Photon-assisted ultrafast electron–hole plasma expansion in direct band semiconductors

2.1 Experimental setup

Our experimental setup (Figure 1) is based on a Ti-sapphire amplified laser system (Spitfire ACE) with 800 nm central wavelength, 40 fs pulse length, 1 mJ pulse energy, and 5 kHz repetition rate. The pulse train is divided into the pump, THz, and sampling branch. The pump pulse (propagating collinearly with the THz probe pulse) is used for the generation of the electron–hole plasma (EHP) at the front surface of the investigated wafer. The THz probe pulse is generated by optical rectification and detected via electro-optic sampling in a pair of (110) 1 mm thick ZnTe crystals.

Figure 1:

Scheme of the time-of-flight variant of the optical pump–THz probe experiment. EM, ellipsoidal mirrors; PBS, pellicle beam-splitter; FS, fused silica beam-splitter with high-reflective dielectric coating for 800 nm. The detail of the pulse propagation in the sample is shown at the right-hand side. The pump pulse excites the front surface of the sample and THz pulse probes the EHP expansion by its internal reflection at the photoexcited surface.

We use a time-of-flight variant of the optical pump–THz probe spectroscopy [10]. The inner surface of the photoexcited EHP acts as a metallic-like mirror; the thickness of the EHP layer is determined from the arrival time of the pulse to the electro-optic sensor. In short, the THz pulse enters the semiconductor slab at its front side before optical excitation. Subsequently, the pump pulse impinges on the front side of the sample, where it is absorbed and generates an EHP with initial thickness l ₀. The THz pulse is partially reflected at the rear side: the directly transmitted pulse serves as a reference of experimental timing while the internally reflected pulse probes the plasma-front state inside the sample. The time of the arrival of the probe pulse to the electro-optic detector carries information about the actual position of the EHP edge. By changing the delay between the pump and the probe pulse, we are able to determine the course of EHP expansion. If the probe pulse waveform is not substantially reshaped, then the accuracy of the time shift of the detected probe pulse is estimated to ∼5 fs (this corresponds to the plasma thickness uncertainty of ∼0.4 µm).

The semiconductor slab was attached behind a metallic aperture with a diameter of 2 mm. The photon fluences of excitation pulses used in the experiments were in the range (0.08–2.4) × 10¹⁶ cm⁻², high enough to saturate absorption close to the front face of the wafer and thus generate a layer of EHP with fluence-dependent initial thickness.

The spatial and temporal properties of the pump pulse can affect several processes of the studied phenomenon. In particular, its duration controls the two-photon absorption (TPA), and its spatial profile determines the distribution of free carriers and thus the efficiency of the stimulated emission. Whereas uncertainties in these characteristics do not affect any qualitative conclusion, they can influence to some extent, e.g., the observed EHP expansion rate. The measurements presented throughout this paper were performed with particular settings of the laser and of the beam path. However, due to the above reasons, the measured values might slightly differ from those observed previously [9], [10].

2.2 Samples

We used samples from common wafers of high-resistivity GaAs and high-resistivity InP with optically polished surfaces (exhibiting standard electron mobilities). The band structures of these direct-gap materials are qualitatively similar (relevant parameters are summarized in Table 1). The excess energy of an electron–hole pair can be tuned from 30 to 130 meV in GaAs, and from 130 to 200 meV in InP as a result of the band gap shrinkage between 20 K and 300 K. This approach thus enables us to directly compare two materials with identical carrier excess energy (although at different lattice temperatures). The energy difference between the split-off band and the valence band in InP is significantly smaller than in GaAs and, consequently, the split-off band can contribute to the absorption of the pump pulse in InP as pointed out in Table 1. The number of states per unit volume estimated in the table corresponds to the number of states accessible by direct optical excitation using 40 fs pulses centered at 800 nm; the bleaching occurs when one half of these states is filled. The number of accessible states in GaAs at 300 K and in InP at 20 K is similar (but not exactly equal, due to the slight difference of effective electron masses and due to the contribution of the split-off band in InP). While the TPA coefficient at 800 nm has been determined for GaAs at room temperature [10], it has been reported neither for GaAs at low temperatures nor for InP at any temperature. The TPA coefficient depends on the joint density of states involved in the two-photon absorption, which should correlate with the joint density of states of the single photon absorption (SPA) at a double frequency. The latter is directly proportional to the imaginary part of the linear permittivity, which is known for GaAs and InP [14]. The correlation between the TPA and SPA is likely to be similar for the two binary semiconductors with similar band structures. Based on these simple arguments, we expect that the TPA in InP could be somewhat weaker than in GaAs but of the same order of magnitude.

4.1 Overview of the theoretical description

In [9], two regimes were proposed to explain the temperature dependence of the observed phenomena in GaAs: an incoherent regime valid at high temperatures and a coherent regime governing the process at low temperatures. We considered two photon fields (incident pulse with frequency ω ₁ and a re-emitted radiation with frequency ω ₂) and two relevant energy levels, which participate in the system dynamics. The upper level 1 (N ₁ is the number of available states per unit volume) is populated by the direct single-photon absorption of the pump laser field (ω ₁).

In the incoherent regime, the level 1 plays a role of the reservoir for the bottom level 2 with a lower number of states per unit volume (N ₂), which is the source of an amplified stimulated emission of “recycled” photons (ω ₂). These photons leave the originally excited EHP region, become absorbed in the unexcited part of the crystal, and contribute to the plasma edge shift in space. The initial ∼ps long plateau, which is always experimentally observed at the beginning of the EHP expansion dynamics, then reflects the build-up of the population inversion at the level N ₂, i.e., a relaxation of carriers from N ₁ to N ₂. The coherence of the system is instantaneously lost during this relaxation process.

In the coherent regime, the two photon fields overlap in frequency, no significant energy relaxation occurs, and the coherent electron–photon interaction leads to the Rabi dynamics connected to a formation of solitons carrying the EHP plasma deep into the sample.

4.2 Coherent regime

The coherent regime takes place when three conditions are met: the photon fields ω ₁ and ω ₂ overlap (within the excitation pulse bandwidth) in frequency, the carrier decoherence time exceeds the pump pulse length, and the pump pulse fluence is sufficient to produce the so called pulse area (i.e., a change in the phase of the population inversion oscillation) of at least 2π, which causes just one Rabi flop in the population inversion [15], [16], [17], [18], [19]. Such coherent electron–photon interaction develops in GaAs at low temperatures; under these conditions, the reflection of the THz pulse monitors the long-distance propagation of a 2π soliton pulse with the speed [20], [21]:

where n _g is the group refractive index and α is the (linear) absorption coefficient in the semiconductor, and t _p is the pump pulse duration.

The coherent regime is demonstrated in GaAs at low temperatures. Indeed, at 20 K and with fluence ϕ = 1.7 × 10¹⁶ cm⁻², the pulse area is, following our estimations, 4 × 2π [9], which means that several copropagating solitons can be formed, each with a spatial thickness of a few micrometers. The THz reflectance is then given by the sheet conductivity of the whole soliton structure. At this fluence, the sheet conductivity is high enough to efficiently reflect the THz radiation as observed in the bottom panel of Figure 2(c). Upon a decrease of the pump fluence, the pulse area decreases with the square root of the excitation fluence. The cut-off fluence below which even a single soliton pulse is not really formed is then ∼ 0.1 × 1 0 16 c m − 2 . This is observed in Figure 3(c), where the reflection on the soliton structure becomes progressively less intense as the pump fluence is diminished, and, finally, in the bottom panel, the soliton does not propagate. At the same time, we observe a reflection on the photoexcited front surface of the wafer; however, we do not observe any propagation of this interface since the density of carriers is not sufficiently high. Similar situation is observed if we heat the sample, Figure 2(c); indeed, as the temperature is increased, the bandgap shrinks and the spectral overlap between the photon fields ω ₁ and ω ₂ becomes progressively smaller. Less carriers can then participate to the coherent interaction and the pulse related to the Rabi dynamics is weak (110 K) or cannot be formed anymore (180 K). At these temperatures, a reflection at the EHP close to the GaAs surface is also observed, and since the overall photon fluence is high, the incoherent regime takes over and the EHP expansion restarts at later times.

Relatively slow propagation of the excitation pulse (i.e., at a speed smaller than expected c/n _g ) is observed also in the initial phase of the EHP build-up before the appearance of the plateau in the dynamics for both GaAs and InP. To address this effect, we estimate the time required by light to propagate through (and thus form) the initial plasma thickness l ₀: l ₀ n _g /c + t ₁ + t ₂. The first term ≲150 fs in GaAs at room temperature; t ₁ is a characteristic time-domain width of the optical pump pulse, during which the light intensity is high enough to saturate the absorption (≲80 fs in our case) and t ₂ is related to the so-called depletion time (time during which the generated plasma fills out an exponentially decaying spatial distribution) [22]. This time can be significant in materials with a long linear penetration depth (e.g., Si excited at 800 nm [11], [22]). However, in our case of GaAs or InP with a sub-micrometer linear penetration depth, it remains quite short (≲45 fs), since the initial plasma propagation is essentially due to the saturation of single photon absorption. The predicted initial plasma formation time is then ≲300 fs, whereas the corresponding observed initial increase of the plasma extent lasts more than a picosecond. The reason for this behavior can be twofold. (i) A reshaping of the picosecond THz pulse is observed during this initial stage since different parts of the THz pulse find the close neighborhood of the EHP front in a different (and nonstationary) state. In other words, we observe a “convolution” of the fast plasma propagation with the two adjacent plateaus (one at negative times before the carriers are generated and the other due to the carrier energy relaxation) leading to a lowering of the apparent speed. (ii) A possible short-lived coherent regime might occur within the directly excited electronic states well above the bandgap (photon field ω ₁) leading to a slower propagation of the optical pump pulse during the absorption process. Such regime would decay rapidly due to the fast momentum relaxation of the charges. An existence of this last contribution is further supported by the dynamics observed in GaAs shown in Figure 2(a). The initial absorption phase becomes progressively longer on cooling from the room temperature while the observed speed (slope of the curves) remains approximately constant and comparable to the soliton speed detected at low temperatures.

For InP in the entire temperature range, the excess energy exceeds the bandwidth of the excitation pulse as well as the optical phonon energy. Under these conditions, a rapid carrier energy relaxation occurs, which prevents the development of the coherent regime for a long time. Indeed, for InP, we always observe the plateau, which is characteristic for the incoherent regime [Figures 2(b) and 3(b)].

4.3 Incoherent regime

Although the band structures of GaAs and InP are very similar, well pronounced differences of the plasma expansion in incoherent regime have been observed between the two materials. First, in InP, the incoherent process is observed down to 20 K and its EHP propagation speed finally saturates upon decreasing the temperature. In contrast, the EHP dynamics is governed by the soliton propagation in GaAs at 20 K. Second, EHP expansion speeds differ by a factor of 2 between GaAs (295 K) and InP (20 K) with the same excitation excess energy (Figure 4). These findings led us to an extension of the kinetic model compared to our original model in [9].

Each electronic state can absorb/amplify electromagnetic radiation, depending on the actual population of this state (more precisely, depending on the inversion of the pair of electromagnetically coupled electron and hole). In a real experiment, the relaxation kinetics and state depletion by photon emission leads to a space-dependent dynamics in which many states within the band dispersion emit and reabsorb photons in a complex cascade. This kinetics, despite its complexity, can be effectively described in terms of an “active level” with the number of states per unit volume N ₂, fed by a “reservoir” with the number of states per unit volume N ₁. Our kinetic equations are shown in the Appendix. The essential extension of the model is an explicit inclusion of the L-valley carriers (n _L ), which are generated through TPA and subsequently relax to the reservoir. Furthermore, we deeply rethought the interpretation of the degeneracy of level 2, N ₂, which reveals to be the crucial parameter for the observed dynamics and its temperature dependence. Indeed, following the estimate of the early time plasma expansion speed u, Eq. (5) in [9]:

we find that u may acquire large values when N ₂ is relatively small (l ₀ is the initial thickness of EHP, τ is the mean relaxation time from level 1 to level 2 and represents the inverse intra-band cooling rate of electrons). This counter-intuitive behavior can be better understood when a real system with a continuum of levels is considered. The distinct energy levels are characterized by different absorption coefficients (due to their different occupancy and degeneracy) and, therefore, by a distribution of expansion speeds. In extremis, photons emitted by electrons with an infinitely small excess energy above the band gap travel through the semiconductor practically unabsorbed, thus causing an infinitely small EHP density to expand at the speed of light in the material. In other words, the experimentally observed motion of the plasma front edge is determined by the fastest speed u N 2 given by (2), where the lowest acceptable value of N ₂ satisfies some physical constraints discussed below.

Our further discussion of the meaning and interpretation of N ₂ is based on the scheme shown in Figure 5. First we discuss the constraints in the plasma expansion region (region B in Figure 5), which impose a minimum (or critical) free electron density n _2,c in the region B (the corresponding level degeneracy is then N _2,c = 2n _2,c ). First, a sufficiently large concentration of electrons n _2,0 must be created in the region B to reflect the THz pulse, defining the first lower limit for the level 2 degeneracy: N _2,0 = 2n _2,0. Second, the propagation of the EHP front is possible only if the optical absorption is saturated. Such situation occurs in the region B when one half of electronic states available for absorption of photons at a given frequency is filled (including those reached by thermal fluctuations). The minimum density of thermalized electrons n _2,T that leads to the saturation of the optical absorption then defines the second lower limit for the degeneracy: N _2,T = 2n _2,T . The critical degeneracy N _2,c (electron density n _2,c ) is then the larger of the two values N _2,0 and N _2,T (n _2,0 and n _2,T ). N _2,0 is important at low lattice temperatures when thermal fluctuations are negligible; N _2,T becomes dominant at higher temperatures. This implies that the number of available states at level 2, N _2,c , is in fact temperature dependent (and that it is also tightly bound to the sensitivity of the experimental technique).

Figure 5:

Scheme of the conduction band in space along the normal to the sample surface after its photoexcitation and during the plasma expansion. The initially excited region A is the part of the sample where the photon field at ω ₁ is absorbed; the plasma expands into the initially unexcited region B (see also [9]). The bottom of the conduction band is renormalized due to the high carrier concentration (n ₁ + n _L + n ₂ in region A, n _2,c in region B). Level 1: yellow, level 2: green, bandgap: gray.

Now we turn our attention to the region A in Figure 5. In fact, given the rather high carrier densities and small excess energies, the bandgap renormalization plays a non-negligible role in our estimations, since it is different in regions A and B due to the different carrier densities in there (Figure 5). Consequently, the division energy line between levels 1 and 2 in the region A should be adjusted appropriately to ensure the critical electron density n _2,c in the region B. This procedure allows us to define the final value N ₂ to be injected into kinetic equations for the EHP expansion and allows us to describe the observed temperature variations of the expansion speeds. Technical details and deeper discussion of the evaluation of N ₂ are provided in the Appendix.

Given these arguments, the difference between the dynamics of InP at 20 K and GaAs at 295 K becomes more apparent. Indeed, the evaluated degeneracies of level 2 differ among the two materials (N ₂ = 1.0 × 10¹⁸ cm⁻³ in InP at 20 K and N ₂ = 2.4 × 10¹⁸ cm⁻³ in GaAs at 295 K) and, therefore, the model predicts different speeds of expansion. In the comparison of the theoretical prediction and experimental results (Figure 4), we observe a very good semiquantitative agreement of the data. We point out that the model predicts a significantly shorter phase of carrier energy relaxation (plateau phase) in InP than experimentally observed (for GaAs the length of the plateau phase is correct). To compare the EHP expansion dynamics, we shifted the origin of the theoretical curve for InP in order to match the onset of the expansion. In this representation, it becomes apparent that the expansion dynamics is well reproduced by the model, which also predicts the correct difference of the expansion velocities in the two materials.

Figure 6 shows a comparison of the temperature dependent plasma expansion in GaAs and InP and the corresponding results of the theoretical model for all the temperatures where the incoherent regime was observed. The overall agreement is very good. In particular, we find a semiquantitative accord of the temperature dependence of EHP expansion in GaAs. The agreement for InP is very good at low temperatures (up to about 180 K); at higher temperatures, the speed of expansion is overestimated by the theory to some extent (even if the decreasing trend of the expansion speed with increasing temperature is preserved). The energy relaxation (plateau) phase in InP is systematically underestimated; for this reason, theoretical curves were shifted in time in Figure 6 to match the onset of expansion with the experiment.

Figure 6:

Comparison of experimental data and results of the model for incoherent regime at several temperatures for GaAs (a) and InP (b). The color code is the same for both plots; the pump fluence was 1.7 × 10¹⁶ cm⁻² in all the cases. Vertical offset of the data was applied for clarity. Since the theory predicts shorter plateaus at the beginning of the dynamics than experimentally observed, we shifted the theoretical curves in time to match the onset of expansion with the experiment.

The relaxation time τ was set to 2 ps in the numerical simulations; the value was selected in order to reproduce expansion speeds and trends in the experimental curves. We, therefore, conclude that the time τ is an effective cooling time of the whole electron–photon system, including the energy redistribution within the inhomogeneously excited plasma, provided by the cascade of photon emission and absorption.

Despite the complexity of this highly nonlinear problem and despite the number of degrees of freedom that can influence the quantitative results, the essence of the observed phenomenon is grasped quite well by the proposed simple model based just on effective populations of the “reservoir” and of the “active level” and relying on (mostly known) properties of the semiconductor band structures. The model then serves as a good quantitative estimate of the system behavior and illustrates well the underlying physical processes. The complex nature of the carrier relaxation is further indicated by the mentioned systematically shorter plateaus in InP; this suggests that the quasi-equilibrium state is established in a more complex process than the simple exponential decay represented by (A1–A5).

4.4 Generalization

The described ultrafast EHP expansion is closely related to the formation of a degenerate EHP and to an efficient stimulated emission and subsequent reabsorption. These particular phenomena should be in principle observable in the majority of common direct-gap semiconductors. The question of observing the ultrafast EHP expansion in other materials is thus much more about the laser pulse fluence and wavelength as discussed below.

The creation of a sufficiently thick degenerate EHP requires high fluence of the excitation field. The higher is the photon excess energy above the band gap, the larger is the excitation pulse attenuation by absorption due to the larger number of available states per unit volume and, in turn, the higher is the required excitation fluence. The highest possible fluence is, however, limited by the available laser technology and also by the damage threshold of the given material (e.g., the highest fluences used in our experiments with GaAs at 800 nm were still safely below its damage threshold [23], [24]).

There may be slight quantitative (not qualitative) differences in the maximum acceptable excess energies due to the particular value of the effective mass of electrons in each semiconductor. However, this influence is not expected to be pronounced since the electron effective masses (which determine the densities of states close to the bottom of the conduction band) do not differ much among common binary direct-gap semiconductors like GaAs (0.067 [13]), InP (0.08 [13]), CdTe (0.096 [25], 0.09 [26]), CdS (0.166 [26], 0.21 [27]), CdSe (0.114 [26], 0.13 [27]), ZnTe (0.11 [25], 0.12 [26]), etc. In the parentheses, we included the relevant effective electron mass at the bottom of the conduction band expressed in free electron mass units. In this context, the observed virtually vanishing EHP expansion in InP at room temperature (Figure 2) leads us to the conclusion that the related excess energy of ∼0.2 eV is a rough maximum for the observation of the effect in the above cited semiconductors at the pump level of the order of 10 mJ/cm². For 800 nm pulses of Ti:sapphire lasers and among common semiconductors, the excess energy condition is satisfied in GaAs, InP, and CdTe only.

It is interesting to notice that apart from the above conditions, the developed interpretation does not involve any details about the band structure of the particular material. One crucial parameter in the incoherent regime [Eq. (2)] is the carrier effective cooling time τ, which is related to the optical phonon emission rate. This parameter is not expected to have a huge spread among common semiconductors. The other fundamental parameter is the product l ₀ N ₁, which, in fact, expresses the areal density of the degenerate electron–hole gas. This product is thus given by the density of photons in the excitation pulse (provided that there is no significant loss channel like, e.g., picosecond carrier lifetime).

All these arguments suggest that the EHP expansion is controlled by the laser pulse fluence and its wavelength (through the excess energy of photoexcited charge carriers), with little differences among individual semiconducting materials.