1 Introduction

The dynamics of photoluminescence (PL) quenching (PLQ) can reveal rich information of carrier transport, scattering, and lattice vibration in semiconductors [1–8]. As a result, the study of PLQ has attracted considerable attention during the past decades due to its potential applications in optoelectrons and photonics, such as PL in quantum wells quenched by far-infrared radiation [1] or by dc electric field [2], and PL in disordered semiconductors quenched by temperature [3]. Recently, an experiment of PLQ in bulk semiconductors induced by ultrashort THz pulse was reported [4] where an 800 nm femtosecond (fs) pulse was used to excite CdTe and GaAs and PL near bandgap was measured. As a single-cycle THz pulse was illuminated on the samples simultaneously, the amplitude of the time-integrated PL was found to be decreased.

When femtosecond (fs) laser pulses irradiate on semiconductors, the electron will be excited from the valence band to the conduction band, creating electron–hole pairs. The PL emission comes from the radiative recombination between the electron near the bottom of conduction band and the holes at top of the valence band. The incident THz waves “accelerate” or “heat” electrons [5] and promote them to high-energy states in the conduction band. The high-energy carriers can induce nonequilibrium phonons which increase the nonradiative centers by phonon–defect coupling [9–11]. Subsequently, the intensity of the radiative recombination of the electron–hole pairs decreases. Finally, we can observe THz-induced PLQ. Based on the viewpoint mentioned above, we combine ensemble Monte Carlo method [8, 12] with nonlinear photon excitation model [13] to describe the transition in bulk semiconductors. Via this model, we systematically investigate the dynamics and THz intensity dependence of quantum efficiency in GaAs and CdTe. Using an experimental setup similar to that in [4], we measure the laser intensity dependence of PLQ in CdTe. The measured results agree well with the calculated ones. Our study on the THz-pulse-modulated PL could potentially lead to applications of a noninvasive ultrafast modulator of light-emitting and THz electro-absorption modulation [14–17].

2 Physical mechanism and model

Scattering of electrons between the different valleys of the conduction band in a semiconductor plays an important role in high field transport process. Under a strong THz field (peak value is above 100 kV/cm), both intrinsic and photo-induced carriers may absorb/emit phonons through intervalley scattering and then intervalley phonons become nonequilibrium. These hot phonons usually have large wave vectors which induce vibration of very few atoms, and their energy can be easily coupled by nonradiative centers such as point defects [18–20]. As a result, PL quantum efficiency quenches.

2.1 Monte Carlo simulation of carriers’ transport and scattering

THz wave can drive carriers before and during photon excitation. The dynamical evolution of the carrier–phonon system can be described by the coupled Boltzmann equations [8, 21, 22]:

$$ \frac{{\partial f_{k} }}{\partial t} = \left. {\frac{{\partial f_{k}^{{}} }}{\partial t}} \right|_{g} {+}\, \left. {\frac{{\partial f_{k}^{{}} }}{\partial t}} \right|_{{c - {\text{ph}}}} {+}\, \left. {\frac{{\partial f_{k}^{{}} }}{\partial t}} \right|_{{c - {\text{impurity}}}} {+}\, \frac{{eE_{t} (t)}}{\hbar }\nabla_{k} f_{k}^{{}} , $$
(1)
$$ \frac{{\partial N_{q}^{{}} }}{\partial t} = \left. {\frac{{\partial N_{q}^{{}} }}{\partial t}} \right|_{{c - {\text{ph}}}} {+}\, \left. {\frac{{\partial N_{q}^{{}} }}{\partial t}} \right|_{{{\text{ph}} - {\text{ph}}}} {+}\, \left. {\frac{{\partial N_{q}^{{}} }}{\partial t}} \right|_{{{\text{ph}} - {\text{nr}}}} , $$
(2)

where \( f_{k} \) and \( N_{q}^{{}} \) are the electron and the phonon distribution function, respectively. \( \left. {\partial f_{k}^{{}} /\partial t} \right|_{g} \) stands for photo induced carriers generation and \( \left. {\partial f_{k}^{{}} /\partial t} \right|_{{c - {\text{ph}}}} \) and \( \left. {\partial f_{k}^{{}} /\partial t} \right|_{{c - {\text{impurity}}}} \) are carrier–phonon and carrier–impurity interaction rates, respectively. The THz-filed-driven process is described by \( eE_{t} (t)\nabla_{k} f_{k}^{{}} /\hbar \), in which e is the unit charge with its sign (e < 0 for electrons and e > 0 for holes), and \( \hbar \) is the Planck constant divided by 2π. Particularly, \( E_{t} \) is equal to \( (2Y_{0} E_{\text{THz}} - Jd)/(Y_{0} + Y_{\text{s}} ) \), and is the THz field transmitted through the thin semiconductor film [23]. Here \( E_{\text{THz}} \) is the incident THz field, d is the laser penetration depth (1 μm for the 800-nm laser pulse), and \( Y_{0} \) and \( Y_{\text{s}} \) are the free-space and sample admittances, respectively. The current density \( J \) is \( (ne^{2} \tau /m)E_{t}. \) Here n is the carrier density, \( \tau \) denotes mean free time between ionic collisions and m is carrier mass. \( \left. {\partial N_{q}^{{}} /\partial t} \right|_{{{\text{ph}} - {\text{ph}}}} \) and \( \left. {\partial N_{q}^{{}} /\partial t} \right|_{{{\text{ph}} - {\text{nr}}}} \) are the phonon decay rates for phonon–phonon and phonon-nonradiative center interactions, respectively. The acoustical phonon, polar-LO phonon and intervalley phonon are taken into consideration. The holes effect is neglected due to the high effective mass. Equation (1) is usually calculated by using ensemble Monte Carlo simulation (EMC) [11]. In order to calculate nonequilibrium phonons exactly, we use fictitious scattering method [24, 25] to obtain the carrier–phonon scattering rates.

In Eq. (2), \( \left. {\partial N_{q}^{{}} /\partial t} \right|_{{c - {\text{ph}}}} \)can be written as \( A\Updelta h_{q} \) [21]. \( \Updelta h_{q} \) is changed at each phonon scattering time, \( A \) is a normalized factor, when THz field is applied in one direction (we can assume that it is Z direction) [26],

$$ A = \pm \frac{{8\pi^{2} }}{{2q_{T} \Updelta q_{T} \Updelta q_{Z} + \Updelta q_{T}^{2} \Updelta q_{Z} }}\frac{n}{{N_{e} }}, $$
(3)

where sign ‘+’ corresponds to the emission and ‘−’ to the absorption of the phonon with the wave vector \( q = \{ q_{Z} ,q_{T} \} ,\) \( q_{Z} \) is the phonon wave vector along Z direction, \( q_{T} = \sqrt {q_{X}^{2} + q_{Y}^{2} } ,\) \( \Updelta q_{T} \) and \( \Updelta q_{Z} \) are the steps of the q-grid and \( N_{e} \) is the number of simulated particles.

2.2 Theoretical model for laser excitation and PL emitting

Since the focal spot diameter of the 0.15 μJ optical pulse is 0.05 mm, i.e., the pump pulse fluence is higher than 5 mJ/cm2, the two-photon absorption (TPA) should become prominent at such excitation levels. The propagation of the optical pump pulse in the direction perpendicular to the surface of the semiconductors is then described by the following pair of differential equations [13, 27]:

$$ \frac{\partial I(z,t)}{\partial z} = - \alpha I(z,t) - \beta I^{2} (z,t) - \frac{1}{{v_{\text{g}} }}\frac{\partial I(z,t)}{\partial t}, $$
(4a)
$$ \frac{{\partial N_{\text{SPA}} (z,t)}}{\partial t} = \frac{\alpha I(z,t)}{\hbar \omega }, $$
(4b)
$$ \frac{{\partial N_{\text{TPA}} (z,t)}}{\partial t} = \frac{{\beta I^{2} (z,t)}}{2\hbar \omega }, $$
(4c)

where \( I(z,t) \) is the pump beam intensity, \( N_{\text{SPA}} (z,t) \) and \( N_{\text{TPA}} (z,t) \) are the density of single-photon absorption (SPA) and TPA excited particles, respectively, \( \omega \) is the angular frequency of the pump pulse, \( \alpha = 1/d \) is the linear absorption coefficient at 800 nm, \( \beta = \beta_{[100]} \propto \text{Im} \chi_{1111}^{3} \) is TPA coefficient and \( \chi_{1111}^{3} \) is the third-order nonlinear tensor of the susceptibility. \( v_{\text{g}} \) is the beam’s group velocity. \( N_{\text{SPA}} (z,t) \) and \( N_{\text{TPA}} (z,t) \) should be distinguished in the Monte Carlo simulation since the electrons generated from SPA and TPA, respectively, have excess energies. The mean concentration in the photo-excited region can be described as \( n(t) = \int_{d} {{\text{d}}z} \frac{{N_{\text{SPA}} (z,t) + N_{\text{TPA}} (z,t)}}{d} \).

After photo-excitation, the time dependence of the excess carrier concentration is governed by the rate equation [28]:

$$ \frac{{{\text{d}}n(t)}}{{{\text{d}}t}} = - \frac{n(t)}{{\tau_{\text{eff}} (t)}}, $$
(5)

where the effective carriers’ lifetime \( \tau_{\text{eff}} (t) = \frac{{\tau_{r} (t)\tau_{\text{nr}} }}{{\tau_{r} (t) + \tau_{\text{nr}} }} \) [29]. The radiative lifetime \( \tau_{r} (t) = 1/Bn(t) \), B (10−9 cm3 s−1 in CdTe) is the radiative rate constant [27]. Since large quantities of hot optical phonons will likely lead to generation of new defects [10, 11], the nonradiative lifetime is assumed to be \( \tau_{\text{nr}} = 1/C\int\nolimits_{{T_{\text{P}} }}^{{}} {{\text{d}}tN_{\text{qne}} (t)/\tau_{\text{ne}} }, \) C is the nonradiative rate constant of defects, we favor a value of C = 2 × 10−7 cm3 s−1 . \( T_{\text{p}} \) is the pump duration, \( \tau_{\text{ne}} \) is the lifetime of nonequilibrium intervalley phonons. \( N_{\text{qne}} (t) = \sum\nolimits_{q} {(N(q,t) - N_{0} )} D(q) \) is the total number of out of equilibrium intervalley phonons [30], \( N_{0} = 1/[\exp (h\omega_{q} /k_{B} T) - 1] \), \( h\omega_{q} \) is intervalley phonon energy. Refer to Eq. (3), phonon density of state \( D(q) = 2q_{T} \Updelta q_{T} \Updelta q_{Z} + \Updelta q_{T}^{2} \Updelta q_{Z}. \) Since the decay nonradiative centers are much longer than \( \tau_{eff}, \) \( \tau_{\text{nr}} \) it is usually treated as time-independent when phonons return to equilibrium.

As we know, PL effect is the consequence of radiative recombination. The time-integrated PL is determined by the total number of radiative carriers \( I_{\text{PL}} \propto N_{r} \),

$$ N_{r} = V\int\limits_{\infty } {{\text{d}}t\frac{n(t)}{{\tau_{r} (t)}}} $$
(6)

V is the size of photon-excited region. After derivation from Eqs. (5) and (6) one can have \( N_{r} \)expression as

$$ I_{\text{PL}} \propto N_{r} = V\int\limits_{0}^{\infty } {{\text{d}}t} \left\{ Bn_{0} n(t) - \int\limits_{0}^{t} {{\text{d}}\tau } n(\tau ) \left(\frac{1}{{\tau_{\text{nr}} }} + Bn(\tau )\right)\right\} $$
(7)

\( n_{0} \) is the carrier concentration at the end of the pump pulse t = 0.

In summary, the simulation procedure for the isotropic electron and phonon distributions is as the following:

At first, initialization of all the parameters is necessary.

In each time step,

  1. 1.

    Calculation of the carrier density \( n(t) \) from Eqs. (4a, 4b, 4c) if \( I(z,t) > 0 \).

  2. 2.

    Calculation of THz field \( E_{t} \).

  3. 3.

    EMC calculation from Eqs. (1) and (2).

  4. 4.

    Refresh the \( \tau \) mean free flight time and nonequilibrium phonon density \( n_{\text{qne}} (t) \).

  5. 5.

    If optical excitation is over, calculate \( I_{\text{PL}} \) from Eqs. (5) and (6).

By using the model above, we calculate the PLQ and THz intensity dependence of PLQ in CdTe and GaAs. These calculated results are compared with previous measurements [4]. Furthermore, we experimentally and theoretically investigate the laser intensity dependence of PLQ in CdTe. The parameters used for calculation are listed. Figure 1 is the scattering rates versus energy in GaAs (a) and CdTe (b) at room temperature in the Γ valley.

Fig. 1
figure 1

Scattering rates versus energy in GaAs (a) and CdTe (b) at room temperature in the Γ valley

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3 Major mechanisms in THz-induced PLQ process

3.1 Reflection after photo-excitation

The high concentration of photo-excited carriers results in a very high THz reflectivity coefficient of the sample interface, thus decreasing the coupling of THz field to the bulk of the carriers.

3.2 Phonons generation via relaxation of intrinsic carriers

The n-doping values of GaAs and CdTe are 1 × 1016–3 × 1016 cm−3 (2 × 1016 cm−3 in calculation) and 8 × 1016–3 × 1017 cm−3 (2 × 1017 cm−3 in calculation). Though the doping values are much lower than laser excitation density (about 2 × 1019/cm3), they play important roles at the initial and mid stage of excitation. Figure 2 is the intervalley phonon distributions induced by intrinsic carriers as a function of time delay t d (<0 means the peak of THz wave is behind optical pulse excitation) between THz wave and laser pulse. The central wave vectors of GaAs and CdTe intervalley phonon are about 107 and 2 × 107/cm, respectively. Their magnitudes are higher than that of nonequilibrium LO phonon and they are more easily absorbed by nonradiative centers.

Fig. 2
figure 2

The calculated the intervalley phonon distribution generated from intrinsic carriers transition in a GaAs and b CdTe in a strong THz field (100 kV/cm)

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3.3 Phonons generation via relaxation of photo-excited carriers

The orange curves in Fig. 3 are the evolution of the mean energy of photo-excited carriers at t d  = 0. The Γ–L separation in GaAs is only 0.29 eV, thus the kinetic energies of photo-induced carriers are sufficient for the intervalley transition. And the hot phonons in GaAs can be generated via the intervalley transition of photo-induced carriers. The blue curve in Fig. 3a shows the generation and decay of these phonons. On the other hand, both the Γ–L separation (1.4 eV) and effective electron mass (0.11 m0) in CdTe are much larger than that in GaAs. Even if the THz field is as strong as 100 kV/cm, as Fig. 3b shows, the photo-excited carriers cannot be accelerated to the necessary energy level. As a result, they will not change the intervalley phonon occupation.

Fig. 3
figure 3

The evolution of mean carrier energy (orange line) and intervalley phonons occupation (blue line) PL at t d = 0 in comparison with the time-domain waveform of the optical pulse (dashed red line)

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4 Results and discussion

Since the intervalley phonon distribution is determined, we can calculate the time-integral PL intensity as a function of time delay t d via Eq. (7) as shown in Fig. 4 in which the THz time-domain waveform measured by electro-optic sampling is plotted together. The shapes of the calculated curves agree well with those of the measured ones (see Fig. 3 in [4]). The calculated curves show that the PL drops as the laser pulse starts to overlap with THz pulse, and recovers back to the level without THz radiation when t d increases from 0. There are two key time delays which proof of the accuracy of this theoretical model: (1) t d = −1 ps (when the peak of THz wave was 1 ps behind optical pulse excitation). The high concentration of carriers results in a very high THz reflectivity. Both calculation and experimental result [4] show that PLQ ratio should be zero when t d = −1 ps; (2) t d = 1 ps. The sample of GaAs exhibits a faster recovery than that of CdTe. Since the strong THz field can drive the photo-induced carriers to Γ−L separation energy before the high THz interface reflection forms in GaAs, the energies of photo-induced carriers are governed by the transient THz wave in the optical excitation duration. As a result, the PL curve is more sensitive to the transient THz intensity; In CdTe, the photo-induced carriers have not enough energy to generate phonons via intervalley transition. As a result, the PLQ effect in CdTe only results from the intrinsic carriers which is more sensitive to time-integrated THz intensity. Therefore, PLQ trace of CdTe has a boarder temporal quenching window than that of GaAs. The calculated PLQ ratios that are about 1 and 10 % in CdTe and GaAs, respectively, at t d is 1 ps also agree well with those of the previous measurement [4].

In order to further verify that PLQ of CdTe is only induced by the nonequilibrium intervalley phonons from intrinsic carriers, the laser intensity dependence of the PLQ ratio is experimentally investigated and compared with the calculation using the model above. Here PLQ ratio = 100 % − I PL(E THz ≠ 0)/I PL(E THz = 0). We expected that the PLQ ratio should monotonically decrease with laser intensity. The physical mechanism is that when the laser intensity becomes stronger, the number of the corresponding photo-induced carriers becomes larger. If those carriers make no contribution to hot intervalley phonons, they will reduce the ratio between nonradiative and radiative recombination rates. Then the PLQ ratio will reduce. In theoretical expression, when excited carriers density n(τ) grows, the damping term 1/τ nr + Bn(τ) in Eq. (7) becomes τ nr-independent when Bn(τ) ≫ 1/τnr (Table 1).

Fig. 4
figure 4

The calculated PL as a function of time delays between THz and optical pulse in CdTe at 831 nm (top) and in GaAs at 854 nm (middle) in comparison with the time-domain waveform of a THz pulse (bottom)

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Table 1 The parameters used for calculation
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In the experiment, the PL emitted from CdTe is filtered by a narrow line interference filter (center transmission wavelength ~840 nm, and bandwidth ~10 nm) and then collected by a photomultiplier tube. The time delay between the THz pulse and laser pulse is fixed at timing where maximal quenching occurs. The diameter of the THz beam and laser beam on the CdTe sample are about 500 and 50 μm, respectively. The peak THz intensity is fixed at 13 MW/cm2. The measured data (circles) and the calculated curve (solid curve) are shown in Fig. 5. As can be seen, PLQ ratio reaches to almost zero at high laser intensity. The experiment agrees well with the expectation. It should be noted that some other mechanisms can also affect PLQ ratio decrease: (1) the high THz reflection reduces PLQ ratio. The pink dots curves are the calculated PLQ ratios when the THz reflection change is not taken into consideration. A deviation occurs when the laser intensity reaches 10 GW/cm2. Under such laser irradiation, the excited carrier’s density is 1019/cm3 which is the threshold density of high THz reflectivity; (2) TPA carriers have much higher excess energies than SPA carriers. They can induce hot phonon markedly without the assist of THz wave, thus reduce the THz modulation efficiency. The dashed green line shows that the error will increase if TPA effect is neglected.

Fig. 5
figure 5

The measured and calculated curves for the laser pulse energy dependence of PLQ ratio in CdTe. Blue circles are experimental result and red solid line is calculation of PL quenching ratio. The green dashed and pink dot lines are the calculated results neglecting TPA and surface reflectivity of THz, respectively. Inset Density of SPA and TPA carriers versus the laser intensity

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PLQ ratio in GaAs is relatively small (only 5 % under the THz-pulse irradiation of 100 kV/cm peak field) providing limited study range. PLQ effect in GaAs is more complex in high pump intensity since photon-excited carriers also contribution to the intervalley phonon generation. In addition, PLQ is not found in ZnTe(E g = 2.23 eV), Zn1−x Cd x Te (E g > 1.65 eV) or GaP(E g = 2.26 eV) [4]. Intervalley scatterings occur without THz wave because of high excess electron energy (3.1 eV − E g) after two-photon absorption in these samples. Thus, if the optical photo energies are adapted to the bandgaps of these semiconductors, PLQ might be observed in these materials.

5 Conclusion

In conclusion, we combine ensemble Monte Carlo method with underlying photon excitation model to describe PLQ in bulk semiconductors induced by intense THz pulses. The PLQ in GaAs results from the photo-excited and intrinsic carriers, whereas the PLQ in CdTe is determined only by the intrinsic carriers. This work provides an effective approach to study PLQ induced by electromagnetic wave pulses in bulk semiconductors and can be potentially used for analyzing PLQ in other materials such as quantum wells and other quenching factors such as dc electric filed. Further study includes investigation of other effects such as temperature and excitation wavelength effect on bulk semiconductor PLQ.