Comparative study of equivalent circuit models for ... - OSA Publishing

Vol. 26, No. 22 | 29 Oct 2018 | OPTICS EXPRESS 29017

Comparative study of equivalent circuit models for photoconductive antennas O. A. CASTAÑEDA-URIBE,1 C. A. CRIOLLO,2 S. WINNERL,3 M. HELM,3 AND A. AVILA2,* 1

Vicerrectoría de Investigaciones, Grupo de Investigación en Ingeniería Biomédica (GIIB), Universidad Manuela Beltrán, Bogotá, DC 110231, Colombia 2 Department of Electrical and Electronic Engineering, Centro de Microelectrónica (CMUA), Universidad de los Andes, Bogotá, DC 11711, Colombia 3 Helmholtz-Zentrum Dresden-Rossendorf, P.O. Box 510119 01314 Dresden, Germany * [email protected]

Abstract: Comparison of equivalent circuit models (ECM) for photoconductive antennas (PCA) represents a challenge due to the multiphysics phenomena involved during PCA operation and the absence of a standardized validation methodology. In this work, currently reported ECMs are compared using a unique set of simulation parameters and validation indicators (THz waveform, optical power saturation, and ECM voltages consistency). The ECM simulations are contrasted with measured THz pulses of an H-shaped 20μm gap PCA at different optical powers (20mW to 220mW). In addition, an alternative two-element ECM that accounts for both space-charge and radiation screening effects is presented and validated using the proposed methodology. The model shows an accurately reproduced THz pulse using a reduced number of circuital elements, which represents an advantage for PCA modeling. © 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction Photoconductive antennas (PCA) play a significant role in THz spectroscopy and imaging systems with several applications in the fields of polymer [1] and food characterization [2], quality control of electronic devices [3], cancer detection [4], and revealing concealed hazardous and illegal elements, such as explosives [5], weapons [6] and drugs [7]. As an emitter, the PCA creates THz pulses when a laser beam excites the antenna gap, causing a generation of charge carriers (electron-hole pairs) in a photoconductive semiconductor substrate, such as low-temperature-grown (LT) GaAs [8], semi-insulating (SI) GaAs [9], InP [10], ZnSe [11], ZnO [12], and 4H/6H-SiC [13]. These carriers are accelerated due to the bias voltage applied across the antenna electrodes inducing a photocurrent peak on a picosecond timescale. As a result, an electric field in the frequency range of THz is created in the antenna [14]. Optimizing the PCA operation is the main challenge for the wide spread use of these THz devices [15] and it requires the development of accurate physical and circuital models that allow tuning and optimizing the emitted THz pulse based on the antenna physical characteristics [16]. In an effort to optimize the PCA operation, several experimental and theoretical studies have been carried out [9,16–24]. The theoretical studies present an electromagnetic physics framework that involves the theory of electromagnetic wave propagation for the incident beam [19] and radiated THz pulse [20], and the photo-generation of charge carriers and transport mobility models for the antenna semiconductor [16,25]. The experimental studies focus on the enhancement of the antenna power efficiency [17,26,27], bandwidth [24] and radiation pattern [23] through the implementation of plasmonic effects on the substrate surface [28], and the variation of physical parameters, such as the semiconductor material [29–31], and the geometry of the electrode arrays [9,32,33]. #340556 Journal © 2018

https://doi.org/10.1364/OE.26.029017 Received 25 Jul 2018; revised 15 Sep 2018; accepted 18 Sep 2018; published 24 Oct 2018


The theory behind PCA operation can be classified into two physical approaches: FullWave Finite-Difference Time-Domain models (FDTD) and equivalent circuit models (ECM) that can be applied if the gap size is smaller than the generated wavelengths of the emitted THz radiation (small gap PCA) [34]. The FDTD approaches estimate the photoinduced current in the PCA based on the drift-diffusion and carrier’s continuity equations, which are numerically solved using Monte Carlo simulations [35,36] or finite element method (FEM) simulations [36–38]. In contrast, the ECM approach seeks to model the PCA operation as a lumped element circuit in which the circuital elements account for all the electromagnetic phenomena. Additionally, the ECMs correlate the antenna physical parameters with the THz emission pulse, which represent an advantage for the design and optimization of photoconductive antennas [39]. Currently reported ECMs [8,39–42] represents the PCA operation in two main electromagnetic phenomena: the photo-generated carrier dynamics and the antenna screening effect, which depending on the model takes into account the radiation and/or the space-charge conditions. In general, the carrier dynamics in the antenna gap is represented using a timedependent impedance, and the radiation and space-charge screening effects with a resistor and a voltage dependent source respectively. The number of passive elements used in the ECMs increase the complexity of the model, which demand a higher computational effort during the simulations. In addition, the space-charge screening effect and the carrier dynamics, both phenomena occurring at the antenna gap, are separated into two different elements in the ECMs. Therefore, the antenna gap can’t be represented as a single element in the circuit, which cumbers the interpretation of the models. As an alternative for the currently reported ECMs, here a two-element model composed by a time-dependent capacitor C(t) and a radiation resistance Ra, which accounts for all the phenomena occurring in the PCA, is presented. Performance of currently reported ECMs has been evaluated using multiple indicators such as the THz pulse shape [8], photocurrent [39,41], optical-power saturation [8,39], and the circuit element voltages [39,41]. The results of each indicator depend on the set of the antenna physical parameters that have been selected for the model simulation, as well as the carrier density equation implemented. These simulations parameters vary from model to model and represent a challenge for the comparison and evaluation of the models’ performance. In this work, we propose a validation methodology for ECMs based on three indicators: the THz pulse waveform, the optical-power saturation value and the ECM voltages consistency. The indicators are used to contrast an experimental data set with the currently reported ECMs using a unique set of simulation parameters (antenna physical parameters and carrier density equation). The proposed methodology represents an attractive tool for the evaluation of ECMs for photoconductive antennas. 2. Methods 2.1 PCA fabrication method A dipole PCA with 20µm gap is fabricated on a low-temperature grown GaAs (LT-GaAs) wafer with an energy gap of 1.43eV and a carrier lifetime below 1ps (from Xiamen Power way Advanced Material Co.). On the wafer surface, an H-shaped gold dipole antenna (see dimensions on Fig. 1(a) is patterned using standard photolithography and metal lift-off processes. The patterned antenna is mounted in a thermally conductive adhesive (TRABOND from Henkel Ag & Co.) on an aluminum holder (2.54cm thickness with an aperture area of 25mm2 from Thorlabs, Inc.) to provide good sink for any heat generated in the LTGaAs substrate [43].


Fig. 1. LT-GaAs based photoconductive antenna. (a) Antenna geometry with an H-shaped dipole structure (not scaled). (b) SEM image of the 20µm antenna gap.

2.2 PCA characterization method The experimental setup for the characterization of the fabricated PCA is shown in Fig. 2. A Ti: sapphire laser with a wavelength of 800nm and a pulse duration of 70fs radiates a beam with a repetition rate of 78MHz and a variable output power from 20mW to 220mW. The laser beam is guided to a beam splitter (BS), which divides the beam into two parts: excitation and reference. The excitation beam passes through a focusing lens with a focal length of 150mm that concentrates the beam at the antenna gap. The antenna biased by a square-wave voltage (frequency 10kHz, amplitude 20V) generates a THz pulsed wave that is collimated and refocused using two off-axis parabolic mirrors (with a reflected focal length of 100 mm). The THz wave is sampled by the pulses from the reference beam using an optical delay stage. Both the THz and reference beams are re-directed to an electro-optic sampling system conformed by a ZnTe crystal (200µm thick -oriented), a quarter wave plate (λ/4), a polarization sensitive beam splitter and a set of two balanced photodiodes. The electro-optic sampling system transforms the THz wave into a voltage signal that is sent to a lock-in amplifier set to same frequency of the antenna bias signal (10kHz) [44]. 2.3 Computational simulation method for ECMs Computational simulations of the ECMs are performed using MATLAB R2015b. The time vector and therefore the time dependent variables, such as voltages and currents are set in a time scale of picoseconds (0-6ps) with a length of 8000 elements (0.75fs of resolution). A time delay is included in the temporal equations to fit the position of the peak for the THz pulse measurements. The function ODE45 is implemented for the numerical solution of nonstiff differential equations with a medium level of accuracy and with a low computational effort. The solution vector from the differential equation is post-processed using a local regression smoothing with a span of 1% in order to filter the oscillations caused by the accuracy of the differential solver.


Fig. 2. Experimental setup for characterization of photoconductive antennas. Beam splitter divides optical pulse in reference and excitation signals. PCA bias signal is modulated at 10KHz. A ZnTe Crystal, a quarter-wave plate, a polarizer beam splitter and two balanced photodiodes are used to detect the emitted THz pulses.

3. Equivalent circuit models for photoconductive antennas Capturing the phenomena of generation of THz pulses involves several phenomena: electromagnetic, carrier dynamics and transport, propagation and absorption of the radiation of the laser beam. Although different models in the literature have investigated to improve the efficiency [39,45] and output power [46]; there has been a challenge to couple different physical phenomena by integrating or connecting electromagnetic, optoelectronic characteristics, electronic transport and lumped circuit models. The THz generation process has been proved to be nonlinear as a function of the optical power [40,47]. At low power, the THz generation tends to increase linearly but saturates at high power [9]. Two sources have been reported to explain this phenomena: radiation screening ScrR [8], and space charge screening Scrsc [48]. The radiation screening is due to radiated near-field THz pulse and the space charge screening accounts for the spatial separation between carriers at the antenna gap. Modeling a PCA using an equivalent circuit requires the implementation of circuit elements that account for the electromagnetic phenomenon, including the screening effects. In the literature, the laser beam propagation, and the photo-generation and mobility of carriers are localized in the antenna gap and typically denoted by a time dependent conductance G(t) and a time dependent capacitance C(t) [39–41,49]. The space charge and radiation screening effects are represented by a resistor R and a voltage dependent source VSC respectively [40]. Depending on the elements representing the antenna gap and the type of screening effect the current ECMs can be classified as: GapG-ScrR [40], GapG-Scrsc [48], GapG-ScrR/SC [8], GapG/C-ScrR/SC [39], and GapG/C/Z-ScrR/SC [41]. A comprehensive description of the circuital models applied for PCA modeling is presented in Fig. 3.


Fig. 3. ECMs and the supporting equations reported applied to model dipole photoconductive antennas. a) GapG-ScrR [40] b) GapG-Scrsc [48] c) GapG-ScrR/SC [8], d) GapG/C-ScrR/SC [39]. e) GapG/C/Z-ScrR/SC [41]. For each equivalent circuit, the main differential equation for modeling PCA has been identified (red dashed line).

The GapG-ScrR is a two-elements model (Fig. 3(a)) that represents the antenna gap with a time dependent conductance G(t) and the screening effect by radiation using a resistance in series Ra [40]. The model uses a voltage divider equation dependent on the conductance (see red inset on Fig. 3(a)). The expression for G(t) mainly depends on the carrier density n(t), and the antenna active area A. In the model [40], n(t) is presented as follows:  T

2

t 

0 Jη T π  4τ c2 −τ c   n(t ) = e 0 e 1 + erf 2Vhvopt 

 t T0    −   .  T0 2τ c  

(1)

Where J is the optical pulse energy, τc is the carrier lifetime, T0 is an approximate value for the laser pulse duration, νopt is the laser frequency, h is the Planck’s constant, V is the antenna active volume, and ηe is the quantum coefficient. However, Eq. (1) mistakenly includes the parameter T0, which impacts the estimation of n(t) at least 12 orders of magnitude (T0 ~10−12 s). Additionally, this circuital model also assumes an antenna active area as the region illuminated by the laser beam confined to the surface of the substrate, this is: A = WL, where W, and L are respectively the width and length of the gap. However, the model uses a conductor G(t) between the antenna electrodes and as a consequence the area that must be


considered is the cross-sectional area instead of the surface area. This implies that the antenna active area of the model must be replaced by A = Wd0, where d0 is the depth of the gap, which is directly related to the laser penetration distance within the antenna substrate, see the inset in Fig. 3(a). In Fig. 3(b) the GapG-Scrsc model is presented. This model is not originally reported as an ECM, instead it is presented as a set of electrostatic equations that explains the antenna THz emission [48]. Using the electrostatic equations, the GapG-Scrsc model is presented here as a two-elements circuit (see Fig. 3(b)) composed by a time dependent conductance G(t) that represents the gap, and a voltage dependent source Vsc(t) that accounts only for the screening effect by space-charge. Solving the differential equation of this model (see red inset on Fig. 3(b)) requires an expression for n(t), and A, which are proposed here in the simulation section. In Fig. 3(c), a three elements model is introduced, the GapG-ScrR/SC. This model connects the two above described models representing the gap with a time dependent conductance G(t) and including both radiation Ra and space-charge Vsc(t) screening effects [8]. The model uses the same expression for the antenna active area reported on the GapG-ScrR model, which is why it has been modified in order to include a cross-sectional area. The GapG/C-ScrR/SC is a four-elements model (see Fig. 3(d)). It represents the antenna gap with a G(t) in parallel with C(t) and includes both radiation Ra, and space-charge Vsc(t) screening effects [39]. This model presents a main differential equation that has been obtained by imposing an expression for the circuital current I(t), which neglect the contribution of the current flowing through the capacitor [50]. In order to simulate this model, the corrected expression of I(t) obtained from the circuital analysis has been applied here. Finally, the GapG/C/Z-ScrR/SC is a five elements model (see Fig. 3(e)) that combines the antenna operation under dark and illuminated conditions [41]. The model represents the gap with an impedance Z that accounts for the dark condition, and a time dependent conductance G(t) and capacitance C(t) for illuminated conditions. In addition, screening effects by radiation and space-charge are included using a resistor Ra and a voltage source Vsc(t) respectively. This model reports a main differential equation (see red inset on Fig. 3(e)) that have been developed considering a time-independent carrier mobility μe in the circuital analysis. However, during the process of solving the differential equation, the μe is replaced by a time-dependent function μe(t) based on a combination of the Arora [51] and CaughyThomas [52] mobility models. Although the reported μe(t) accounts for the carrier mobility dependency on both carrier concentration (Arora model) and electric field across the gap (Caughy-Thomas model), it should not be used on the reported differential equation of the model. This is because a time-dependent mobility affects most of the circuital elements in the model (G(t), C(t) and Vsc(t)), causing a modification in the main differential equation of the circuit and therefore a distortion in the waveform of the calculated photo-current. In order to use this model for comparison purposes, here an average value for μe based on the physical parameters of the antenna used in the experimental section is assumed. 4. Results 4.1 ECMs simulation parameters The main current and voltages of each equivalent circuit model from Figs. 3(a)-3(e) were calculated by computationally solving the main differential equations listed per model in Fig. 3. For comparison purposes the carrier density, antenna gap conductance and the antenna physical parameters (Table 1) were set equals for all the models. The carrier density of Drude-Lorentz is defined by [48]: dn(t ) n(t ) =− + g (t ), dt τc

(2)

where τc is the carrier lifetime, and g(t) is the generation rate of carriers by laser pulses [41]:


g (t ) =

2ηe P (t , r ), Vhvopt

(3)

where P(t,r) is the optical laser beam power contained within a radius r, h is the Plank’s constant, νopt is the laser frequency, the antenna active volume is V = WLd0, and ηe is the quantum efficiency [40,41]:

ηe = (1 − R )(1 − e−α d ),

(4)

0

where R is the reflection coefficient in air-semiconductor interface, and α is the optical absorption coefficient. Assuming that the laser behaves as a Gaussian beam, the P(t,r) can be expressed as [39]:  2r2   2t 2   −   − 2   w2    0    τl   P (t , r ) = P0 1 − e e ,    

(5)

where P0 is the laser peak power, w0 is the laser beam waist, and τl is the laser pulse duration. Finally, using Eqs. (2)–(5), the carrier density can be written as: P0 (1 − R )τ l

π

 2r2   −   2  1 − e w02   1 − e −α d0 n(t , r ) =  (WLd 0 )hvopt   

(

)

  τ l 22 − t     e 8τ c τ c   1 + erf    

 t 2 τl 2  −    . (6) 4τ c    τl

All the model simulations were performed assuming that r = w0 and w0 = L, which indicates that the laser is centered at the antenna gap with a spot size bigger than the gap area (in agreement with the experimental conditions). In addition, the gap conductance were also a common term for all the ECMs and it was calculated using the following expression [40]: G (t ) = q μe n(t )

A , L

(7)

where μe is the carrier mobility, q is the electron charge, L is the antenna gap length, and A is the antenna active area. Conductance is here represented as a region of induced charge between the electrodes, which implies a cross-sectional area of A = Wd0. Table 1. Laser excitation and photoconductive antenna parameters. Simulation Parameter Carrier Recombination Lifetime [48] Carrier Mobility [53] Antenna Resistance [40] Permittivity [54] Screening Factor [41] Electron Charge Antenna Gap Width Antenna Gap Length Antenna Gap Depth Bias Voltage Laser Wavelength Laser pulse duration Laser Beam Waist Laser repetition rate Carrier lifetime Optical Absorption Coefficient [55] Reflection Coefficient in Air-GaAs Interface [40]

Value τr = 100 ps μe = 350 cm2V−1s−1 Ra = 120 Ω ε = 1.1417 x 10−10 F/m k = 900 q = 1.6 x 10−19 C W = 20 μm L = 20 μm d0 = 1 μm Vbias = 20 V λ = 800 nm τl = 300 fs w0 = 20 μm νopt = 78 MHz τc = 0.1 ps α = 6000 cm−1 R = 0.3


4.2 THz behavior of the fabricated PCA In order to validate the ECMs a PCA was fabricated and characterized using a single pulse measurement for different optical powers: from 20mW to 220mW in steps of 20mW. The THz pulses resulting from the characterization exhibit a typical bipolar waveform [40], [44] in which the positive peak is slightly wider and higher than the negative peak, as shown on Fig. 4(a). The small oscillations of the THz transients in the time interval from 4 to 6 ps are attributed to the absorption and reemission of THz radiation by water vapor in the unpurged experimental setup. The complete sets of THz pulses have been normalized using the maximum value of the highest THz peak and no-other pre-processing has been applied. The THz pulse saturation curve on Fig. 4(b) shows a non-linear behavior between the maximum peak of the THz pulse and the optical power of the incident laser beam. This nonlinearity is widely reported [9,47,48,56] and it is attributed to the screening effect of the antenna. The data fits well to a typical saturation equation: ETHz =

k

1+ e

1 − k2 ( P − Psat )

,

(8)

where P is the optical power of the incident laser beam, Psat is the saturation power, k1 is a proportionality constant and k2 is the steepness of the curve. For the case of the experimental data, the saturation curve fitting presents a Root Mean Square Error (RMSE) of 2.6% with a Psat = 69.26mW, k1 = 0.96 and k2 = 2.78. The tolerance on the confidence boundaries of the fitting curve is mainly attributed to the THz pulse at 180mW, which represent an outlier in the collected data.

Fig. 4. a) Measured THz pulses of a PCA with a 20μm gap and 20Vpeak bias voltage at different optical powers. b) THz pulse saturation curve for the PCA. The blue dots represent experimental data, which are compared with a saturation fit equation.

4.3 ECMs performance The complete set of ECMs reported in the literature is confronted to experimental data in Fig. 5. Here the model accuracy is evaluated based on: i) The THz pulse waveform, ii) the optical power saturation, and iii) the ECM voltages consistency (Vbias = Vg(t) + VR(t) + Vsc(t)). Figure 5 presents the results of the validation indicators. Comparing the THz pulse waveforms from the models with the measurements presents several differences as shown on Figs. 5(a)-5(e). For the GapG-ScrR, GapG-ScrR/SC, and GapG/C/Z-ScrR/SC models, the characteristics of the THz pulses are in good agreement with the


measurements. A different result arises from applying the GapG-Scrsc model. In this model the width of the negative part of the pulse is noticeable smaller than for the measurements. In contrast, the GapG/C-ScrR/SC model does not reproduce the experimental THz pulse. In this model the equation used for the calculation of the circuit current is: I (t ) = Vg (t )G (t ) + C (t )

dVg (t ) dt

+ Vg (t )

dC (t ) . dt

(9)

As shown in Eq. (9), I(t) depends on the current flowing through the capacitor, which is a differential equation. Then, the derivate terms on the current equation present an inversion of the waveform as well as an increment on the amplitude of at least one order of magnitude compared to the non-derivative term. As a result, the total current waveform of I(t) is inverted causing an inversion on the THz pulse as shown in Fig. 5(d). Pulses here have been shifted adding a time delay of 2.3ps in the carrier density equation to facilitate the pulses shape comparison. The performance of ECMs can be quantitatively measured by estimating saturation power (Psat) based on the THz pulse saturation curves on Figs. 5(f)-5(j). From the fitting curves (red lines) it can be concluded that the GapG-ScrR, GapG-ScrR/SC, and GapG/C/Z-ScrR/SC models present the best accuracy (Exp: 69.3mW / Mod: 69.0 mW). In contrast, the GapG-ScrSC model shows the lowest accuracy (Exp: 69.3 mW / Mod: 125 mW) followed by the GapG/C-ScrR/SC model, which presents a low experimental adjustment (Exp: 69.3 mW / Mod: 64.1 mW).


Fig. 5. Proposed validation indicators to evaluate the performance of ECMs for a PCA with a 20μm gap and 20Vpeak bias voltage. a-e) Calculated (discontinuous line) and experimental (continuous line) THz pulse waveforms for optical powers varying from 20mW to 220mW.The calculated THz pulses are shifted using a time delay of 2.3ps. f-j) THz pulse saturation curves compared with experimental data. Inset: the optical power saturation values are calculated according to Eq. (8). k-o). Calculated temporal evolution of the circuit voltages Vg, VR and Vsc for each ECM at 220mW of optical power.

The consistency of the circuit voltages is presented in Figs. 5(k)-5(o). This indicator allows to identify possible errors on the loop equations. Although all the models present a


loop consistency, which indicates a correct implementation of the circuital equations in the computational modeling, there are two important features that have to be described in the circuit voltages. The first feature corresponds to the small oscillations on the Vsc for the GapGScrR/SC, and GapG/C/Z-ScrR/SC models, which are attributed to the accuracy level of the ODE45 solver implemented in the computational simulations. The second feature is the excessive amplitude of the Vsc and Vrad for the GapG/C-ScrR/SC model that is a consequence of including the contribution of the current flowing through the capacitor into the calculation of the total circuit current. The performance of the reported models in Fig. 5 can be explained by analyzing the elements used to represent the carrier dynamics and screening effect during PCA operation. In general, the screening by radiation is predominant on the PCA operation compared to the space-charge screening effect, therefore neglecting the radiation screening in a model results in a low performance, as shown in the GapG-ScrSC model. Moreover, considering both screening effects in a model increases the accuracy only if the carrier dynamics in the gap is adequately modeled. For the case in which the ECM uses a time dependent capacitor, the carrier dynamics in the gap depends on the current flowing through the capacitor Ic(t). In order to obtain a good performance of the model, Ic(t) must be minimized as in the case of the GapG/C/Z-ScrR/SC model, which uses a diffusion current. Using a conventional capacitor current as in the case of the GapG/C-ScrR/SC model leads to an inversion of the total photocurrent (see Eq. (9)). This current inversion, although mathematically correct, it cannot be explained in terms of physical phenomena, making this model unphysical. For the case in which the ECM uses only a time-dependent conductance the carrier dynamics is largely influenced by the carrier density function n(t). Using an appropriate n(t) leads to a good performance as shown in the GapG-ScrR/SC, and GapG-ScrR models. Interestingly, models GapG-ScrR/SC and GapG/C/Z-ScrR/SC exhibit a similar behavior. This is attributed to the small diffusion current flowing through the capacitor (GapG/C-ScrR), which for the parameters selected in the simulation can be neglected. As a consequence, the capacitor works as an open-circuit leading to a simplified version of the model that represents the carrier dynamics with a time-dependent conductance only (GapG-ScrR/SC). 4.4 Proposed ECM In Fig. 6(a) an alternative ECM that accounts for the total screening effect (space-charge and radiation) without other elements such as time-dependent conductance and a voltage dependent source is presented. The proposed ECM thereafter identified as the GapC-ScrR model is constituted by two main elements: a time dependent capacitance C(t) and a resistor Ra. Each element of the GapC-ScrR model represents a phenomenon occurring in the PCA. The resistor accounts for the screening effect by radiation, while the capacitance represents the dynamics of the charge carriers that have been generated in the semiconductor substrate due to the excitation energy applied by the laser beam. The mathematical expression of the capacitance follows the form: C (t ) =

εA Lsc

+ τ r q μe n(t )

A , L

(10)

where τr, is the carrier recombination time, Lsc is the space-charge region thickness, ε is the semiconductor permittivity, q is the electron charge, μe is the carrier mobility (only electrons are considered in order to simplify the model), A is the cross-sectional area of the antenna, and n(t) is the density of carriers.


Fig. 6. a) Proposed equivalent circuit model for PCA as a THz emitter. b). Calculated temporal evolution of circuit voltages using following parameters: Vbias = 20V, Gap = 20μm and optical power = 220mW. c) Measured and calculated THz pulses of a PCA with a 20μm gap and 20Vpeak bias voltage at different optical powers. c) Optical-Power saturation curves of the PCA compared with experimental data. Inset: the optical power saturation values are calculated according to Eq. (8).

The capacitance is composed by a time dependent term τrqμen(t)A/L and a timeindependent term εA/Lsc. The time dependent term of C(t) is a free-carrier capacitance that accounts for the carrier generation and carrier dynamics. The time independent term is a space-charge capacitance that represents the space-charge screening effect. This capacitance is formed by a region of charge carriers near the electrodes forming a layer with a thickness, commonly known as the space-charge layer thickness Lsc. The Lsc mainly depends on the physical parameters of the semiconductor material and the magnitude of the applied electric field [57]. For the case of study presented here (LT-GaAs), there has not been reported in the literature a value for Lsc, however this parameter has been set to 2.8nm in order to fit the experimental data. The selected value is within the range of the reported values in the literature for similar semiconductors [57,58]. The results of the GapC-ScrR model validation are presented in Figs. 6(b)-6(d). From the figures it can be seen that the model exhibits the same characteristic waveform of the experimental THz pulse (Fig. 6(b)). Based on the THz pulse saturation curve (Fig. 6(c)), the proposed model presents a good agreement with the experimental data (Exp: 69.3 mW / Mod: 69.3 mW), which is comparable to the estimated for the GapG-ScrR, GapG-ScrR/SC and GapGC/Z-ScrR/SC models. This behavior is attributed to the time-dependent capacitor, which represents both the carrier dynamics (photo-generation and mobility) and the space-charge screening effect in a single element. The proposed ECM model is initially restricted to classic H-type dipole, small gap apertures and considers only the PCA operation under illuminated conditions at the environmental experimental conditions (22 °C and 70% RH). Moreover, the model allows to control the optical pulse and antenna physical parameters by modifying the carrier’s density equation, which offers the possibility of studying more complex PCA designs such as


interdigitated antennas with novel photoconductive semiconductor substrates and different optical excitation conditions. 5. Conclusions In the presented work the accuracy of the most reported ECMs for PCA has been estimated by contrasting THz pulse measurements of a fabricated PCA with three indicators: THz pulse waveform, optical-power saturation value, and circuit voltages consistency. The comparison of the models has revealed that the screening effect by radiation is the predominant effect that accounts for the saturation of the PCA, although including the screening by space-charge increases the accuracy of the models. The circuital elements conforming the models depend on the optical excitation, and antenna physical parameters, which are mainly represented by the antenna active area, and the carrier density and mobility. Variations in these parameters modifies the form of model’s main differential equation causing a variation on the total circuit current, thus affecting the characteristic of the modeled THz pulse. In addition, we have also proposed an alternative ECM, denominated GapC-ScrR. The introduced model reduces the number of circuital elements required to accurately reproduce the THz pulse, which decreases the computational effort during simulation. The model exhibits a good performance (Exp: 69.3 mW / Mod: 69.3 mW), which is in the range of accuracy of the state of the art ECMs. This result is attributed to the inclusion of a time dependent capacitance that accounts for the antenna’s carrier dynamics and space-charge screening effect at the same time. The validation methodology and equivalent circuit model proposed in this work represent a new alternative to systematically study photoconductive antennas with different constitutive materials while increases the accuracy on the simulation of emitted THz pulses. Funding Universidad de los Andes (Engineering faculty seed research grants 2014); HelmholtzZentrum Dresden-Rossendorf. Acknowledgments Authors are grateful for the support of Sebastian Arévalo for the antenna fabrication and Martin Mittendorff from Helmholtz-Zentrum Dresden-Rossendorf for contributions with the antenna characterization. Additionally, authors acknowledge the support provided by the Electrical and Electronic Department at Universidad de los Andes for supporting a Master students Internship. Disclosures The authors declare that there are no conflicts of interest related to this article. References 1. 2. 3. 4. 5. 6. 7.

N. Krumbholz, T. Hochrein, N. Vieweg, T. Hasek, K. Kretschmer, M. Bastian, M. Mikulics, and M. Koch, “Monitoring polymeric compounding processes inline with THz time-domain spectroscopy,” Polym. Test. 28(1), 30–35 (2009). A. A. Gowen, C. O’Sullivan, and C. P. O’Donnell, “Terahertz time domain spectroscopy and imaging: Emerging techniques for food process monitoring and quality control,” Trends Food Sci. Technol. 25(1), 40–46 (2012). T. Bowman and M. El-Shenawee, “Nondestructive imaging of packaged microelectronics using pulsed terahertz technology,” Int. Symp. Microelectron., vol. 2017, no. 1, pp. 709–714, Oct. 2017. H. Cheon, H. J. Yang, S.-H. Lee, Y. A. Kim, and J.-H. Son, “Terahertz molecular resonance of cancer DNA,” Sci. Rep. 6 (1), 37103 (2016). M. C. Kemp, P. F. Taday, B. E. Cole, J. A. Cluff, A. J. Fitzgerald, and W. R. Tribe, “Security applications of terahertz technology,” 2003, vol. 5070, p. 44. J. F. Federici, B. Schulkin, F. Huang, D. Gary, R. Barat, F. Oliveira, and D. Zimdars, “THz imaging and sensing for security applications - explosives, weapons and drugs,” Semiconductor Science and Technology, vol. 20, no. 7. IOP Publishing, pp. S266–S280, 01-Jul-2005. A. G. Davies, A. D. Burnett, W. Fan, E. H. Linfield, and J. E. Cunningham, “Terahertz spectroscopy of


8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36.

explosives and drugs,” Mater. Today 11(3), 18–26 (2008). G. C. Loata, M. D. Thomson, T. Löffler, and H. G. Roskos, “Radiation field screening in photoconductive antennae studied via pulsed terahertz emission spectroscopy,” Appl. Phys. Lett. 91(23), 232506 (2007). M. Tani, S. Matsuura, K. Sakai, and S. Nakashima, “Emission characteristics of photoconductive antennas based on low-temperature-grown GaAs and semi-insulating GaAs,” Appl. Opt. 36(30), 7853–7859 (1997). T.-A. Liu, M. Tani, M. Nakajima, M. Hangyo, K. Sakai, S. Nakashima, and C.-L. Pan, “Ultrabroadband terahertz field detection by proton-bombarded InP photoconductive antennas,” Opt. Express 12(13), 2954–2959 (2004). X. Ropagnol, F. Blanchard, T. Ozaki, and M. Reid, “Intense terahertz generation at low frequencies using an interdigitated ZnSe large aperture photoconductive antenna,” Appl. Phys. Lett. 103(16), 161108 (2013). S. Ono, H. Murakami, A. Quema, G. Diwa, N. Sarukura, R. Nagasaka, Y. Ichikawa, H. Ogino, E. Ohshima, A. Yoshikawa, and T. Fukuda, “Generation of terahertz radiation using zinc oxide as photoconductive material excited by ultraviolet pulses,” Appl. Phys. Lett. 87(26), 261112 (2005). X. Ropagnol, M. Bouvier, M. Reid, and T. Ozaki, “Improvement in thermal barriers to intense terahertz generation from photoconductive antennas,” J. Appl. Phys. 116(4), 043107 (2014). Y.-S. Lee, “Generation and detection of broadband terahertz pulses,” in Principles of Terahertz Science and Technology, Boston, MA: Springer US, 2009, pp. 1–66. B. Ferguson and X.-C. Zhang, “Materials for terahertz science and technology,” Nat. Mater. 1(1), 26–33 (2002). E. Moreno, M. F. Pantoja, F. G. Ruiz, J. B. Roldán, and S. G. García, “On the numerical modeling of terahertz photoconductive antennas,” J. Infrared Millim. Terahertz Waves 35(5), 432–444 (2014). C. W. Berry and M. Jarrahi, “Terahertz generation using plasmonic photoconductive gratings,” New J. Phys. 14(10), 105029 (2012). N. Zhu and R. W. Ziolkowski, “Photoconductive THz antenna designs with high radiation efficiency, high directivity, and high aperture efficiency,” IEEE Trans. Terahertz Sci. Technol. 3(6), 721–730 (2013). F. Miyamaru, Y. Saito, K. Yamamoto, T. Furuya, S. Nishizawa, and M. Tani, “Dependence of emission of terahertz radiation on geometrical parameters of dipole photoconductive antennas,” Appl. Phys. Lett. 96(21), 211104 (2010). M. R. Sang-Gyu Park, M. R. Melloch, and A. M. Weiner, “Analysis of terahertz waveforms measured by photoconductive and electrooptic sampling,” IEEE J. Quantum Electron. 35(5), 810–819 (1999). D. Turan, S. C. Corzo-Garcia, N. T. Yardimci, E. Castro-Camus, and M. Jarrahi, “Impact of the metal adhesion layer on the radiation power of plasmonic photoconductive terahertz sources,” J. Infrared Millim. Terahertz Waves 38(12), 1448–1456 (2017). N. T. Yardimci, S. Cakmakyapan, S. Hemmati, and M. Jarrahi, “A high-power broadband terahertz source enabled by three-dimensional light confinement in a plasmonic nanocavity,” Sci. Rep. 7(1), 4166 (2017). X. Li, N. T. Yardimci, and M. Jarrahi, “A polarization-insensitive plasmonic photoconductive terahertz emitter,” AIP Adv. 7(11), 115113 (2017). S. Kono, M. Tani, P. Gu, and K. Sakai, “Detection of up to 20 THz with a low-temperature-grown GaAs photoconductive antenna gated with 15 fs light pulses,” Appl. Phys. Lett. 77(25), 4104–4106 (2000). P. J. Hale, J. Madeo, C. Chin, S. S. Dhillon, J. Mangeney, J. Tignon, and K. M. Dani, “20 THz broadband generation using semi-insulating GaAs interdigitated photoconductive antennas,” Opt. Express 22(21), 26358– 26364 (2014). J. Krause, M. Wagner, S. Winnerl, M. Helm, and D. Stehr, “Tunable narrowband THz pulse generation in scalable large area photoconductive antennas,” Opt. Express 19(20), 19114–19121 (2011). A. Mingardi, W.-D. Zhang, E. R. Brown, A. D. Feldman, T. E. Harvey, and R. P. Mirin, “High power generation of THz from 1550-nm photoconductive emitters,” Opt. Express 26(11), 14472–14478 (2018). S. Lepeshov, A. Gorodetsky, A. Krasnok, E. Rafailov, and P. Belov, “Enhancement of terahertz photoconductive antenna operation by optical nanoantennas,” Laser Photonics Rev. 11(1), 1600199 (2017). S. Winnerl, F. Peter, S. Nitsche, A. Dreyhaupt, B. Zimmermann, M. Wagner, H. Schneider, M. Helm, and K. Köhler, “Generation and detection of THz radiation with scalable antennas based on GaAs substrates with different carrier lifetimes,” IEEE J. Sel. Top. Quantum Electron. 14(2), 449–457 (2008). H. Roehle, R. J. B. Dietz, H. J. Hensel, J. Böttcher, H. Künzel, D. Stanze, M. Schell, and B. Sartorius, “Next generation 1.5 µm terahertz antennas: mesa-structuring of InGaAs/InAlAs photoconductive layers,” Opt. Express 18(3), 2296–2301 (2010). A. Jooshesh, F. Fesharaki, V. Bahrami-Yekta, M. Mahtab, T. Tiedje, T. E. Darcie, and R. Gordon, “Plasmonenhanced LT-GaAs/AlAs heterostructure photoconductive antennas for sub-bandgap terahertz generation,” Opt. Express 25(18), 22140–22148 (2017). S. Winnerl, “Scalable microstructured photoconductive terahertz emitters,” J. Infrared Millim. Terahertz Waves 33(4), 431–454 (2012). N. Vieweg, M. Mikulics, M. Scheller, K. Ezdi, R. Wilk, H. W. Hübers, and M. Koch, “Impact of the contact metallization on the performance of photoconductive THz antennas,” Opt. Express 16(24), 19695–19705 (2008). D. H. Auston, “Impulse response of photoconductors in transmission lines,” IEEE J. Quantum Electron. 19 (4), 639–648 (1983). M. Lundstrom, Fundamentals of Carrier Transport, 2nd edn, vol. 13, no. 2. IOP Publishing, 2002. E. Castro-Camus, J. Lloyd-Hughes, and M. B. Johnston, “Three-dimensional carrier-dynamics simulation of terahertz emission from photoconductive switches,” Phys. Rev. B Condens. Matter Mater. Phys. 71(19), 195301


(2005). 37. C. A. Criollo and A. Ávila, “Simulation of photoconductive antennas for terahertz radiation,” Ing. Invest. 35(1), 60–64 (2015). 38. N. Burford and M. El-Shenawee, “Computational modeling of plasmonic thin-film terahertz photoconductive antennas,” J. Opt. Soc. Am. B 33(4), 748 (2016). 39. N. Khiabani, Y. Huang, Y.-C. Shen, and S. Boyes, “Theoretical modeling of a photoconductive antenna in a terahertz pulsed system,” IEEE Trans. Antenn. Propag. 61(4), 1538–1546 (2013). 40. G. C. Loata, “Investigation of low-temperature-grown GaAs photoconductive antennae for continous-wave and pulsed terahertz generation,” Johann Wolfgang Goethe-Universität, 2007. 41. J. Prajapati, M. Bharadwaj, A. Chatterjee, and R. Bhattacharjee, “Circuit modeling and performance analysis of photoconductive antenna,” Opt. Commun., vol. 394, no. November 2016, pp. 69–79, 2017. 42. I. Malhotra, P. Thakur, S. Pandit, K. R. Jha, and G. Singh, “Analytical framework of small-gap photoconductive dipole antenna using equivalent circuit model,” Opt. Quantum Electron. 49(10), 334 (2017). 43. C. Criollo, A. Avila, and S. Winnerl, “Estudio de dispositivos terahertz para futuras aplicaciones en biomédica,” Universidad de los Andes (Salta) (2014). 44. M. Beck, H. Schäfer, G. Klatt, J. Demsar, S. Winnerl, M. Helm, and T. Dekorsy, “Impulsive terahertz radiation with high electric fields from an amplifier-driven large-area photoconductive antenna,” Opt. Express 18(9), 9251–9257 (2010). 45. S. Lepeshov, A. Gorodetsky, A. Krasnok, N. Toropov, T. A. Vartanyan, P. Belov, A. Alú, and E. U. Rafailov, “Boosting terahertz photoconductive antenna performance with optimised plasmonic nanostructures,” Sci. Rep. 8(1), 6624 (2018). 46. A. Dreyhaupt, S. Winnerl, T. Dekorsy, and M. Helm, “High-intensity terahertz radiation from a microstructured large-area photoconductor,” Appl. Phys. Lett. 86(12), 121114 (2005). 47. D. S. Kim and D. S. Citrin, “Coulomb and radiation screening in photoconductive terahertz sources,” Appl. Phys. Lett. 88(16), 161117 (2006). 48. P. U. Jepsen, R. H. Jacobsen, and S. R. Keiding, “Generation and detection of terahertz pulses from biased semiconductor antennas,” J. Opt. Soc. Am. B 13(11), 2424 (1996). 49. N. Khiabani, Y. Huang, Y. Shen, and S. Boyes, “Time variant source resistance in the THz photoconductive antenna,” in 2011 Loughborough Antennas & Propagation Conference, 2011, pp. 1–3. 50. J. Prajapati, V. K. Boini, M. Bharadwaj, and R. Bhattacharjee, “Comments on ‘Theoretical modeling of a photoconductive antenna in a terahertz pulsed system,’,” IEEE Trans. Antenn. Propag. 64(6), 2583–2584 (2016). 51. N. D. Arora, J. R. Hauser, and D. J. Roulston, “Electron and hole mobilities in silicon as a function of concentration and temperature,” IEEE Trans. Electron Dev. 29(2), 292–295 (1982). 52. D. M. Caughey and R. E. Thomas, “Carrier mobilities in silicon empirically related to doping and field,” Proc. IEEE 55(12), 2192–2193 (1967). 53. M. C. Beard, G. M. Turner, and C. A. Schmuttenmaer, “Subpicosecond carrier dynamics in low-temperature grown GaAs as measured by time-resolved terahertz spectroscopy,” J. Appl. Phys. 90(12), 5915–5923 (2001). 54. T. G. Sanchez, J. E. V. Perez, P. M. G. Conde, and D. P. Collantes, “Five-valley model for the study of electron transport properties at very high electric fields in GaAs,” Semicond. Sci. Technol. 6(9), 862–871 (1991). 55. S. M. Sze and K. K. Ng, Physics of Semiconductor Devices. Hoboken, NJ, USA: John Wiley & Sons, Inc., 2006. 56. R.-H. Chou, C.-S. Yang, and C.-L. Pan, “Effects of pump pulse propagation and spatial distribution of bias fields on terahertz generation from photoconductive antennas,” J. Appl. Phys. 114(4), 043108 (2013). 57. T. M. Wallis and P. Kabos, “Dopant profiling in semiconductor nanoelectronics,” in Measurement Techniques for Radio Frequency Nanoelectronics (Cambridge: Cambridge University, 2017), pp. 203–221. 58. D. Ban, E. H. Sargent, S. J. Dixon-Warren, I. Calder, A. J. SpringThorpe, R. Dworschak, G. Este, and J. K. White, “Direct imaging of the depletion region of an InP p–n junction under bias using scanning voltage microscopy,” Appl. Phys. Lett. 81(26), 5057–5059 (2002).