The dielectric function of PbS quantum dots (Qdots) with diameters of 3.5-5.0 nm in glass matrix is determined from transmission measurements by Maxwell-Garnett effective medium theory combined with iterative Kramers-Kronig analysis. The algorithm used provides real and imaginary part of the dielectric function in the 200-1800 nm spectral range, for both Qdot-doped glasses as well as the PbS Qdots alone. The latter data are compared with the results obtained from colloidal PbS quantum dots and, within the limits of the experimental error, agreement is found.
© 2012 OSA
In recent years lead-salt quantum dots (Qdots), more specifically PbS Qdots, have attracted increasing interest in both basic  and applied research. Their direct band structure promotes optoelectronic applications such as light emitters , gain media , detectors , solar cells , saturable absorbers , biomarkers , and nanoscopic temperature sensors . Their narrow (bulk) band gap energy Eg = 0.41 eV offers utmost flexibility for tuning the fundamental resonance of the Qdots by varying their size. In contrast to colloidal Qdots, Qdots in a glass matrix offer an additional practical advantage: The glass almost perfectly protects the semiconductor material against the ambient environment, therefore this type of Qdots can be treated, handled, and machined like pure glass rather than a semiconductor wafer for instance. Moreover, the glass matrix can serve as preform for pulling fibers. This option further widens the range of applications to fiber amplifiers or fiber-based near infrared light sources with customizable spectrum. However, the design of the corresponding devices requires precise knowledge of the spectrum of the dielectric constant (dielectric function) of the Qdots as well as the effective optical constants of the Qdot-doped glasses.
In this communication we provide such knowledge via spectroscopic analysis of the optical properties of the glass, either undoped or containing PbS Qdots with a diameter d = 3.5-5.0 nm, confined within the glass matrix. The analysis is based on transmittance measurements leading to spectra of the absorbance A and absorption coefficient α = A/L (L: sample thickness). A newly developed algorithm, which previously has been applied to colloidal Qdots , is now used for the Qdot-doped glasses. Based on the Kramers-Kronig (KK) relations, this approach allows for the determination of the real and imaginary part of the dielectric function of the Qdots as well as the Qdot-doped glasses in the 200-1800 nm spectral range. Finally, we provide a brief comparison of our results with the properties of colloidal PbS Qdots. Within the accuracy of our approach, no difference has been detected.
The samples were grown according to the method of Borrelli and Smith . Glass samples without Qdots and with four different Qdot sizes were synthesized (samples A-D), with following compositions: SiO2: 60.2; Al2O3: 4.2; Na2O: 11.7; NaF: 4.4; ZnO: 12.4; ZnS: 3.1; PbO: 3.0; PbS: 1.0 (A-C) and SiO2: 60.3; Al2O3: 4.3; Na2O: 11.7; NaF: 4.4; ZnO: 13.3; ZnS: 2.0; PbO: 3.0; PbS: 1.0 (D), all percentages by mass. The mixture of the reagent grade materials was molten for 90 min in an alumina crucible at 1380°C (A-C) and 1360°C (D), respectively. Subsequently, it was casted and re-molten for an additional 30 min in order to further improve homogeneity. Glass plates about 3.5 mm thick were obtained, which we then annealed as follows: sample A at 470°C for 22 h and at 523°C for 45min; B at 475°C for 22 h and at 523°C for 70 min; C at 475°C for 22 h and at 523°C for 85 min; D at 470°C for 22h and at 530°C for 105 min. Reference samples have been molten without any sulfur, hence no PbS Qdots were obtained. From each sample, we prepared and carefully polished several coplanar plates ranging from L = 100 µm to 3 mm. The transmittance T was measured in the 200-1800 nm range using a Perkin Elmer Lambda 900 spectrophotometer.
3. Results and discussion
The UV absorption onset of the glass was observed near 350 nm. At longer wavelengths, T, obtained for L = 100 µm, allows to determine the refractive index of the host glass ng by assuming that A is zero and all losses are due to reflections at the air-glass interfaces:11]; see Fig. 1(a) . A fit to the ng-data using a Sellmeier equation,Fig. 1(b), spectra normalized to one at 400 nm]. In the NIR spectral region, the first absorption peak yields the ground-state transition, from which we calculate d using an empirical sizing curve; see Table 1 . In the visible, spectra almost match. This is in line with results obtained for colloidal PbS Qdots and other II-VI, IV-VI and III-V material systems (see the discussion in ) and it implies that optical transitions in this range are no longer influenced by quantum confinement. At 400 nm a theoretical intrinsic Qdot absorption coefficient αi = 2.19 × 105 cm−1 is calculated from bulk PbS optical constants (ε = 4.53 + i·26.4) and ng = 1.67 according to the Maxwell-Garnett (MG) effective medium theory [12,13]. The experimental α of the Qdot-doped glasses then yields the Qdot volume fraction f = α/αi; see Table 1.
The absorbance spectra, together with the refractive index spectrum of the glass, now allow to calculate the Qdot dielectric function εQD = εR + i·εI via KK-analysis of α . Experimental data are used for λ>400 nm, while below this wavelength bulk properties are assumed. Hence α is known over the entire spectral range, a prerequisite for the KK-analysis. However, MG theory implies that α is determined by both real and imaginary part of εQD, therefore a straightforward calculation of the optical constants as for bulk materials, is not possible. We circumvented this issue by applying an iterative scheme to calculate εQD . Starting from an initial guess for the imaginary part of the dielectric function, εI,0 (the bulk PbS εI makes a good approximation), we calculate εR,0 via the KK-relations. Both then yield an initial guess α0 for the absorption coefficient. From a linearization of the ratio α0/α, we calculate a first-order correction to εI,0 and obtain εI,1 = εI,0 + ΔεI . This scheme is repeated until the calculated αk is converged to the experimental α.
Figures 2(a) and 2(b) show the spectra of the Qdot refractive index n and the extinction coefficient k as obtained from the resulting εQD = n2-k2 + i·2nk. In compliance with , an error of about 10 percent is quoted. Two important optical properties, the oscillator strength of the ground state transition and the static dielectric constant ε0, were further derived from εQD to compare the PbS Qdots dispersed in a glass matrix to data obtained on colloidal PbS Qdots . Figures 2(c) and 2(d) show the results. The size-independent ε0, see Fig. 2(c), results in an average value of ε0 = 14.5 ± 1.8 for the colloidal Qdots . The Qdot-doped glass samples yield on average ε0 = 16.8 ± 2.1, i.e. within experimental error, no significant difference is observed. Furthermore, ε0 is again close to the bulk value of 17.5, in agreement with previous results . The oscillator strengths are also comparable; see Fig. 2(d). Hence we can conclude that both types of Qdots yield highly similar optical properties.
From the intrinsic Qdot optical constants we now come to the effective optical constants neff and keff of the Qdot-doped glasses, i.e. to the properties of our actual samples. They are obtained from the complex effective dielectric function (MG-theory, low-f limit ):12]. Figure 3(a) shows the keff-spectra of the four samples. Interestingly, the resulting keff, see full circles in Fig. 3(a),agrees well with values directly calculated from the absorption coefficient α = (4π/λ)keff, see full black lines. This provides further justification to our approach. Figure 3(b) displays the neff-spectra of sample A and of the pure glass. Due to the incorporation of Qdots, with a higher dielectric constant than the glass, neff of samples A-D exceeds ng of the undoped reference sample by 0.5-3.5∙10−3; see Fig. 3(c). Notice that the enhancement of neff calculated for sample A-D follows exactly the volume fractions f given in Table 1.
In summary we present spectra of the dielectric function of PbS Qdots in glass matrix in the 200-1800 nm range. The transmittance data are processed by a newly developed algorithm, which previously has been applied to colloidal Qdots. Based on the KK-analysis, this approach allows for the determination of the real and imaginary part of the dielectric function of the Qdots. A comparison of Qdots in glass matrix and colloidal Qdots reveals that optical properties are comparable. By calculating both the intrinsic Qdot refractive index and extinction coefficient and data on the effective optical constants of the synthesized Qdot-doped glasses, we provide important input for the modeling of photonic devices employing these novel materials.
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