Systematic analysis of semiconductor photoconductivity dynamics under different laser excitations: two- and three-level models

A numerical study of a model photoconductor under pulsed and continuous-wave laser excitation is presented. From the solution of a system of partial differential equations related to the populations of the conduction and valence bands, we analyze the time behavior of the photoexcited concentrations of electrons and holes where the diffusion phenomena and the presence of electric field are neglected. Additionally, for the three-level model, the population of trap levels is also introduced and analyzed as a third variable. The dynamics of electrons, holes and traps levels concentrations are studied in the presence of different types of pulse and harmonic laser photoexcitations. We show that an improved physical insight is obtained through a systematic analysis of the influence of the main physical parameters, such as generation and recombination rates, between the available energy levels.


Introduction
The photoconductive phenomenon, i.e., the physical mechanism which creates electron-hole pairs from the absorption of electromagnetic radiation, thus changing the electrical conductivity, has been extensively studied for its interest in fundamental material properties and for several practical applications [1]. Depending on the specific material, the transitions of electrons and holes take place either directly between valence band (VB) and conduction band (CB) or through trap levels (TL) situated in the energy gap. A careful analysis of the dynamics of these transitions is necessary for the optimization of different radiations detectors, photoconductive switches and other photoelectric devices [2,3].
As a recent example, semiconductor-based photoconductors are more and more studied for high-frequency generation of electromagnetic waves with rapid progress over the past decade [4,5]. In this context, high-speed photodiodes, e.g., uni-traveling carrier photodiodes [6,7], are emerging as essential devices for high data-rate communication and related methods such as beam-steering [8][9][10][11]. Indeed, the photoconductive-based sources are considered as promising emitting devices because they can be coherent, widely tuneable and cost effective by operating at room temperature [12][13][14][15].
While the generation of short pulses using optical excitation from ultrafast lasers has been studied in the literature [16], the case of continuous-wave (CW) generation by downconverting two single-mode laser signals or a dual-mode laser signal has received less attention [14] even if such an analysis is mandatory to model correctly photomixer sources [10,17,18]. Moreover, even in the case of pulsed excitations, available studies are usually limited to a specific situation using a single set of physical parameters.
To fill this lack of knowledge, in this paper, we propose a simple but systematic numerical analysis of the physical processes that take place in a model semiconductor material excited by pulse (Gaussian and square temporal functions) and CW harmonic laser signals. The physical model is based on a system of coupled differential equations describing the band-to-band generation and recombination processes possibly coupled to the release and capture rates associated with the transitions between the TL and the bands. In particular, we present two models both taking into account different excitation scenarios of interest for practical situations. The former one is a two-level model which considers only the transitions between the VB and the CB for different pulse durations and harmonic frequencies of laser excitation. The latter one is a more complex three-level model which includes the presence of TL between the VB and the CB. In this case, we analyze the effect of changing the release and capture rates as well as the characteristic times and frequencies of the laser excitations.

Physical model
This section describes the physical model used to study the photoconductive dynamics of a model semiconductor bulk sample excited by a laser emitting photons at wavelength (and frequency ) and producing electron-hole pairs. The spatio-temporal dynamics of the photo-excited electrons and holes can be modeled by a system of coupled equations describing the evolution of the photoexcited electron n(t) and hole p(t) concentrations, within the CB and VB, respectively, as functions of time t. By assuming a negligible electric field and neglecting diffusion phenomena, one can consider that the carrier population and dynamics are only affected by the generation-recombination (GR) processes. This assumption is usually employed in the study of bulk samples but it may become not rigorously valid for more complicated samples. The dynamics of photo-excited carrier are calculated by the electron and hole continuity equations when the drift and diffusion contributions are neglected [19]: and where, for electrons and holes, respectively, • G n and G p are the generated densities of carriers per unit time ; • R n and R p are the recombinated densities of carriers per unit time. (1) For most of semiconductor materials, it has been shown that an important role can be played by traps whose energy levels are located within the energy gap [1]. Such traps can be intrinsically present as defects in the semiconductor bulk material or intentionally introduced in order to decrease substantially the carriers lifetimes for high-frequency applications [19][20][21]. These levels can be occupied by electrons and/or holes, depending on their positions within the band gap. In this work, we consider the case of a single-level trapping both electrons and holes. We assume that the traps are distributed in a homogeneous way in the semiconductor with a density sufficiently low to be represented by a single TL. Since the TL concerns both electron and holes, we describe the trap population through the trapped electron density only n t (t) , and we note N t the available electron states density, so that the trapped hole density at time t is equal to N t − n t (t) . In these conditions, the trap electron population n t depends only on the generation-recombination processes, so that where G t and R t are the generated and recombinated density of trapped electrons per unit time.
The generation-recombination processes associated with Eqs. (1)-(3) are schematically depicted in Figs. 1 to 3 showing the possible transitions of photoexcited carriers among the VB, the CB and the TL. Electrons and holes are indicated, respectively, with circles and squares. The transitions between energy levels are reported as vertical arrows, and the involved carriers leave an empty symbol to reach a closed one. We can remark that, for each population, there are two generation and two recombination processes. Fig. 1 Generation-recombination processes involving the electron density n(t) (only the electron transitions between different energy levels are presented). The processes from 1 to 4 correspond to bandto-band electron generation (from VB to CB), band-to-band electron recombination (from CB to VB), electron release (from TL to CB) and electron capture (from CB to TL), respectively. Closed circles represent final occupied electron states and open circles final empty electron states, respectively 1 3

Generation-recombination processes
We discuss separately the balance equations for the three variables n(t), p(t) and n t (t) describing, respectively, the CB, VB and traps levels populations. In order to make explicit the GR terms of Eqs. (1)-(3), we consider that each transition induces a variation of the density equal to the product between (i) the associated transition rate, (ii) the density of carriers of the initial state (that is able to make the transition) and (iii) the available density of final states. For what concerns n(t), we consider the following GR processes, ordered with the numbers corresponding to those of Fig. 1 : 1. the band-to-band generation rate. If we note N and P the densities of states of the CB and VB, respectively, and E vc the transition rate in cm 3 s −1 , the time variation of n(t) due to this process can be written as 2. the band-to-band recombination with a rate E cv expressed in cm 3 s −1 , so that 3. the electron release from TL to CB, which constitutes a second generation term. It involves the density of trapped electrons n t and of available CB states N − n with a transition rate E tc , so that: 4. the electron capture from CB to TL, which constitutes a second recombination term, involving the density of electrons n and the density of available states in the TL, N t − n t . If we note E ct the transition rate, we obtain: The PDE (1) describing the time evolution of n(t) can then be explicitly written as The GR processes associated with the hole density p(t) are depicted in Fig. 2 and correspond to the following processes: band-to-band recombination rate H vc , band-toband generation rate H cv , the hole capture rate H vt and the hole release rate H tv .
By following the derivation method of electrons, we obtain the VB hole population PDE as: G n 2 = E tc n t (N − n), Finally, the balance equation for n t (t) describes the following processes, depicted in Fig. 3 with the corresponding numbers: • the electron release from TL to CB described with a transition rate E tc through the expression • the electron capture from CB depending on the available electron states within the TL and on the free electron density through the transition rate E ct with the expression Fig. 2 Generation-recombination processes involving the hole density p(t) (only the hole transitions between different energy levels are presented). The processes from 1 to 4 correspond to band-to-band hole recombination (from VB to CB), band-to-band hole generation (from CB to VB), hole capture (from VB to TL) and hole release (from TL to VB), respectively. Closed squares represent final occupied hole states and open squares final empty hole states, respectively Generation-recombination processes involving the trapped electrons density n t (t) (only the transitions between bands and traps level are presented). The processes from 1 to 4 correspond to electron release (from TL to CB), electron capture (from CB to TL), hole capture (from VB to TL) and hole release (from TL to VB), respectively. Processes 3 and 4 can be alternatively represented as electron release (from TL to VB) and electron capture (from VB to TL), respectively. Closed circles/squares represent final occupied electron/hole states and open circles/squares final empty electron/hole states, respectively • the hole capture from VB to TL, which can be viewed as an electron release from TL to VB, described as • the hole release from TL to VB, which can be considered as an electron trapping from VB to TL described with the transition rate H tv as Therefore, we obtain the following balance equation for n t (t)

Assumptions and simplifications
To obtain a suitable system of equations, some simplifications of the Eqs. (8), (9) and (14) can be performed.
1. As concerns the photo-generation processes, we remark that electrons and holes are always involved by pairs, so that E vc = H cv . Moreover, if we consider that the photogenerated population is negligible in comparison with the available states (that is the case of a weak photogeneration regime and a non-degenerate material), we can write that P − p ≃ P and N − n ≃ N . Since N and P have constant values, the generation depends only on the transition rate, which is essentially on the power of the laser and on the photo-excited volume. Then, we can write where G is the electron and hole photo-generation rate, given in cm −3 s −1 . 2. The band-to-band recombination rate for electrons is usually written in the form and the band-to-band recombination rate for holes in the form: where the common band-to-band recombination bb is introduced [1]. It should be noticed that, under lowlevel excitation, the value of bb is constant while under high-level excitation (i.e., when the concentration of (11) G t 1 = E ct n (N t − n t ), photoexcited carriers is higher than the equilibrium concentration) it may become time dependent, thus introducing an additional source of nonlinearity in the system of equations. In the model here presented, we assume constant values for bb in order to better separate the role of this parameter with respect to other characteristics times. It is rather straightforward to introduce a time-dependent band-to-band recombination time in the framework of the same model. Moreover, if we simulate a situation where n = p (e.g., the two-level model presented hereafter) we can assume that R n = R p , while for the case of more complicated scenarios (e.g., the three-level model presented hereafter) since n is not necessarily always equal to p, the two recombination rates may differ. 3. As concerns the electron release from the trap level to the CB, the weak photo-generation regime allows to write that and, since N is constant, we introduce the electron untrapping rate g n = E tc N given in s −1 . 4. The same remark is made for the case of the hole untrapping, so that where g p = H tv P is the hole untrapping rate given in s −1 . 5. As concerns the CB electron capture, we introduce an electron relaxation rate 1∕ n given in s −1 by 1∕ n = E ct N t , so that 6. We do the same for the VB hole capture by introduction a hole relaxation rate given in s −1 by 1∕ p = H vt N t , so that Then, the PDE equations are rewritten as Depending on the type of laser excitation the three photogeneration rates g n , g p and G, which are proportional to the optical power, can be appropriately adjusted [19]. The solution of the differential equations is obtained by a numerical method based on the MATLAB solver using Runge-Kutta 4th order algorithm.

Two-level model
It is useful to analyze the GR processes in a simplified twolevel model where the TL are absent, which corresponds to a perfect intrinsic semiconductor. This situation can be also used as a validation model to analyze the interplay between the different characteristic times of the system. In this case, the Eq. (24) for n t disappears and Eqs. (22) and (23) become and where it is trivial that n(t) and p(t) will exhibit the same temporal dynamics determined by the interplay between the generation processes, modeled by the G term, and the recombination processes controlled by the band-to-band recombination time bb . This latter, depending on the chosen semiconductor, can vary within a quite large domain of values [22,23]. In our case, we choose a reasonable value bb = 1 s corresponding to some typical semiconductor as GaAs, InP, InGaAs and amorphous Si [24] but it is evident that, from a physical point of view, what matters is just the comparison between the value of bb and the time scale of generation processes, i.e., G. Therefore the model can be easily generalized to other situations where the value of bb can be significantly different from 1 μs.

Gaussian and square excitations
As a first situation, we consider the case of a pulse excitation where the generation rate takes a Gaussian form where G 0 represents the maximum of the function, t 0 its center and 2 its variance. Indeed, in many practical cases the temporal behavior of a laser pulse is approximated by a Gaussian function [25]. Here we choose a value G 0 = 2 × 10 16 cm −3 s −1 . We report in Fig. 4 the behavior of n(t) for different values of . We remark that when (i.e., a quantity that can be used to characterize the time duration of the Gaussian pulse) is larger than bb , the concentration of photoexcited electrons and holes follow the behavior of G(t) and keep also a Gaussian shape reaching a maximum value determined by the stationary solution n(t) = G bb = 2 × 10 10 cm −3 as shown in Fig. 4a. However, when becomes comparable or smaller than bb , we observe two main modifications: the density of photoexcited electrons and holes does not exhibit a Gaussian shape anymore and its maximum value progressively decreases as the duration of the excitation decreases (see Fig. 4b).
A similar behavior is obtained if the generation rate is described by a rectangular function of the type where t 0 represents the center of the function and its duration. The results are shown in Fig. 5 for different values of . Indeed, we observe that if the duration of the laser excitation is longer than the band-toband recombination time the density of electrons and holes keeps a shape similar to G(t) even if the rise and fall parts are not sharp but governed by bb as shown in Fig. 5(a). However, when the duration of the pulse becomes comparable or smaller than bb , the shape of n(t) is progressively modified in a similar way as observed in the case of a Gaussian perturbation, i.e., the response is no longer rectangular and the maximum value of n(t) decreases with decreasing value of (see Fig. 5b).
It is interesting to analyze the maximum density of photoexcited electrons and holes as a function of the pulse duration (i.e., for the Gaussian case and for the rectangular case) as shown in Fig. 6. We remark that, independently of the temporal shape of the excitation, when its characteristic duration becomes smaller than bb the photoexcitation is progressively less effective and the maximum concentration of photoexcited carriers is smaller than expected due to the fact that the system cannot reach the stationary conditions.
In Fig. 7, we have reported the value of G 0 that should be employed to recover the initial values of electrons concentration n(t) which is determined by the stationary condition and equal to n(t) = 2 × 10 10 cm −3 .
For very short pulse durations the results shown in Fig. 7 demonstrate that the amplitude of the laser excitation should be increased of almost two orders of magnitude in order to recover the same concentration of photocarriers.

Harmonic excitation
Another type of excitation can be obtained by superposing two CW lasers with close frequencies thus producing a laser beating where the envelope of the two signals behave as an harmonic function oscillating at the difference frequency [26]. This technique is typically used for frequency down-or up-conversion as, for instance, in the case of CW terahertz generation [27,28]. In the harmonic excitation, the generation rate is expressed by the form G = G 0 (sin(2 f )) 2 . Figure 8 reports the temporal behavior of n(t) for two beating frequencies: the former one ( f = 10 7 Hz) is comparable to the inverse of the band-to-band recombination time (1/ bb = 10 6 Hz) while the latter one ( f = 10 12 Hz) is much bigger. In all cases, after an initial transient, the time behavior of n(t) is harmonic with an oscillating frequency corresponding to the beating frequency of the lasers. However, the amplitude of the oscillations decreases when the beating frequency increases and becomes greater than 1∕ bb .  This effect is shown in Fig. 9 which reports the maximum and minimum values of the concentration of photoexcited carriers as functions of the beating frequency: We observe that at increasing beating frequencies the amplitude of the oscillations becomes significantly reduced.
This effect should be taken into account since it can reduce the up-conversion efficiency. One possibility to counteract the effect in the reduction of the amplitude of the harmonic oscillations is to increase the generation rate which, from a practical point of view, means to increase the power of the laser.
It is interesting to notice that, in this case, the amplitude of the response at 1 THz is only divided approximately by a factor 4, whereas in the case of pulse excitation the maximum response is reduced by more than 2 orders of magnitude suggesting that the harmonic excitation seems to be more robust in terms of optic-to-electric conversion. Figure 10 shows the amplitude of the continuous wave G 0 that should be applied for obtaining the initial electron concentration (the maximum value of n(t) in Fig. 9 is 2 × 10 10 cm −3 ).
In the continuous wave regime, the results have shown (see Fig. 9) that the response is weakened for the frequencies f ≥ 10 7 Hz. For obtaining the initial value of concentration n(t) the amplitude of the harmonic signal G 0 should be increased with the increasing frequencies. During the short time of harmonic excitation, we observe a slight increase of amplitude G 0 which leads to generate more electrons in the energy levels and therefore increase the response of the electrons after several transitions processes (see Fig. 10).

Three-level model
In this section, we consider the GR dynamics in the presence of TL which are supposed to be located in a generic position within the band gap and to act as traps for electrons as well  as for holes, as illustrated in Fig. 3. The study of the transitions of electrons and holes through the TL is here presented in two steps: at first, we analyze the transient behavior of the different concentrations starting from an initial arbitrary state in the absence of laser excitation, then, in a second time and when the stationary state is achieved, we discuss the effect of the presence of different types of laser excitations. For the simulations, if not specified, the following values have been used: traps density N t = n 0 = 10 11 cm −3 , band-toband recombination time bb = 10 −6 s and hole recombination time p = 4 × 10 −7 s.

Transient behavior in the absence of photoexcitation
For the study of the transient behavior, we consider the initial state corresponding to all energy levels empty, i.e., n(t) = p(t) = n t (t) = 0 . We stress that, even if the transient behavior does not correspond necessarily to a real situation, it is of great interest for evidencing the intrinsic characteristic times of the system. Before analyzing the results of simulations, it is worthwhile to remark that analytical solutions for the stationary values of the different concentrations can be obtained. Indeed, by solving the system of coupled equations (22), (23) and (24) without band-to-band generation G, we get the following expressions: and where which can be compared to the numerical results. Figure 11 shows the transient behavior of the different concentrations when the generation rates of electrons and holes are equal.
If, in addition to the condition that g n = g p , both quantities are almost equal to 1∕ n = 1∕ p , the time evolutions of electrons and holes concentrations remain nearly the same and exhibit an exponential increasing behavior (see Fig. 11a). In this situation only one time scale determines the transient behavior. The decreasing of the untrapping rates g n = g p to a value equal to 1∕ bb (i.e., the inverse of the band-to-band recombination time) modifies , significantly the transient behavior of n(t) and p(t) which are no longer equal at any time as in the previous case (see Fig. 11b). Here, before reaching an equal stationary value, p(t) exhibits an overshoot behavior evidencing the presence of two time scales.
In both cases the stationary value of n t (t) is equal to N t ∕2 which indicates that the TL are half occupied and half empty.
It is interesting to study the effects induced by a change in the generation and recombination rates on the different concentrations. In order to keep the analysis sufficiently simple, we keep n = p when discussing the effect of a change in the generation rate and g n = g p when the recombination rate is varied as shown in Figs. 12a, b, respectively.
On one hand, when the electron generation rate is increased to a value equal to 10g p = 10 bb ,which favors the transitions of electrons from TL to the CB, we observe a quite similar increase of n(t) and p(t). However, an increase of g n is accompanied by a significant decrease of the TL concentration n t (t) as evidenced in Fig. 12a and confirmed by Eq. (29). In addition, in this situation, the recombination processes between the band and the TL (a) (b) Fig. 11 Electrons n(t), holes p(t) and traps n t (t) concentrations as functions of time in the absence of photoexcitation with n = p . We take a g n = g p =7×10 7 cm −3 s −1 and b g n = g p = 10 6 cm −3 s −1 (given by the case p = n ) are dominant compared to the band-to-band processes (governed by bb ).
On the other hand, when the electron recombination time is slowed down to a value n = 10 p but keeping equal generation rates ( g n = g p ), the transient of n(t) toward its stationary value becomes longer and a more significant difference is observed on the time evolutions of n(t) and p(t) as reported on Fig. 12b. Moreover, the TL become more filled by electrons (i.e., the stationary value of n t (t) increases) while band-to-band processes do not have a significant influence.
Indeed, the effect of band-to-band recombination on the time evolution of the different concentrations is analyzed in Fig. 13 where calculation are performed for the case g n = 1∕ n = 10g p and g p = 1∕ p .
We remark that in this case Eqs. (27), (28) and ( . Indeed when the band-to-band recombination becomes faster, there is a weaker probability that TL are occupied thus leading to the decrease of n t (t) shown in Fig. 13b.
In parallel, the concentration of electrons and holes as well as the duration of the transient increase when the band-to-band recombination processes are slow, as shown in Fig. 13a. In contrast, the recombination processes from CB, VB and TL are much less important for smaller values of 12 Electrons n(t), holes p(t) and traps n t (t) concentrations calculated for different electrons generation rates g n and recombination times n with g p = 10 6 cm −3 s −1 . We take a g n = 10g p with n = p and b n = 10 p with g n = g p

Gaussian photoexcitation
After reaching their stationary values, we analyze here the effect of a pulse band-to-band photoexcitation having a Gaussian form of the same type as studied in Sect. 3. Moreover, since this previous study has evidenced that there is no significant physical difference in the responses to a square with respect to a Gaussian excitation, we have here limited our analysis only to the second case.
In Fig. 14 we present the time evolutions of the three concentrations n(t), p(t) and n t (t) resulting from a Gaussian photoexcitation of different durations (i.e., different standard deviations) superimposed to the respective stationary values. Again, to simplify the analysis, we have considered only two situations: the former one corresponding to g n = g p and the latter one corresponding to g n = 10g p .
The stationary photocarriers concentrations (electron n and hole p) increase from 1.4×10 11 cm −3 to 3.6×10 11 cm −3 when the electron generation rate increases from g n = g p to g n = 10g p (see Fig. 14a); however their time evolutions remain similar in both cases. Indeed, as long as the duration of the pulse is comparable or longer than the recombination times (and inverse generation rates), the time behaviors of n(t) and p(t) follow the shape of the photoexcitation impulse.
When the duration of the pulse becomes shorter, we observe a progressive decreasing of the maximum values of n(t) and p(t) and significant deviations from the Gaussian shapes which become strongly asymmetric (see curves labeled 2 and 3 in Fig. 14a). As a result, the spectral content of the photoconduction will be significantly modified.
In parallel, high values for the TL occupation can be observed in the case of g n = 10g p (see Fig. 14b). while it keeps a constant value for the case g n = g p , which means that no additional filling of TL is observed in the second case. For this reason, the curves of n t (t) are reported in Fig. 14b only when g n = 10g p . In summary, for high electron generation rate ( g n = 10g p ) and longer photoexcitation time ( = 10 −6 s), the generation processes of photocarriers in CB, VB and TL are favored more than the recombination process toward de TL, and a sharp pic in n t concentration is obtained. Figure 15 illustrates the time evolutions of n(t), p(t) and n t (t) for the same standard deviations as in Fig. 14 but calculated for two different situations: equal recombination times n = p and faster electron recombination time n = 10 p but, in both cases, keeping equal generation rates g n = g p . To avoid superimposition of the curves, the central time of the Gaussian photoexcitation has been shifted in Fig. 15a.
Analogously to the results shown in Fig. 14, the stationary values of n(t) and p(t) depend on the values of the recombination times and they are actually equal to 2.1 × 10 11 cm −3 (for n = p ) and 1.4 × 10 11 cm −3 (for n = 10 p ).
Despite a slight difference in the time evolutions of n(t) and p(t), which has been already evidenced in the study of the transient behavior (see for instance Fig. 12b), the results obtained by varying the recombination time are qualitatively similar to those obtained by varying the generation rates previously shown in Fig. 14. The main difference concerns the behavior of the occupied traps concentration n t (t) . Indeed, while for the case n = p no change in n t (t) is observed (not reported in Fig. 15(b)), an increase of the electron recombination time to a value n = 10 p favors a decrease of the population of electrons on the TL because they spend more time in the CB. As a consequence, n t (t) exhibits the reversed behavior shown in Fig. 15b evidencing that a larger effect is obtained for a longer duration of the Gaussian photoexcitation (see the case = 10 −6 s in Fig. 15b). In contrast, a faster pulse

Harmonic photoexcitation
In this section, we consider a harmonic excitation of two CW lasers oscillating at the beating frequency f applied to the three-level bulk semiconductor where the stationary condition is already achieved. This harmonic excitation is assumed to pump the electrons from the VB to the CB. This band-to-band generation is modeled by introducing the generation rate G = G 0 sin 2 (2 ft) in Eqs. (22) and (23).
The effect of electron generation rate g n = 10g p and electron recombination time n = 10 p on the temporal behavior of electron n(t), hole p(t) and traps n t (t) is reported in Fig. 16a, b, respectively. A harmonic response corresponding to the beating frequency of the laser photoexcitation is observed in the response of all concentrations. In addition, the amplitude of the oscillations is found to depend, on the one side, on the excitation frequency and, on the other side, on the values of the generation rates and recombination times.
To get a better physical insight into these processes, we have focused the attention on the maximum and minimum values of the concentrations oscillations.
The effect on the amplitude of the concentrations oscillations of a progressive increase of the frequency of the photoexcitation from f = 10 4 Hz (i.e., a value much smaller than the different generation rates and inverse recombination times) to f = 10 12 Hz (i.e., a value much greater than the different generation rates and inverse recombination times) is analyzed in Fig. 17. As a general trend, we observe that, in the low-frequency region (here corresponding approximately to f ≤ 10 7 Hz where the oscillations behavior of photocarriers appears), the amplitude of the oscillations of all concentrations is great, which is a consequence of the fact that n(t), p(t) and n t (t) follow the time behavior of G(t) as if the system was near to quasi-stationary conditions. Moreover a slightly higher amplitude of the oscillations is found in the electrons and holes concentrations for an increased electrons    Fig. 17a) than for an increased electrons generation rate (squares and circles in Fig. 17a). This effect is reversed if the oscillations in the traps concentration is observed as reported in Fig. 17b. When the frequency of the photoexcitation becomes comparable to the shortest characteristic time of the system (either a recombination time or an inverse generation rate), the amplitude of all the concentrations oscillations is significantly reduced, since the system is no longer able to follow to excitation signal.

Comparison with experiments
The model developed in the previous sections can be validated by comparison with experimental results. It must be noticed that the majority of available experiments concerns the pulse regime where a bulk material or a device is photoexcited by a laser pulse whose time behavior can be approximated by a Gaussian function. To emphasize the flexibility of the model, we have chosen the experimental results obtained on two very different systems: an InGaAs sample and a WS 2 monolayer presented in Refs. [21] and [29], respectively. In Ref. [21], an optical pump-probe technique with a Gaussian pulse is used to characterize the conductivity spectra of In 0.53 Ga 0.47 As films irradiated with heavy ions (Br + ). In order to compare the experimental data of Fig. 6a sample A in Ref. [21], we use the numerical model of section 4 to reproduce the normalized electron concentration. The model considers an optical pulse with Gaussian duration = 0.1 ps in the absence of band-to-band recombination processes. The numerical calculations consider a free carrier concentration n 0 = 1.9 × 10 16 cm −3 , a traps concentration N t = 3.1 × 10 15 cm −3 an electrons capture time n = 110 ps, and a holes capture time p = 290 ps. The electron and holes release rates are described by the Shockley-Read model [21] as: and where E g = 0.74 eV is the energy band gap of InGaAs, E t = 0.3 eV is the position of the trap level below the conduction band minimum and N e,p is the effective density of states of electrons and holes.
As a second example of photoconductive material, we consider monolayer WS 2 which has been studied by experimental method in Ref. [29]. To compare with experiments, we have considered the following parameters in the numerical calculations: the position of traps level in the middle of band gap ( E g = 2.05 eV) E t = 0.5 eV with traps concentration N t = 10 10 cm −3 , initial concentration n 0 = 1.9 × 10 16 cm −3 , electrons recombination time n = 5 ps and holes recombination time p = 6 ps. In the case of Fig. 5a in Ref. [29], the pump fluence equal to 115 J/cm 2 corresponds to a pulse duration = 3 ps. Despite the differences in the time scales and in the material parameters, Fig. 18 shows a good agreement between the numerical and the experimental results for both InGaAs and WS 2 materials.
These results suggest that the model here proposed can characterize the dynamics of photocarriers in ultrafast materials of different types through the traps contribution to the potoconductivity. Moreover, the agreement supports the assumption that drift-diffusion terms may be neglected in the study of bulk samples.

Conclusion
In this paper, we have studied the semiconductor photoconductivity dynamics under pulse and harmonic laser excitations in the frameworks of two-level and three-level models. The analysis is based on the numerical solution of the system of partial differential equations for the concentrations of electrons, holes and occupied trap levels considering all the physical processes responsible for the transitions of photocarriers between the bands and the traps level. In order to analyze the role of generation-recombination phenomena only, the presence of an electric field and diffusion phenomena are neglected. This approximation remains acceptable for most of photoexcitation phenomena taking place in bulk samples.
In the first simplified situation (two-level model), the calculations consider the generation-recombination processes between the valence and conduction bands with different scenarios of laser excitation. The pulse temporal behavior of a laser was approximated by Gaussian and square functions where the influence of the time duration of the pulse on the electrons and holes concentrations has been investigated. While the intrinsic characteristic times of the system are comparable or longer than the time duration of the excitation, the carriers concentrations follow the shape of the optical pulse, significant deviations are observed in the opposite case in terms of shape and maximum amplitude of the response. Similar effects have been found in the case of harmonic excitation even if the effect in the amplitude of the response is much smaller, thus suggesting that the harmonic excitation is more robust with respect to an increase in the frequency of the excitation.
In the second situation (three-level model), due to the simultaneous presence of many characteristic times, the dynamics of electrons, holes and occupied traps concentrations become more complicated and they are discussed in two steps for a limited set of parameters. In the former one, we considered the transient behavior starting from empty bands and traps levels in the absence of any laser excitation and we studied the influence of the generation and recombination characteristic times. While in the stationary regime,the electrons and holes concentrations reach equal values, this is not true in the transient regime, where, for some choice of the characterized times, significant differences are found evidencing a behavior similar to the wellknown velocity overshoot (here concentration overshoot) typical of the transient transport in III-V materials. In the latter one, we have considered the presence of pulse or harmonic photoexcitations while the influence of their duration and frequency on the stationary behaviors of electrons, holes and occupied traps concentrations has been analyzed with respect to the intrinsic characteristic times of the system. Similarly to the case of the two-level model, the time response of the carriers concentrations is found to deviate from the shape of the photoexcitation when the intrinsic characteristic time scales are shorter that those of the external excitation. However, the trap levels can be populated or emptied depending on the choice of the generation and recombination times from the trap levels to the valence and conduction bands. In the case of a harmonic excitation, the progressive attenuation of the effect of the photoexcitation with frequency has been found to affect not only the carriers but also the occupied traps concentration.
The model has been validated by a comparison with some experimental data obtained in InGaAs and WS 2 samples in the presence of Gaussian photoexcitation.
The proposed models are sufficiently flexible to be adapted to several practical situations and also to simulate future scenarios not corresponding to present materials. This work paves the way for a better understanding of photoconductivity dynamics in semiconductors under optical regime and could be useful to predict the performances of present and future photoconductive devices and systems.  Fig. 18 a Normalized electron concentration as a function of time calculated by the numerical model (line) and experimental data (circles) obtained in InGaAs samples from Ref. [21]. b Photoconductivity as a function of time calculated by the numerical model (line) and experimental data obtained in monolayer WS 2 from Ref. [29]. The values of the parameters are given in the text Funding Author(s) warrant that any financial interests, direct or indirect, that exist or may be perceived to exist for individual contributors in connection with this manuscript have been disclosed in the covering letter. Furthermore, sources of financial support of the project are named in the covering letter as well as the Acknowledgement.
Data availability All authors undersigned have made a substantial contribution to the manuscript and are willing to take public responsibility for its contents. Author(s) attest that all persons designated as authors qualify for authorship and all those who qualify are listed.