Optical tuning of the terahertz response of black phosphorus quantum dots: effects of weak carrier confinement

2.1 Sample preparation and characterization

In brief, by using N-methyl-2-pyrrolidone (NMP) as solvent for LPE, we produced large quantities of BP-QDs. The concentration of the NMP dispersion of BP-QDs was about 0.1 mg/mL. Then, 0.15 mL of the dispersion was dropped onto a sapphire substrate. After spin coating and drying in a vacuum dryer, the sample of BQ-QDs was obtained. The details of our preparation process can be found in Appendix A. It should be noted that another bare sapphire substrate was used as the reference in our THz measurements. The thicknesses of the substrate under the BP-QDs and of the reference (bare) substrate are defined as d _sam and d _ref, respectively. These two substrates were cut apart from the same sapphire plate, which is highly plane parallel and possesses very flat surfaces. The thicknesses of these two substrates were experimentally determined as d _sam = d _ref = 497 μm. Accordingly, an error in the relative phase change of the THz wave transmitted through the BP-QDs, which will be introduced by the difference between d _sam and d _ref, can be neglected.

Next, to characterize the basic properties and quality of the sample, transmission electron microscopy (TEM), atomic force microscopy (AFM), and Raman spectroscopy were performed. The morphology and crystallinity of the BP-QDs are shown in Figure 1(a). Most of the BP-QDs are non-percolated and well-distributed on the substrate. Furthermore, clear lattice fringes can be observed from the high-resolution TEM (HR-TEM) image. For example, the spacing of 2.02 Å corresponds to (022) plane of a BP crystal, which is consistent with the BP lattice parameter [39]. The uniform lattices suggest that BP in the form of QDs retains the original crystalline state. Figure 1(c) shows the AFM image of a larger area of the BP-QD sample. Based on the statistical TEM and AFM analysis of quantitative BP-QDs, i.e., Figure 1(b) and (d), the average length and average thickness are obtained as ∼3.0 nm and ∼4.2 nm, respectively.

Figure 1:

Basic characterization. (a) TEM and (c) AFM images of the BP-QDs and the corresponding statistical analysis of (b) the lateral length and (d) thickness. (e) Raman spectrum of the BP-QDs. The inset to (a) shows the HR-TEM image of lattice fringes.

Although AFM can measure the lateral lengths (L) and thicknesses (t) of nanoparticles simultaneously, two limitations should be noted. (i) Due to the finite size of AFM tips, the lateral lengths of nanoparticles would be usually overestimated by AFM [27]. Thus, we utilized the AFM image only for thickness analysis. (ii) Since the resolution of the AFM used here is lower than that of the TEM, some of the BP-QDs cannot be well resolved and thereby the thickness statistical analysis is not as precise as the length statistical analysis. In addition, one can find from the AFM image that the agglomerations of a small amount of BP-QDs (less than 5 %) result in false particle signals with lengths of 30∼60 nm. Nevertheless, we can hardly find particle signals for L = 7–30 nm in both of the TEM and AFM images. Therefore, these signals (7–60 nm) are not included in our statistical analysis for the BP-QDs. The QD agglomerations can respond the THz wave, but this contribution is much weaker than that of a large number of isolated BP-QDs. Further evidence and discussion can be found in Section 2.2.

Moreover, three characteristic peaks of the BP-QDs can be detected by Raman spectroscopy, as illustrated in Figure 1(e). The peak located at 361.9 cm⁻¹ can be ascribed to out-of-plane phonon modes ( A g 1 ), whereas the other two peaks at 438.8 and 465.8 cm⁻¹ are attributed to in-plane phonon modes (B _2g and A g 2 ), respectively. These peaks correspond to three of the characteristic Raman peaks of bulk BP crystal but blue-shift slightly. It demonstrates that the BP-QDs still exhibit the unique puckered hexagonal layered structure but possess few layers [14, 21]. In addition, the integrated intensity ratio of A g 1 / A g 2 can be kept larger than 0.3 for a long time, indicating that the degeneration/oxidation level of our sample is very low and the BP-QDs obtained by LPE are stable enough [13]. This is because the solvation shell acts as a barrier to prevent oxidative species reaching the BP-QD surface and edge [20].

2.2 Optically tunable THz response of BP-QDs

Based on a THz-TDS system in conjunction with a continuous-wave (CW) laser, we measured the THz response of BP-QDs excited by visible light (445 nm), as shown in the inset of Figure 2(b). Furthermore, since the THz response could not be strong due to the small amount of BP, we should ensure that the laser-induced change of the THz response and the spectral noise can be well distinguished. Accordingly, we only discuss the THz properties within the effective bandwidth [40] (0.5–1.0 THz). Details can be found in Figure 7(a) in Appendix B. To improve the reliability of the measured data, the CW optical pump-THz probe experiment was carefully performed for three times.

Figure 2:

THz response of BP-QDs under laser excitation. The (a) real and (b) imaginary parts of the effective optical conductivity of the BP-QDs. Note that the THz frequency is defined as f = ω/2π and σ _eff(ω) is normalized by the universal conductivity σ ₀ = e ²/4ℏ. The inset to (b) shows the schematic of the experimental geometry for CW optical pump-THz probe measurements. (c) The real part of the frequency-average value σ ̄ eff as a function of the pump power. The imaginary part of σ ̄ eff is also presented in the inset of (c). Error bars in these figures correspond to the standard deviation of the average value obtained from data accumulations within multiple measurements. (d) Schematic band diagrams under weak and strong laser excitation. The valence band and conduction band of the BP-QDs are denoted by VB and CB, respectively. The carriers (electrons) associated with the THz absorption are illustrated as yellow balls. We use blue and red arrows to represent the laser absorbed by the BP-QDs and the corresponding photon-induced inter-band transition, respectively. The thin/thick purple arrow represents the weak/strong THz absorption by carriers.

At a given pump power P, the THz pulses transmitted through the sample (BP-QDs/sapphire) and through the bare sapphire substrate were measured as the sample and reference signals, respectively. By the Fourier transformation of these two time-domain signals, we can derive the complex electric fields in frequency-domain for the sample and reference, i.e., E _sam(ω) and E _ref(ω), respectively. Therefore, the relative THz transmission through the BP-QDs can be obtained as T(ω) = E _sam(ω)/E _ref(ω), where E _ref(ω) is found to be nearly independent to P. Then, we calculate the relative amplitude |T(ω)| and phase change arg[T(ω)] for the BP-QDs excited by different laser power (see Figure 8 in Appendix B). As mentioned in Section 2.1, we can safely neglect the error in arg[T(ω)] introduced by the difference between d _sam and d _ref.

Because the THz wavelength is much larger than the average lateral length of the BP-QDs and the average spacing between them, the THz wave cannot resolve the details of the BP-QDs and sees only an average THz response [41, 42]. This response can be described by an effective optical conductivity σ _eff(ω). It should be noted that σ _eff(ω) characterizes not only the intrinsic response of BP inside the QDs but also the effects of finite size of these nanoparticles. Based on T(ω), we can extract the effective optical conductivity through the Tinkham relation [43]:

in which n ̃ = 3.05 is the refractive index of the sapphire substrate measured by our THz-TDS system (see Figure 7(b) in Appendix B), Z ₀ = 377 Ω is the impedance of the free space.

For a thin film pumped/excited by visible light, multiple reflections of the light in this film can enhance the photoexciting effect. Thus, the THz response of a photoexcited film may be overestimated [44]. Based on this effect, the Tinkham approximation may lead to inaccurate THz conductivity of the film. In fact, this situation will be definitely encountered if the wavelength of the excitation/pump light is comparable to (or smaller than) the film thickness and the optical penetration length. In contrast, the Tinkham approximation is only valid when the optically excited layer fulfill k ₀ t _p n _p ≪ 1, where k ₀ = ω/c is the wave vector of vacuum, t _p is the thickness of the photoexcited layer in a film, and n _p is the complex refractive index of the photoexcited layer. In our experiment, the THz response of the BP-QDs can be approximately equivalent to that of an effective film with a thickness of about t ̄ = 4.2 nm and an optical conductivity of σ _eff(ω). Since the thickness is much smaller than the wavelengths of the pump laser (445 nm) and the THz wave (300–600 μm), the multiple reflections of the laser and the THz wave can be neglected. Furthermore, the pump laser can definitely penetrate the effective film of BP-QDs, leading to t p = t ̄ . By considering the fact that n _p is impossible to be very large, we can safely obtain k ₀ t _p n _p = 10⁻⁵ × (4.4 ∼ 4.8)n _p ≪ 1. Therefore, the effective optical conductivity of the BP-QDs can be extracted accurately by using the Tinkham relation of Equation (1).

Although we have verified the condition of d _sam = d _ref, it is known that an error in the experimental determination of d _sam may also lead to an uncertainty in the optical parameter of the BP-QDs [45]. We denote this error by Δd _sam here. Specifically, since the substrate is nearly lossless (κ = 0) within the frequency band of research, Δd _sam can only result in an uncertainty Δn in the real part of the complex refractive index ( n ̃ ) of the substrate. Thus, an error (Δσ _eff) will be introduced in σ _eff(ω) due to the uncertainty of n ̃ in Equation (1). Based on the given substrate parameters, it is easy to roughly estimate the accuracy of σ _eff(ω). For example, if we assume a case of Δd _sam = ±10 μm (i.e., d _sam = d _ref = 497 ± 10 μm), which is a rather large error and not common in THz experiments, Δn is calculated as about ±0.04 and thereby the ratio between Δσ _eff and σ _eff(ω) can be obtained as only about 1 %. In fact, the realistic situation is |Δd _sam|<<10 μm. It means that σ _eff(ω) is almost robust against the error in the determination of d _sam.

As shown in Figure 2(a) and (b), the measured data is mainly expressed by the average values and their errors for clarity. The error bars correspond to the standard deviation of the average value obtained from data accumulations within multiple measurements (three times of measurements), indicating the measurement accuracy. Interestingly, we find no spectral features of discrete energy levels in σ _eff(ω), which means that the strong quantum confinement hardly affects the THz response of BP-QDs. This is very different from the situations of the UV-visible measurements of BP-QDs [22, 27, 28]. According to the overall trend, one can see from the figures that Re(σ _eff) and Im(σ _eff) enhance with increasing P at most of the frequencies. Moreover, Im(σ _eff) is apparently smaller than Re(σ _eff) at a given power, which means that the BP-QDs affect the amplitude of the transmitted THz wave more significantly than affect the phase change of it.

By comparing the THz response of the BP-QDs with the results obtained in some previous studies on other nanoparticles [46–48], we find some differences in the shape and magnitude of the optical conductivity spectra. Because the form, bandgap, and distribution of our BP-QD sample is different from those of the nanoparticles reported in the aforementioned works, these differences are entirely possible and reasonable. For example, the optical conductivity spectra of our sample are relatively smooth and have no oscillations, which can be attributed to the absence of the multiple reflections of THz wave due to the ultra-small t. Obviously, t is much thinner than the plates compressed with nanoparticle powders and the cuvettes containing nanoparticle dispersion [46, 47], so the thickness of the effective BP-QD layer is deep-subwavelength and thereby the multiple reflections of THz wave cannot occur. Furthermore, comparing with HgTe nanocrystals [48], the bandgap of BP is much larger and the BP-QD distribution is not highly dense. Therefore, the spectral contribution of a few QD agglomerations (30–60 nm) will not lead to significantly enhanced THz response via larger particle size. In other words, the THz response of these agglomerations could not dominate over that of the real BP-QDs.

From the view of overall behavior of the THz response, our results are consistent with the previous studies on other nanoparticles. One can see that the real part of σ _eff(ω) does not decrease monotonically with increasing ω, while the imaginary part is negative. These are typical evidences of weak carrier confinement rather than features of intra-band-like transport of free carriers. The effect of weak carrier confinement in nanoparticles or QDs can be well described by the modified Drude–Smith model [47, 49, 50], which presents the blue shift of the plasmonic peak at ω = 0 in the Drude formula [30]. In short, the features of σ _eff(ω) are akin to the THz response of other kinds of nanoparticles, such as non-percolated semiconduction nanocrystals [47]. It should be noted again that the strong quantum confinement in nanoparticles (including our BP-QDs) may give rise to discrete quantum levels, but the spectral features of these discrete levels have been hardly observed in the THz regime. A theoretical model that describes the effective optical conductivity will be discussed in detail in Section 2.3.

Next, one can see from Figure 2(a) and (b) that σ _eff(ω) can be effectively modulated by the laser. By calculating the integral of σ _eff in the frequency range of 0.5–1.0 THz, the pump power dependence of the frequency-average value of σ _eff is depicted in Figure 2(c). Note that the frequency-average value σ ̄ eff can be expressed as σ ̄ eff = ∫ ω σ eff ( ω ) d ω / Δ ω with Δω being the frequency bandwidth. We can observe nonlinear increase of Re ( σ ̄ eff ) and Im ( σ ̄ eff ) with P. It is well known that the absorption coefficient is proportional to the real part of optical conductivity. Therefore, the result of Re ( σ ̄ eff ) demonstrates that the THz absorption of the BP-QDs can be modulated/tuned by purely optical means instead of electric approach. In the power range of 0–150 mW, the modulation depth of Re ( σ ̄ eff ) or THz absorption can reach 24 % at most.

The interpretation of this phenomenon is illustrated in Figure 2(d). Note that the conventional band diagram of BP is used here for brief and approximate discussion, because there are no spectral features of discrete levels induced by strong quantum confinement in our results. The physical mechanism can be summarized as follows: (i) Since the photon energy of THz wave is undoubtedly below the band gap of BP (0.3–1.7 eV) [24, 26], only intra-band transition of carriers can be induced by the absorption of THz probe pulse. (ii) When the pump/excitation laser with photon energy of 2.79 eV irradiates the BP-QDs, photogenerated carriers emerge in large numbers via the process of inter-band transition. (iii) The photogenerated carriers can absorb more THz photons, resulting in the increase of Re ( σ ̄ eff ) . (iv) Due to the limited number of electrons in the valence band as well as the effects of the laser on the characteristic properties of carriers, the THz absorption associated with the photogenerated carriers nonlinearly increases with P and then gradually saturates. The microscopic origin of this phenomenon will be discussed in detail later. Obviously, the performance of THz absorption and the optical modulation of it can be further improved by preparing samples via high-concentration dispersion of BP-QDs, which is flexible and low-cost. Moreover, the method of purely optical modulation can overcome the shortages of electrically tuning and find applications in all-optical THz devices based on 2D BPs.

2.3 Mechanism and characteristic parameters

In previous works, the THz response of 2D BP (nanosheet or large-size sheet) was usually described by the Drude formula [30], which is based on free-electron-gas approximation. In this formula, the real part of optical conductivity should decrease monotonically with increasing ω, while the imaginary part should be positive. However, most of the experimental data shown in Figure 2(a) and (b) does not obey this law, so the conventional Drude model cannot describe the THz response of BP-QDs rightly. Meanwhile, because no sharp rising edges or peaks are observed in the spectra of Re(σ _eff), the absorption of phonon/exciton/plasmon resonances has no contribution in the range of 0.5–1.0 THz. Therefore, the Drude-Lorentz formula including Lorentzian terms is also not suitable here. In essence, since our sample is a typical nanomaterial consisting of nanostructures (i.e., QDs), the weak carrier confinement induced by diffusive restoring current should be considered [49], which leads to the modified Drude–Smith formula:

in which τ′ is the intrinsic electronic scattering time, t _diff is the diffusion time of carriers, and ω p = N e 2 / ε 0 m * is the plasma frequency with N and m* being the carrier density and effective electron mass, respectively. The factor C = [0, 1] is used to describe the contribution of weakly confined charge carriers. This parameter is dependent on the size of a single nanostructure and its boundary reflectivity. In our sample, most of the quantum dots are distributed and isolated, so the boundary reflectivity should be 100 %. Thus, here C is mainly decided by the average size of the BP-QDs, especially the average lateral size. Moreover, as mentioned before, all these parameters characterize the average/effective response of the BP-QDs located in the area of THz illumination rather than the response of a single QD.

Figure 3(a)–(f) show the comparison between the experimental results and the fitting ones. We provide the fitting data as well as the experimental data with error bars. The fitting data is obtained by reproducing the average value of σ _eff(ω) calculated from data accumulations within multiple measurements. To show the stability of the fitting process, we also present the modulus of the maximum deviation (i.e., δ _max) between the fitting curve and the average value of measured data in Figure 3(g) to (l). Most of the experimental data can be reproduced well by Equation (2). However, for lower pump power such as P ≤ 10 mW, one can find that the model deviates from the experimental data relatively obviously. This discrepancy can be attributed to two reasons: (i) The effective carrier density N is relatively small, leading to weak response of carriers. (ii) For weak laser excitation, the average lateral length L ̄ may approach (or even smaller than) the carrier mean free path l ≈ v _th τ′ or the carrier diffusion length Z diff ≈ v th t diff τ ′ ≥ l , where v th = k B T / m * is the thermal velocity [49, 50]. In this situation, the effect of strong quantum confinement may affect the carrier behaviors by a complex way, which should be discussed in a quantum mechanical system rather than a classical model such as Equation (2).

Figure 3:

Comparison of the measured and fitted results. The measured effective optical conductivity (average value) is fitted by the modified Drude-Smith formula for (a) P = 0, (b) 10 mW, (c) 20 mW, (d) 50 mW, (e) 100 mW, and (f) 150 mW. Error bars show the standard deviation of the average value obtained from data accumulations within multiple measurements. To clearly illustrate the stability of the fitting, the modulus of the maximum deviation between the fitting curve and the experimental data is provided for (g) P = 0, (h) 10 mW, (i) 20 mW, (j) 50 mW, (k) 100 mW, and (l) 150 mW.

As the laser power increases, the response of carriers enhances with increasing N, while the limitation of L ̄ > Z diff > l can be fully satisfied due to the decrease of the mean free path and diffusion length. Obviously, δ _max tends to decrease with increasing P, also indicating more significant effect of weak carrier confinement for relatively strong excitation. Therefore, Figure 3(c)–(f) shows excellent agreement between the experimental and fitting data. Furthermore, we can roughly estimate the boundary values of l, Z _diff, and m* in this experiment based on the aforementioned discussion.

Through the fitting process, the effective characteristic parameters as functions of the pump/excitation power are extracted, as illustrated in Figure 4. Based on the stability analysis of the fitting via δ _max, we believe that the characteristic parameters extracted from the average value of σ _eff(ω) are representative and reliable. In Figure 4(a), the quasi-linear increase of ω _p with P can be understood by the mechanism shown in Figure 2(d). Stronger laser illumination can excite more photogenerated carriers, which results in the increasement of the plasma frequency, i.e., ω p ∝ N ∝ P . Thus, the contribution of the Drude term in Equation (2) can be enhanced, as mentioned previously. It is also known that the THz absorption is usually proportional to the carrier density N in a semiconductor, so one may expect the relation of Re ( σ ̄ eff ) ∝ N ∝ P 2 for the BP-QDs. However, we haven’t observed this kind of quadratic relation between Re ( σ ̄ eff ) and P in Figure 2(c). The strong effects of the pump light on the kinematic characteristic parameters of carriers (e.g., t _diff and τ′) can mainly give rise to this discrepancy.

Figure 4:

Characteristic parameters. The effects of the pump power on the (a) effective plasma frequency, (b) confinement factor, (c) intrinsic scattering time, and (d) diffusion time. The dotted curves are drawn to fit the experimental data or guide the eye.

Figure 4(b) illustrates that the confinement factor C increases nonlinearly with P and then saturates at strong laser excitation. The saturation value is about C = 0.8 instead of 1. For relatively larger nano-particles, e.g., L ̄ ∼ 100 nm [50], the boundary reflectivity of 100 % usually results in C = 1. This difference can be attributed to the effect of small particle size. As mentioned previously, C depends on the boundary reflectivity and lateral size of nanostructure, whose accurate functional relation and the corresponding mechanism are not clear until now [49]. Therefore, further investigations on this topic are interesting and necessary but beyond this work.

One can see from Figure 4(c) and (d) that the intrinsic scattering time and diffusion time can be nonlinearly tuned by P. Specifically, the intrinsic scattering time decays exponentially as τ ′ = τ 1 ⁡ exp − P / P 0 + τ 0 , in which P ₀ = 108.2 mW, τ ₁ = 6.4 fs, and τ ₀ = 8.4 fs. The decrease of τ′ is associated with the enhanced probability (or rate) of electron scattering for increased carrier density, but the enhancement of the scattering probability (or rate) may gradually saturate for high carrier density. Therefore, the scattering time remains around 8.4 fs when the laser power exceeds 150 mW.

In contrast, the diffusion time t _diff does not change monotonically with the pump power but can be fitted by a third order polynomial of P. It reaches a maximum of 51.4 fs at about 53.0 mW. This behavior is a result of the competition between the increase of N and the decrease of τ′. Moreover, to ensure the validity of our model, we can estimate the boundary values of l, Z _diff, and m* based on t _diff and τ′. Under the limitation of L ̄ > Z diff > l , one can derive the lower boundary of the effective electron mass m b * of our BP-QDs:

in which the product of t _diff τ′ should be its maximum. Equation (3) means that if m* is smaller than m b * , the quantum effects cannot be neglected in the BP-QDs and thereby the fitting performance of Equation (2) may become very poor. Figure 5 shows the values of l and Z _diff for different m*, in which m* = 0.15m _e and m* = 0.7m _e correspond to the effective electron masses in the AC-direction and ZZ-direction for monolayer BP [30], respectively. It should be emphasized that the discrepancy between the experimental and fitting data for smaller P is not only associated with L ̄ < Z diff or L ̄ < l but also related to the low carrier density.

Figure 5:

Estimated characteristic lengths. The power-dependent (a) mean free path and (b) diffusion length. The yellow region denotes the average sizes of the BP-QDs, whereas the dotted curves are drawn to guide the eye.