering time:
$\begin{eqnarray}&&{\tau }_{h}=80\,{\rm{fs}},\end{eqnarray} \tag{ 16 }$
calculated from a hole mobility of 300 cm² V^–1 s^–1 and an effective mass of 0.45 m₀.

Alloy scattering and electron–electron scattering have been neglected as they are much weaker than the other scattering parameters. However, energy transfer by neglected scattering mechanisms may result in early side valley scattering and may therefore be a plausible cause for the underestimation of the transit time. A more thorough study would further require 3D Monte-Carlo simulations where the velocity profile in all three directions is considered.

A.4. Carrier velocities

Electron saturation velocity in the high field limit [27]:
$\begin{eqnarray}&&{v}_{{\rm{sat}},e}=1.3\times {10}^{7}\,{\rm{cm}}\,{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 17 }$
Hole saturation velocity in the high field limit [28]:
$\begin{eqnarray}&&{v}_{{\rm{sat}},h}=5\times {10}^{6}\,{\rm{cm}}\,{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 18 }$
The electron velocity in the Γ-valley is calculated by a quasi-ballistic transport model. For ${E}_{{\rm{kin}}}(x,v)\lt {E}_{{\rm{LO}}}$ we only allow for LO phonon absorption:
$\begin{eqnarray}&&{v}_{e}(t+{\rm{d}}{t})={v}_{e}(t)+\left(-\displaystyle \frac{{e}_{0}E}{{m}_{e,{\rm{Kane}}}(x)}-{v}_{e}(t){{\rm{\Gamma }}}_{{\rm{LO}},{\rm{abs}}}\right){\rm{d}}{t},\end{eqnarray} \tag{ 19 }$
where E is the accelerating field. For ${E}_{{\rm{kin}}}(x,v)\geqslant {E}_{{\rm{LO}}}$ we further allow for LO phonon emission:
$\begin{eqnarray}&&\begin{array}{l}{v}_{e}(t+{\rm{d}}{t})={v}_{e}(t)+\left(-\displaystyle \frac{{e}_{0}E}{{m}_{{\rm{eff}},e}(x)}-{v}_{e}(t)({{\rm{\Gamma }}}_{{\rm{LO}},{\rm{abs}}}\right.\\ \quad +\,{{\rm{\Gamma }}}_{{\rm{LO}},{\rm{em}}})){\rm{d}}{t}.\end{array}\end{eqnarray} \tag{ 20 }$
At ${E}_{{\rm{kin}}}(x,v)={E}_{{\rm{\Gamma }},L}(x)$ the electron is abruptly decelerated to ${v}_{{\rm{sat}},e}$ and described as L-valley electron from then on.
The electron velocity in the L-valley is determined by the Drude-model:
$\begin{eqnarray}&&{v}_{e}(t+{\rm{d}}{t})={v}_{e}(t)+\left(-\displaystyle \frac{{e}_{0}E}{{m}_{L}(x)}-\displaystyle \frac{{v}_{e}(t)}{{\tau }_{e,L}}\right){\rm{d}}{t},\end{eqnarray} \tag{ 21 }$
and limited to a maximum velocity of ${v}_{{\rm{sat}},e}$ .
The hole velocity is determined by the Drude-model:
$\begin{eqnarray}&&{v}_{h}(t+{\rm{d}}{t})={v}_{h}(t)+\left(\displaystyle \frac{{e}_{0}E}{{m}_{h}(x)}-\displaystyle \frac{{v}_{h}(t)}{{\tau }_{h}}\right){\rm{d}}{t},\end{eqnarray} \tag{ 22 }$
and limited to a maximum velocity of ${v}_{{\rm{sat}},h}$ .

Terahertz generation with ballistic photodiodes under pulsed operation

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Abstract

1. Introduction