Observation and Modeling of Near-Bistable Dark-Mode Current-Voltage Characteristics in Semi-Insulating Gallium Arsenide With Implications for Photoconductors

In this work, we demonstrate and model the deep-level defect physics of semi-insulating gallium arsenide bulk photoconductive semiconductor switches (PCSS) with gap size of <inline-formula> <tex-math notation="LaTeX">$10 ~\mu \text{m}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$25 ~\mu \text{m}$ </tex-math></inline-formula> in dark-mode operation. Experimental measurements up to biasing field of 10 kV/cm show near-bistable characteristics in the dark-mode current-voltage relations for the PCSS, which cannot be reproduced through commercial Technology Computer-Aided Design simulations. Thus, we model the PCSS by solving for homogeneous non-equilibrium steady-state of the PCSS trap dynamics, where we introduce two semi-analytical models both involving two deep levels with impact ionization effects. Both models have an excited deep-level that can capture electrons from or emit electrons to the conduction band. The two models differ, however, by the fact that one has a ground state with capture and emission, whereas the other does not include such mechanisms but instead includes electron excitation and relaxation processes directly between the ground state and the excited state without interactions with the conduction band. We find that the former does not fit with experimental near-bistable features while the latter achieves a good match with the same total number of fitting parameters. Further measurements of bias upto 50 kV/cm on one <inline-formula> <tex-math notation="LaTeX">$10 ~\mu \text{m}$ </tex-math></inline-formula> PCSS confirms the validity of the second model as well. Finally, a brief discussion of the implications on the illuminated operation of the PCSS is also given to illustrate the importance of including defect interactions and defect avalanche effects.

is increasing interest in obtaining useful information on the deep-level traps of SI semiconductors from the output measurements of PCSS [13].
However, many of the existing studies of trap properties in relaxation semiconductors and in PCSS are under optical illumination, [14], [15], [16]. While such studies capture the trap physics under high-level injection (HLI), measurements under the dark mode are more suitable to directly extract the trap properties and optimize the device design for its use in the illuminated mode.
In this work, we theoretically study the dark-mode currentvoltage (I-V) characteristics of SI GaAs PCSS in which near-bistablility has been experimentally observed at room temperature. Bistability refers to an S-shaped Negative Differential Conductivity (SNDC) where two stable solutions of the free electron concentration are possible at the same applied electric field provided that the system has at least two deep-level defect states, so during a quasi-static measurement voltage sweep, hysteresis in current is observable [17]. When certain carrier transfer rates between the deep-level defect states overwhelm others, the semiconductor will no longer manifest true SNDC. Nevertheless, the semiconductor will display near-bistable I-V characteristics with a sudden increase in current at a critical electric field [18]. Such a near-bistable operation has been observed in the dark-mode measurements of the GaAS PCSS reported in this paper.
The experimental dark-mode I-V characteristics of GaAS PCSS discussed here cannot be reproduced via commercial Technology Computer-Aided Design (TCAD) simulations that incorporate only one trap level without avalanche effects. To resolve the discrepancy between TCAD results and experimental data, we implement two distinct custom numerical models for trap dynamics involving two trap levels, as well as trap-assisted impact ionization or avalanche. Our key finding is that only one of the two custom models in which the excited state participates in electron emission and capture processes with the conduction band can accurately replicate experimental results. Finally, we take advantage of the analysis conducted in dark mode with trapassisted avalanche processes to draw conclusions regarding the PCSS performance in the on-state. Our dark-mode results would be useful for the illuminated PCSS because the trap kinetics influence the electrostatic screening and thus the photo-current of the device. Besides, trap-assisted avalanche processes might take place on a time-scale much faster than the low-field trapping and de-trapping of carriers. Thus, the concurrence of avalanche and optical injection processes would be crucial to understand the physics and design of the PCSS.

A. EXPERIMENT
Measurements were taken on GaAs PCSS switches that followed a similar fabrication process and design that was reported in our previous work [13]. To summarize the fabrication, semi-insulating 2-inch GaAs wafer with 380 µm thickness from AXT Inc. was cleaved and cleaned in 1:1 HCl:H 2 O to create 1-inch quarter. The PCSS patterns were defined with direct-write ultraviolet lithography using a Heidelberg maskless aligner and negative photoresist, three patterns total. The ohmic contacts with metal stacks of Ge (20 nm)/Au (40 nm)/Ni (10 nm)/Au (20 nm)/Ti (10 nm)/Au (200 nm) were deposited with electron beam evaporation and rapidly thermally annealed. An approximately 10 µm thick polyimide layer was spun and cured to provide dielectric protection and support for an optical aperture, made of Ti/Au, for pulse compression photoconductive switch operation, not considered in this particular study, but described in our earlier work [13]. The polyimide was then coated with a thick positive resist and etched to expose the ohmic contacts for testing.
The devices tested in dark current mode implemented as co-planar wave-guide structures with varying gaps in the signal trace (60-µm wide) as shown in Fig. 1. The gaps tested were 10 and 25 µm. For each gap, five different devices were tested. The dark current measurements were taken in a probe station with a metal enclosure using a Keithley 4200A parameter analyzer. DC probes were placed on the open pads on both sides of the signal trace, and voltage was swept from equilibrium to 10 kV/cm equivalent average fields, in a positive direct first, and then repeated for the negative direction. An additional measurement on the 10 µm gap device was conducted up to 50 kV/cm in order to verify the high-field near bistability operation, as predicted by our models.

B. COMMERCIAL TCAD SIMULATION
The structure of the GaAs PCSS we simulated here is shown in Fig. 2. The separation (L) between the anode and cathode is varied, while other dimensions are fixed. First, we analyze the steady-state behavior of the device under darkmode operation. The temperature is fixed at 300 K, and the corresponding material properties of GaAs can be found in Table 1. We consider shallow-donor-shallow-acceptor-deepdonor (SDSADD) compensation using silicon, carbon and EL2 to achieve SI properties in GaAs. This scheme allows the sample to be slightly n-doped while also maintaining resistivity close to 10 8 .cm [19]. Table 2 shows the doping details.
Governing equations for the drift-diffusion simulations are the Shockley equations with spatial and time dependence not explicitly shown: Equation (1a) is Gauss' Law, where q is the elementary charge, p is free hole concentration, n is free electron concentration, N + sd is the ionized shallow donor concentration, N − sa is the ionized shallow acceptor concentration, N + dd is the ionized deep donor concentration, r is the relative permittivity, and 0 is the vacuum permittivity. We use Fermi-Dirac carrier statistics for higher accuracy in TCAD simulations, and we use Maxwell-Boltzmann carrier statistics in analytical and numerical calculations. Maxwell-Boltzmann distribution is justified in the dark-mode for a SI GaAs substrate because the Fermi level, E F , is far from the band edges (i.e., E C for the conduction band and E V for the valence band). At equilibrium, (1a) is equated to zero on the left-hand side to ensure charge neutrality and using Maxwell-Boltzmann statistics we obtain where p 0 is the equilibrium free hole density, and n 0 is the equilibrium free electron density. The equilibrium solution of E F and the carrier densities is obtained by iteratively solving (2a)-(2f). Our equilibrium solutions confirm that the low-field electron mobility in SI GaAs is 4588 cm 2 /V.s, which is close to the wafer-specifications of 4000 cm 2 /V.s. The sample resistivity, ρ, of 5.5 × 10 7 .cm is also close to 10 8 .cm as found in typical SI GaAs substrates [26], as well as close to the value reported in the wafer specification sheet. Equations (1b) and (1c) are the electron and hole continuity equations, respectively, where − → J n and − → J p are the respective electron and hole current densities. R n SRH , R n rad , R n Auger are the net electron recombination rates for the Shockley-Read-Hall processes, the radiative processes, and the Auger processes. Likewise, R p SRH , R p rad , R p Auger are the corresponding terms for holes. We neglect R Auger in darkmode due to the low carrier concentrations. For simplicity, R n SRH = R p SRH = R SRH , and R n rad = R p rad = R rad . Then, where τ h0 = τ e0 = 229 ps [27], and C rad = 2 × 10 −10 cm 3 s −1 [28]. Equations (1d) and (1e) describe the electron and hole current densities, respectively, due to drift and diffusion, where E is the electric field, μ n (D n ) is the electron mobility  (diffusivity), and μ p (D p ) is the hole mobility (diffusivity). The doping-dependent carrier mobility [29] is given as where μ low is the low-field mobility after taking doping into consideration; μ min , μ d , and ξ are fitting parameters whose values can be found in Table 3.
To model high-field transport, we consider a fielddependent electron mobility [30] given as where μ 1 , v sat , E 0 , E 1 , ν, β and γ are fitting parameters whose values are given in Table 4, and F d is the driving force, i.e., the average electric field, E. This is otherwise known as the Transferred Electron Effect (TEE) model [31]. It is a consequence of electron velocity overshoot when small angle forward scattering takes place with polar optical phonons. With optical deformation potential removing excess electron momentum, electrons return to steady state in the conduction band valley corresponding to a larger effective mass, which reduces their velocity at higher fields. We note that adopting the TEE model for high-field electron mobility in GaAs does not contradict the physics of the Gunn effect [32]. The formation of Gunn domains require the product, n × L, to exceed 10 12 cm −2 in GaAs [33] (n is the free carrier density, while L is the sample length). Since this condition is not met in the dark-mode GaAs switches analyzed here, we do not observe the N-shaped Negative Differential Conductivity (NNDC) in any of our measurements. Trap dynamics have also been included in TCAD simulations for the deep level EL2, where σ n = 1.8×10 −16 cm 2 is the electron capture cross section arbitrarily set close to theoretical values [34] and gives electron life time of approximately 1.5 ns, v n th = 4.4 × 10 7 cm/s is the electron thermal velocity, F 1/2 stands for Fermi integral of order 1/2, and F n is the electron quasi-Fermi level. Because time dependence of the trap dynamics is unimportant at steady state, we will not write time dependence explicitly for analytic expressions that appear in Section II-C. The result of trap dynamics can then be substituted into (1a) to determine φ, as well as E in the PCSS at steady state.

C. SEMI-ANALYTICAL TRAP MODELING
To resolve an apparent drawback in the commercial TCAD simulations, discussed in Section III-B, we consider two semi-analytical models for trap dynamics and seek homogeneous steady-state solutions for different biases and fit to the experimental I-F d characteristics. Unlike TCAD simulations that involve only one deep level, in both of our custom trap dynamics models, we incorporate two deep level traps. Moreover, for highly energetic electrons under large applied bias, impact ionization via deep level traps becomes important and is thus included in our model. We do not consider trap dynamics or avalanche for shallow dopants since they are almost fully ionized at room temperature if not subject to large optical injection. The first custom model for trap dynamics is referred to as the Doubly Model. In this model, we consider a doublycharged ground state (gs) N 2+ dd , a singly-charged excited state (es) N + dd , and a neutral state, N 0 dd , of the EL2 trap in GaAs. The latter two states are capable of capturing (c) an electron from and emitting (e) an electron to the conduction band, while the ground state communicates, via electron capture and emission, with the singly-excited state. These processes are illustrated in Fig. 3 in detail. Through capture of an electron, the positively charged state of the defect decreases, whereas through emission of an electron, the positively charged state of the defect increases. In addition to the capture and emission processes, both the neutral state and the excited state can be impact ionized to provide electrons to the conduction band and subsequently transition to the next more positively charged state. For the neutral state transitioning into the excited state, we describe avalanche rate as α es , while for the excited state transitioning into the ground state, we describe avalanche rate as α gs .
In the second model, referred to as the Singly Model [18], we consider a singly-charged excited state with concentration N es of EL2, as well as an arbitrary ground state with concentration N gs . Both these states can be positively charged or neutral depending on their electron occupancy. While the excited state participates in capture/emission processes from/to the conduction band, the ground state does not, and is thus missing c gs and e gs as seen in Fig. 4. The rate of electron transfer from the ground state to the excited state is . Arrows indicate change of states and direction of electron transfer. c es is electron capture rate of the excited state, c gs is electron capture rate of the ground state, e es is electron emission rate of the excited state, e gs is electron emission rate of the ground state, α es is impact ionization rate to the excited state, and α gs is impact ionization rate to the ground state. denoted as X, while Y denotes the rate of electron transfer from the excited state to the ground state. Like the Doubly Model, the Singly Model also has impact ionization processes governed by avalanche coefficients of ground and excited states denoted as α gs and α es , respectively.
Originally, this Singly Model was proposed by Schöll for low temperature regime bistability. However, it must be noted that the previous work only studies the trap-assisted avalanche process for very low electric fields, in the range of few volts per cm, for which precise temperature conditions must be realized. On the other hand, in this work, both theoretically and experimentally, we have considered field regime up to tens of kilovolts per cm. Only when the electric fields are large enough, carriers can gain sufficient kinetic energy to cause trap assisted avalanche process. That is, the kinetic energy gained by carriers over a mean free path due to the applied electric field, KE, must exceed the ionization energy of the lattice for avalanche. This is adequately expressed by the following equation: where E(x) is the electric field, and λ is the mean free path. Because λ decreases at higher temperatures due to stronger carrier-phonon scatterings, the electric field must be much higher to observe the avalanche process at higher temperatures. Thus, in our samples which are measured at 300 K, we need much higher electric fields (around 3 kV/cm) than those required in Schöll's analysis of trap-assisted avalanche. Several other works in the literature that have explored the room temperature applications of this model. For example, in [17], authors showed that the trap-assisted avalanche process at 300 K in a GaAs MESFET was the leading cause of the observed SNDC. In [35], authors used the same model to explain the switching behavior of TiN/GeTe 6 /TiN threshold devices occurring at roughly 20 kV/cm fields at 300 K. Our most important advance with this model is its application to photoconductive switches, and we explain the details of modeling as follows.

C.1. EQUILIBRIUM SOLUTION
To begin with, we first find the equilibrium solution for the various state variables. This case corresponds to the zero bias situation where we do not observe any impact ionization of traps. In the Doubly Model, the total deep donor concentration is defined as Applying the charge balance condition and ignoring the minority holes, we obtain Using Maxwell-Boltzmann statistics, the ionized concentration for the various charged states of EL2 is given as where g a = 4 is the shallow acceptor degeneracy, g d = 2 is the shallow donor degeneracy, g 0 = 1 is the N 2+ dd degeneracy, g 1 = 2 is the N + dd degeneracy, and g 2 = 1 is the N 0 dd degeneracy. Using E V as the reference for energy terms, transition energy E 1 = 10 − 00 = 0.52 eV is difference between the N + dd states and the N 2+ dd states. Transition energy 65 eV is the energy difference between the conduction band and the N + dd states [36]. For the Doubly Model, the compensation occurs for N sd = 2 × 10 15 cm −3 , N sa = 2 × 10 16 cm −3 , N dd = 1 × 10 17 cm −3 , and the equilibrium E F = E C − 0.62 eV with n 0 = 2.3 × 10 7 cm −3 .
For the Singly Model, the total deep donor concentration is defined as where N + dd is the total singly-charged empty donor concentration, N 0 gs is the neutral ground state concentration, and N 0 es is the neutral EL2 excited state concentration. The equilibrium solution is obtained by solving the following equations: where E es = E C − 0.65 eV is the energy location of the EL2 excited state, same as in the case of the Doubly Model.
For obtaining the equilibrium solution in the Singly Model, we neglect the ground state in the charge balance equation. To avoid inaccurate final results, we only use the charge balance equations to obtain an initial guess for the equilibrium concentration, which is then used to identify the ratio X/Y that yields an optimal fit to the measured I-F d characteristics of the PCSS. Once X/Y is extracted, the equilibrium concentrations can be automatically determined by setting F d = 0. The compensation for the Singly Model is obtained for N sd = 1 × 10 15 cm −3 , N sa = 1.9 × 10 16 cm −3 , N dd = 9.3 × 10 16 cm −3 , E F = E C − 0.61 eV, and n 0 = 2.48 × 10 7 cm −3 .

C.2. TRAP DYNAMICS
After obtaining the equilibrium concentrations for both the Doubly Model and the Singly Model, we solve for the trap dynamics at steady state in the presence of trap-assisted avalanche in order to match the model-generated current with the experimentally measured current in the PCSS.
For the Doubly Model, the trap dynamics equations are Similarly, the Singly Model has dn dt = α gs nN 0 gs + e es N 0 es − c es N + dd + α es nN 0 es , (17a) In both models, we assume the van Overstraeten model [37] for avalanche rate with external bias, which gives where a es , b es , a gs and b gs are fitting parameters to the avalanche model. By setting all time derivatives in both (16) and (17) to zero, we obtain steady-state homogeneous solutions for the concentrations.
Knowing the steady-state concentration, n, the current versus field in the PCSS can be obtained using where μ n is again modeled using the doping-and fielddependent mobility models described in (5) and (6), respectively, and A eff is the effective area of the PCSS contacts, which is related to their transfer length [38] and is estimated from the measurements. In the Doubly Model, the electron capture cross sections σ es and σ gs are extracted from calibrating the model against measurements, where the standard definition for electron capture cross section (σ n ) from any deep donor state has already been given in (8) and (9). In contrast, we fix σ es = σ gs = 1.81×10 −16 cm 2 for the Singly Model. To summarize, the seven fitting parameters for the Doubly Model are σ es , σ gs , a es , b es , a gs , b gs , and A eff . The seven fitting parameters for the Singly Model include a es , b es , a gs , b gs , X Y , Y, and A eff .
To understand the effect of X and Y on I versus F d in the Singly Model, we vary X/Y, while keeping all other parameters fixed. To do this, Y is fixed to an arbitrary value close to 1 × 10 −11 s −1 while X is varied. A change in X/Y implies changing the relative rate of relaxation and excitation processes between the ground state and the excited state. When X/Y < 1, relaxation dominates over excitation and more defects are in their ground state, while excitation dominates relaxation for X/Y > 1. As shown in Fig. 5, an increase in X/Y leads to a larger deviation of I versus F d from ohmic response. This is because the probability of carriers to be excited and thus impact ionized increases with an increase in X/Y. On the other hand, if X and Y were simultaneously increased, while keeping X/Y constant, the rate of excitation of carriers from the ground state to the excited state would increase. In this case, the knee in I versus F d would occur at a lower field for a larger X and Y as shown in Fig. 6.

VOLUME 11, 2023
Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.  Here, X/Y is fixed at 0.87, while Y is perturbed by ±50% from an arbitrary value close to 1 × 10 −11 s −1 . As a result, X also varies by ±50%.

A. EXPERIMENT
The three most reliable measurements out of the respective five devices of 25 µm and 10 µm gap are shown in Fig. 7. The random error in these measurements is lower as they are obtained on devices fabricated on the same wafer. Hereafter, we refer to these measurements as case 1, case 2, and case 3 for both 25 µm and 10 µm devices. The current is presented with respect to the average applied electric field, which can be calculated from the applied voltage divided by the device gap. At approximately 3-3.5 kV/cm electric field, the current increases sharply by about an order of magnitude. This unusual non-ohmic response of the dark-mode current motivated further investigation of the underlying physics via TCAD simulation.

B. COMMERCIAL TCAD SIMULATION
We compare and contrast our TCAD results with case 1 of L = 25 µm and L = 10 µm. The TCAD framework SDSADD compensation scheme is discussed in Section II-B.  As evident in Fig. 8, the commercial TCAD can fit to the low-field portion of the I-F d curves relatively well. In this region, the device shows an ohmic response and by appropriately choosing electron mobility and concentration, the error between the TCAD results and experimental data can be reduced. However, the TCAD models are unable to capture the increase in I at F d ≈ 3 kV/cm. The experimental observations cannot be explained by invoking the bulk avalanche process as it takes place at a significantly higher electric field in GaAs [39]. For similar reasons, we can also rule out Poole-Frenkel effects in our sample [40], [41]. Another common theory with regard to the SNDC is current filamentation in GaAs PCSS [42]. However, current filamentation is oftentimes observed in situations where optical carrier injection is present [43], [44], and the carrier density needs to be as high as 10 17 cm −3 for filamentation at relatively low electric fields [45]. In the dark mode, the carrier concentration of GaAs is quite low as the sample is highly resistive and nearly compensated such that the essential condition to observe current filamentation cannot be satisfied in our samples. Alternatively, it has also been shown in [46] that the electric field for filamentation in GaAs can be greater than 250 kV/cm, which is also an order of magnitude higher than the electric fields considered in our work. Instead, we focus on impact ionization of traps, commonly present in SI GaAs, as the leading cause of the observed sharp increase in I at a certain electric field. Moreover, we need to invoke two traps in our models as the experimental I-F d curves show two turning points with respect to the electric field.

C.1. DOUBLY MODEL
We apply the Doubly Model to analyze the 25 µm device for which results are shown in Fig. 9. The parameters in the Doubly Model that can be tweaked to match the measured data include σ es , σ gs , A eff , a es , b es , a gs , and b gs . The parameters σ es , σ gs , and A eff are field-independent and thus enable scaling the magnitude of I for a given F d . On the other hand, parameters a es , b es , a gs , and b gs tune the I-F d curve in two ways. These parameters shift the location of the first abrupt increase in I due to the avalanche of the excited state and can also adjust the slope of the I-F d curve following the first abrupt increase.
Although in the Doubly Model, we could effectively control the appearance of the second knee in I-F d curves by adjusting the avalanche parameters, this model does not allow us to control the magnitude of the current at the lower F d (i.e., the first knee in the measurement). The reason for the lack of tunability in the Doubly Model is because any process between the three deep donor EL2 states in Fig. 3 requires some interaction with the conduction band and thus influencing n and I. This means that in the Doubly Model, the increase in output current due to the transition from low-field region to the region after avalanche of the excited state is dependent on the electric field magnitude at which the first knee of I-F d occurs. Once we have a set field at which the first knee arises, the balance between c and e combined with α is already fixed, so the avalanche electron density and avalanche current that is in addition to the ohmic current are also fixed quantities. Thus, the independent tunability of the low field and high field characteristics is lacking in the Doubly Model. In other words, the adjustment of the onset electric field of the excited state will necessarily modify the electron concentration due to the interaction of the EL2 states with conduction band. Based on this analysis, we conclude that the Doubly Model cannot capture the essential features of the fabricated PCSS. The Doubly Model allows for observation of near-bistability but does not fit to experimental with quantitative accuracy. As a result, for the remainder of this paper, we will focus on the Singly Model in which the physics of carrier excitation and relaxation between the ground and excited states, captured via the X and Y parameters, provides a useful knob to control the current over a broad F d range.

C.2. SINGLY MODEL
The Singly Model was first applied to analyze the I-F d profile of the L = 10 µm device. For case 1 corresponding to L = 10 µm, we experimentally measure the dark-mode PCSS response at higher electric fields by conducting quasistatic measurements up to 50 kV/cm. However, due to the limitations of our measurement setup, for the remainder cases, we have acquired data only for fields up to 10 kV/cm. To explain experimental measurements, we use the Singly Model where all seven model parameters are adjusted to capture the increase in the measured current at around 3 and 30 kV/cm in Fig. 10. Although in our experimental setup, the electric field is restricted to ≤ 50 kV/cm, our model can be applied to predict the PCSS response at even higher fields, where saturation in the current is expected to emerge as shown in Fig. 10(c).
To obtain the optimal model parameters, the least-square fitting process was adopted, where the error is defined as square error = log 10 (I fit ) − log 10 I exp 2 .
Here, I fit denotes the model-generated data, while I exp refers to the data from experiments. The reason we define error in log-scale is due to the fact that the output current has values over several orders of magnitude. Once the optimal   parameters in the Singly Model are identified, the model is able to capture the measured I-F d data accurately, both in the linear scale ( Fig. 10(a)) and the log-scale ( Fig. 10(b)). In this model, an optimal X/Y can be identified that allows for a precise control of the magnitude of increase in I around the first knee. We note that X/Y is manually tweaked for the minimum root mean square error (RMSE) and then kept constant throughout. The X/Y parameter is the key to accurately capture experimental near-bistability because X and Y are direct interactions between the two defect states without involving the conduction band. As a result, the magnitude of the first abrupt increase no longer depends on the knee field location. The magnitude now depends on how fast the ground state can supply electrons to the excited state as opposed to extracting them. The ratio between X and Y thus plays the role to set the magnitude of observed avalanche current that is in addition to the ohmic current. Once the ratio is fixed, we then independently tune the location of the knee through the onset avalanche field parameter of the excited state, which allows us to fit to measurements with high accuracy. Values of the fitted parameters can be found in Table 5.
Our calibrated model clearly shows five distinct operation regimes in Fig. 10: the ohmic region (F d < 2 kV/cm), the first abrupt increase (F d ≈ 3 kV/cm), the transition region [F d ∈ (3, 20) kV/cm], the second abrupt increase (F d ≈ 40 kV/cm), and the saturation region (F d > 50 kV/cm). Using X/Y = 4.95 as determined from the measurement of 50 kV/cm case, we calibrate the Singly Model for L = 10 µm up to 10 kV/cm fields as shown in Fig. 11-13. The value of X/Y greater than unity can be appreciated because the ground state needs to supply more carriers to the excited state rather than extracting them in order to assist the avalanche process of the excited state. Only when the field is large enough will the ground state also get impact ionized and produce the second abrupt increase not seen in the 10 kV/cm measurements. Fitting to the very high field and very low field regions are obviously worse in the 10 kV/cm cases, but our model can accurately capture the first abrupt increase and the transition region observed in measurements. Lack of sufficient measurement data in the low-field (high-field) regime for 50 kV/cm (10 kV/cm) fields somewhat reduces the fidelity of the model predictions. Yet, the main feature related to the trap-assisted avalanche is readily reproduced in the model for all analyzed devices. Table 5 and Fig. 14 show that the extracted values of a gs and b gs parameters have a wider variation across different measurements, unlike other parameters that are more tightly controlled. As mentioned previously, this is related to the lack of measurement data at high fields. We also note that A eff acts as a multiplicative factor for the current and must be smaller than the physical area of the contacts, which is respected in our parameter extraction. We have observed in TCAD simulations of illuminated PCSS that transfer length of electrons is at most one tenth of the physical length of the electrodes. For dark-mode, the conductivity is smaller, so the transfer length decreases-this is consistent with our fitted A eff values that are in the range of 1/100 to 1/1000 size of physical area.
The Singly Model is also applied to the L = 25 µm device. The model-generated results are compared with measured data in Figs. 15-17, where all seven parameters in the model are again optimized to minimize the least-square fitting error. Once more, we observe good fitting in the first knee and the transition region. Table 6 summarizes the parameter values of all three cases corresponding to L = 25 µm device. The extracted parameter values, reported in Fig. 18, indicates that there is a large variation in the avalanche parameters of both the ground and excited states. Compared to all our L = 10 µm samples, measurement data for L = 25 µm have larger differences around the first knee, so the extracted parameters are unavoidably also more spread out.

D. ILLUMINATED PCSS
The dark-mode characteristics of the PCSS shed light on the underlying physics of trap-assisted avalanche mechanisms that are present in GaAs substrate. To extend our understanding to a PCSS under illumination, we perform transient homogeneous simulations of the L = 10 µm device. Toward this, we consider parameters from the 50 kV/cm case fitted with the Singly Model, and add an optical generation rate to (17a).   For simplicity, we use a square-wave pulse with optical generation rate G opt and time duration t. We solve the system from time t = 0 to some t t when the concentration of electrons have reached a new steadystate after the optical pulse. We restrict our analysis by limiting the product G opt t to ensure that the shallow VOLUME 11, 2023 395 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.  dopants are still close to being fully ionized, i.e., E F is still far from the band edges, and our assumption of Maxwell-Boltzmann statistics is still valid. As illustrated in Fig. 19, we have considered three cases with combinations of G opt = 10 25 cm −3 s −1 & t = 1 ps, G opt = 10 25 cm −3 s −1 & t = 10 ps, and G opt = 10 26 cm −3 s −1 & t = 1 ps. We find that at fixed G opt & t combination, increasing the voltage will escalate the avalanche processes in the PCSS, which leads to a monotonic increase of the peak carrier concentration, n, as well as the full-width at halfmaximum (FWHM) of the n waveform. We also find that the n waveforms of Fig. 19(b) and Fig. 19(c) look almost identical. This is because is that we are solving for homogeneous solution of the illuminated PCSS, and effects of channel length, contact length, dielectric relaxation, and recombination have been neglected. Therefore, turn-off is a much slower process than turn-on, and it is almost entirely dominated by avalanche when capture and emission in the defect level are field-independent. We also observe that N 0 es and N 0 gs do not change with time because the optical injection is not strong enough to excite a significant fraction of electrons in the occupied/unionized excited states. Thus, N 0 es and N 0 gs are only bias-dependent, but time-independent.
Figures 20 shows a comparison of the temporal profile of n for t = 1 ps, while Fig. 21 presents a comparison of the FWHM versus F d for different optical injection characteristics.
We show that for a stronger injection condition, i.e., G opt t, more free electrons are available in the system, so the influence from impact ionization of deep levels will be more detrimental to the PCSS turn-off performance, especially in the negative differential mobility (NDM) regime for which F d > 4 kV/cm. In fact, the devices with F d ≥ 8 kV/cm cannot be turned off unless we remove the electrical bias in addition to removing the optical bias. As a result, when designing a GaAs PCSS with active trap-assisted avalanche processes, low-field operation would be preferred. However, if the device is a pulse compression photoconductive switch [13], [47] that must be operated at high fields that access the NDM regime, the total laser power must be kept below a certain maximum threshold to achieve faster turn-off characteristics.

IV. CONCLUSION
In this paper, we demonstrated dark-mode I-F d characteristics of six SI-GaAs PCSS devices, three of which have dimension L = 25 µm while the other three have L = 10 µm Experiments show a sudden increase in I when F d is close to 3-3.5 kV/cm, which could not be captured adequately within commercial TCAD solvers. The characteristic feature in the measured current could not be explained via bulk avalanche or Poole-Frenkel emission processes as those require a much higher field in GaAs. We also exclude the possibility of filamentation since the devices examined here are not operating in the high gain mode that is featured in illuminated GaAs PCSS samples.
To explain experimental measurements, we implemented two semi-analytical models for trap dynamic-Doubly Model with a doubly-charged ground state, a singly-charged excited state, and a neutral state, and Singly Model where both ground state and excited state could be either neutral or singly-charged. In the Doubly Model, capture, emission and avalanche processes of electrons, involving the conduction band, are included. In the Singly Model, the ground state only has the avalanche process, whereas the excited state could have capture, emission and avalanche. The Singly model also includes a direct electron relaxation from the excited state to the ground state and direct electron excitation from the ground state to the excited state. We showed that only the Singly Model could faithfully reproduce the measured I-F d profile for different length devices. This confirmed that the device exhibits at least two deep-level traps with direct electron transfer between the ground state and the excited state of the trap, without involving the conduction band. The Singly Model was used to simulate the timedomain response of the PCSS under homogeneous optical excitation. The illuminated PCSS exhibited a super-linear FWHM versus applied bias due to the avalanche process, and some of the devices with severe trap-assisted avalanche   effects do not turn off. To identify the actual energy levels of defects as well as information on more minor defects, the model presented here will be complemented with Arrhenius measurements, which is part of our prospective work.