当前位置：
X-MOL 学术
›
Chem. Mater.
›
论文详情
Our official English website, www.x-mol.net, welcomes your
feedback! (Note: you will need to create a separate account there.)
Comment on “Size Dependent Optical Properties and Structure of ZnS Nanocrystals Prepared from a Library of Thioureas”
Chemistry of Materials ( IF 7.2 ) Pub Date : 2022-06-22 , DOI: 10.1021/acs.chemmater.2c01526 Tangi Aubert 1 , Aleksandr A. Golovatenko 2 , Anna V. Rodina 2 , Zeger Hens 3
Chemistry of Materials ( IF 7.2 ) Pub Date : 2022-06-22 , DOI: 10.1021/acs.chemmater.2c01526 Tangi Aubert 1 , Aleksandr A. Golovatenko 2 , Anna V. Rodina 2 , Zeger Hens 3
Affiliation
In their recent article, Owen et al. published a new data set for ZnS quantum dots, linking an optical band gap with a nanocrystal diameter. (1) Such band gap/size data sets have been very helpful in the field of quantum dots to build calibration or sizing curves. These sizing curves are widely used to quickly determine nanocrystal sizes from absorption measurements, and they have been reported for a large variety of materials. The sizing of ZnS quantum dots, however, has always suffered from the low contrast this material provides in bright-field transmission electron microscopy (TEM), which is the most common technique for measuring nanocrystal dimensions. In this regard, the work presented by Owen and co-workers is remarkable, both from a synthesis and a characterization perspective. On the one hand, quasi-spherical ZnS quantum dots are formed across a broad diameter range and sized by high-angle annular dark-field scanning transmission electron microscopy (STEM) and a pair distribution function (PDF) analysis of X-ray scattering. On the other hand, the optical band gap of the nanocrystals was determined from their UV–vis absorption spectra as it is routinely done for this purpose. Finally, the authors fitted their data set using a function that adds the inverse of a second-order polynomial to the bulk band gap energy. Although this type of function may fit well experimental data and provide reliable size calibration within the limits of the fitted data range, we recently pointed out the limitations of this approach, which often makes use of several adjustable parameters without physical meaning. (2) To address this shortcoming, we proposed a general expression to describe size quantization in semiconductor nanocrystals, in an effort to standardize nanocrystal size determination based on optical band gap measurements. (2) We validated this expression for 10 different materials, including Cd and Pb chalcogenides (CdS, CdSe, CdTe, PbS, PbSe, and PbTe), a III–V semiconductor (InP), indirect (Si) and negative (HgTe) band gap materials, and nonspherical nanocrystals (cube-shaped CsPbBr3). Thanks to the extensive data set from Owen et al., we can now show that this model is in good agreement with the experimental data for ZnS as well, and we provide here a sizing curve for ZnS in line with this standardized approach. The initial motivation to develop a general, analytical expression for a sizing function was the need for meaningful extrapolations outside of experimental data ranges. To this end, we considered the expected behavior at small and large diameters and integrated a correction for the impact of nonparabolic energy bands on the quantum dot band gap. This ultimately led to a parameter-free model, capable of predicting the optical band gap (E1), as a function of the nanocrystal diameter (d), based on bulk semiconductor parameters only.
(1) Here, E0 is the bulk optical band gap, which is the difference between the single-particle gap Eg and the exciton binding energy Eex, Ry is the Rydberg constant (13.606 eV), a0 is the Bohr radius of hydrogen (0.053 nm), ε∞ is the high-frequency dielectric constant of the material, and d0 is seen as the largest diameter where strong quantization dominates. For consistency between different materials, d0 was systematically identified with the exciton Bohr diameter, calculated using the high-frequency dielectric constant.
(2) In eq 2, ε0 is the vacuum permittivity, ℏ is the reduced Planck constant, m0 is the free electron mass, e is the elementary charge, and μ is the reduced effective mass of an electron–hole pair. In the process of establishing eq 1, α was initially introduced to compensate for approximations made in the calibration procedure. Our study of the 10 aforementioned materials later showed α to be material-independent, and a single value of α equal to 0.7 was determined, although the physical meaning of this factor remains unclear. Inspired by the work published by Owen et al., we used eq 1 to predict the optical band gap energy of ZnS nanocrystals using the parameters listed in Table 1. As shown in Figure 1a, the results show very good agreement with the data set from Owen et al. For this prediction, we used averaged parameters between the zinc blende and wurtzite phases of ZnS. As for CdSe, (2) these two polymorphs can be difficult to discriminate in nanocrystals and even coexist within the same sample; this point was noted by Owen et al. in their PDF analyses. Thus, using parameters that can approximate both polymorphs can be more relevant in practice. Room-temperature values of the parameters were used by default when available. When multiple values were reported, averaged or rounded values were used. Average between zinc blende and wurtzite values. Density-of-states heavy-hole values were used for mh. Figure 1. (a) Optical band gap energy predicted by eq 1 for ZnS (blue continuous line), i.e., using the Bohr diameter as calculated from bulk parameters, and compared to the experimental data from Owen et al. (1) (b) Log–Log representation of Efit as a function of 1/ε∞d0, comparing the new value obtained for ZnS (full black marker) with the materials investigated in our previous work (open gray markers). (2) The red line depicts the linear regression with α = 0.7. (c) Optical band gap energy obtained as best fit of eq 4 to ZnS data set (red continuous line), i.e., using the Bohr diameter as an adjustable parameter, which resulted in a dfit value of 2.0 nm. (a, c) Only the data used by Owen et al. in their sizing curve fit are shown. The bulk optical band gap is indicated with gray dashed lines. While Figure 1a is already a good confirmation for the predictive power of our model, we can also use this additional data set to verify the α-value. To this end, the experimental data were fitted to the following expression.
(3) Here, Efit is the only adjustable parameter. The resulting value (Table 1) is plotted as a function of 1/ε∞d0 and compared to the other materials in Figure 1b. Interestingly, ZnS, being at the low dielectric constant/small Bohr diameter extreme of the materials set considered, still shows good agreement with the previously observed trend. In fact, a linear fit including all the materials in Figure 1b still yields a best-fit value of α = 0.7. Despite the good correspondence between eq 1 and the new band gap/size data for ZnS, some of the bulk semiconductor parameters used to calculate E1 through eq 1 are not always accurately known. This holds true, in particular, for the Bohr diameter, whose calculation involves multiple parameters that each have their own uncertainties. Thus, in order to further improve the sizing curve, d0 is replaced by an adjustable parameter dfit in our general expression.
(4) Fitting the experimental data of Owen et al. to eq 4 (Figure 1c) yielded a dfit of 2.0 nm, a figure very close to the calculated d0 value of 2.3 nm (Table 1). This result neatly illustrates the advantage of a sizing function that allows one to assess bulk semiconductor parameters. In this regard, our results strongly suggest that ZnS has a relatively small Bohr diameter, smaller than the value of 5 nm that is often mentioned in the literature. (1) In practice, sizing curves are mostly used to determine the nanocrystal diameter from the optical band gap. To this end, eq 4 can be inverted to the following expression of the sizing curve.
(5) As a final point, Figure 1a,c shows that, both in the case of the prediction with eq 1 and the fit with eq 4, the experimental sizes for the two largest samples deviate significantly from the sizing curve. Possibly, this problem results from the way the optical band gap of quantum dots is routinely obtained from absorption spectra. To illustrate the issue, Figure 2 reproduces the UV absorption spectra of these two largest samples as provided by Owen et al., which correspond to diameters of 3.81 and 4.47 nm, respectively, as determined by STEM. As can be seen, both spectra lack a clear feature, such as a Gaussian peak, that helps one to identify the band gap. Hence, transition energies estimated from such spectra will only provide a rough estimate of the band gap. If done systematically, such energies can still be used for the purpose of size calibration, but one should take care when calling such values the optical band gap. In fact Owen et al. had to treat the bulk optical band gap as an adjustable parameter in order to fit their experimental results. This point is well-illustrated here for ZnS, which has a very small Bohr diameter, but the same holds true for other close-to-bulk nanocrystals of, for instance, CdSe or InP. On the one hand, this point hints at an intrinsic limitation of sizing curves and could motivate the search for a more systematic approach to experimentally determine optical band gaps as well. On the other hand, one could also consider reverting the approach and use an experimentally determined nanocrystal size, for example, by TEM, to estimate its band gap based on our model. Figure 2. UV absorption spectra of the two largest samples used by Owen et al. (sample 13, estimated diameter 3.81 nm; sample 17, estimated diameter 4.47 nm) in their fit, and the position of their extracted first optical transition (reproduced from the Supporting Information of ref (1)). In conclusion, we found that the band gap/size data set recently published by Owen et al. is in good agreement with the predictions of the parameter-free sizing curve we published recently. When using the high-frequency dielectric constant, the best agreement is obtained for a refined Bohr diameter of 2.0 nm, a figure considerably smaller than often published values, but close to values calculated using bulk parameters in-between those of zinc blende and wurtzite ZnS. This correspondence provides an additional validation of our generalized approach to quantum dot sizing curves, and we therefore believe that the optimized parameters reported here for size quantization in ZnS can be valuable to spread a standardized approach to sizing curves in quantum dot studies. This article references 3 other publications. This article has not yet been cited by other publications. Figure 1. (a) Optical band gap energy predicted by eq 1 for ZnS (blue continuous line), i.e., using the Bohr diameter as calculated from bulk parameters, and compared to the experimental data from Owen et al. (1) (b) Log–Log representation of Efit as a function of 1/ε∞d0, comparing the new value obtained for ZnS (full black marker) with the materials investigated in our previous work (open gray markers). (2) The red line depicts the linear regression with α = 0.7. (c) Optical band gap energy obtained as best fit of eq 4 to ZnS data set (red continuous line), i.e., using the Bohr diameter as an adjustable parameter, which resulted in a dfit value of 2.0 nm. (a, c) Only the data used by Owen et al. in their sizing curve fit are shown. The bulk optical band gap is indicated with gray dashed lines. Figure 2. UV absorption spectra of the two largest samples used by Owen et al. (sample 13, estimated diameter 3.81 nm; sample 17, estimated diameter 4.47 nm) in their fit, and the position of their extracted first optical transition (reproduced from the Supporting Information of ref (1)). This article references 3 other publications.
更新日期：2022-06-22
(1) Here, E0 is the bulk optical band gap, which is the difference between the single-particle gap Eg and the exciton binding energy Eex, Ry is the Rydberg constant (13.606 eV), a0 is the Bohr radius of hydrogen (0.053 nm), ε∞ is the high-frequency dielectric constant of the material, and d0 is seen as the largest diameter where strong quantization dominates. For consistency between different materials, d0 was systematically identified with the exciton Bohr diameter, calculated using the high-frequency dielectric constant.(2) In eq 2, ε0 is the vacuum permittivity, ℏ is the reduced Planck constant, m0 is the free electron mass, e is the elementary charge, and μ is the reduced effective mass of an electron–hole pair. In the process of establishing eq 1, α was initially introduced to compensate for approximations made in the calibration procedure. Our study of the 10 aforementioned materials later showed α to be material-independent, and a single value of α equal to 0.7 was determined, although the physical meaning of this factor remains unclear. Inspired by the work published by Owen et al., we used eq 1 to predict the optical band gap energy of ZnS nanocrystals using the parameters listed in Table 1. As shown in Figure 1a, the results show very good agreement with the data set from Owen et al. For this prediction, we used averaged parameters between the zinc blende and wurtzite phases of ZnS. As for CdSe, (2) these two polymorphs can be difficult to discriminate in nanocrystals and even coexist within the same sample; this point was noted by Owen et al. in their PDF analyses. Thus, using parameters that can approximate both polymorphs can be more relevant in practice. Room-temperature values of the parameters were used by default when available. When multiple values were reported, averaged or rounded values were used. Average between zinc blende and wurtzite values. Density-of-states heavy-hole values were used for mh. Figure 1. (a) Optical band gap energy predicted by eq 1 for ZnS (blue continuous line), i.e., using the Bohr diameter as calculated from bulk parameters, and compared to the experimental data from Owen et al. (1) (b) Log–Log representation of Efit as a function of 1/ε∞d0, comparing the new value obtained for ZnS (full black marker) with the materials investigated in our previous work (open gray markers). (2) The red line depicts the linear regression with α = 0.7. (c) Optical band gap energy obtained as best fit of eq 4 to ZnS data set (red continuous line), i.e., using the Bohr diameter as an adjustable parameter, which resulted in a dfit value of 2.0 nm. (a, c) Only the data used by Owen et al. in their sizing curve fit are shown. The bulk optical band gap is indicated with gray dashed lines. While Figure 1a is already a good confirmation for the predictive power of our model, we can also use this additional data set to verify the α-value. To this end, the experimental data were fitted to the following expression.(3) Here, Efit is the only adjustable parameter. The resulting value (Table 1) is plotted as a function of 1/ε∞d0 and compared to the other materials in Figure 1b. Interestingly, ZnS, being at the low dielectric constant/small Bohr diameter extreme of the materials set considered, still shows good agreement with the previously observed trend. In fact, a linear fit including all the materials in Figure 1b still yields a best-fit value of α = 0.7. Despite the good correspondence between eq 1 and the new band gap/size data for ZnS, some of the bulk semiconductor parameters used to calculate E1 through eq 1 are not always accurately known. This holds true, in particular, for the Bohr diameter, whose calculation involves multiple parameters that each have their own uncertainties. Thus, in order to further improve the sizing curve, d0 is replaced by an adjustable parameter dfit in our general expression.(4) Fitting the experimental data of Owen et al. to eq 4 (Figure 1c) yielded a dfit of 2.0 nm, a figure very close to the calculated d0 value of 2.3 nm (Table 1). This result neatly illustrates the advantage of a sizing function that allows one to assess bulk semiconductor parameters. In this regard, our results strongly suggest that ZnS has a relatively small Bohr diameter, smaller than the value of 5 nm that is often mentioned in the literature. (1) In practice, sizing curves are mostly used to determine the nanocrystal diameter from the optical band gap. To this end, eq 4 can be inverted to the following expression of the sizing curve.(5) As a final point, Figure 1a,c shows that, both in the case of the prediction with eq 1 and the fit with eq 4, the experimental sizes for the two largest samples deviate significantly from the sizing curve. Possibly, this problem results from the way the optical band gap of quantum dots is routinely obtained from absorption spectra. To illustrate the issue, Figure 2 reproduces the UV absorption spectra of these two largest samples as provided by Owen et al., which correspond to diameters of 3.81 and 4.47 nm, respectively, as determined by STEM. As can be seen, both spectra lack a clear feature, such as a Gaussian peak, that helps one to identify the band gap. Hence, transition energies estimated from such spectra will only provide a rough estimate of the band gap. If done systematically, such energies can still be used for the purpose of size calibration, but one should take care when calling such values the optical band gap. In fact Owen et al. had to treat the bulk optical band gap as an adjustable parameter in order to fit their experimental results. This point is well-illustrated here for ZnS, which has a very small Bohr diameter, but the same holds true for other close-to-bulk nanocrystals of, for instance, CdSe or InP. On the one hand, this point hints at an intrinsic limitation of sizing curves and could motivate the search for a more systematic approach to experimentally determine optical band gaps as well. On the other hand, one could also consider reverting the approach and use an experimentally determined nanocrystal size, for example, by TEM, to estimate its band gap based on our model. Figure 2. UV absorption spectra of the two largest samples used by Owen et al. (sample 13, estimated diameter 3.81 nm; sample 17, estimated diameter 4.47 nm) in their fit, and the position of their extracted first optical transition (reproduced from the Supporting Information of ref (1)). In conclusion, we found that the band gap/size data set recently published by Owen et al. is in good agreement with the predictions of the parameter-free sizing curve we published recently. When using the high-frequency dielectric constant, the best agreement is obtained for a refined Bohr diameter of 2.0 nm, a figure considerably smaller than often published values, but close to values calculated using bulk parameters in-between those of zinc blende and wurtzite ZnS. This correspondence provides an additional validation of our generalized approach to quantum dot sizing curves, and we therefore believe that the optimized parameters reported here for size quantization in ZnS can be valuable to spread a standardized approach to sizing curves in quantum dot studies. This article references 3 other publications. This article has not yet been cited by other publications. Figure 1. (a) Optical band gap energy predicted by eq 1 for ZnS (blue continuous line), i.e., using the Bohr diameter as calculated from bulk parameters, and compared to the experimental data from Owen et al. (1) (b) Log–Log representation of Efit as a function of 1/ε∞d0, comparing the new value obtained for ZnS (full black marker) with the materials investigated in our previous work (open gray markers). (2) The red line depicts the linear regression with α = 0.7. (c) Optical band gap energy obtained as best fit of eq 4 to ZnS data set (red continuous line), i.e., using the Bohr diameter as an adjustable parameter, which resulted in a dfit value of 2.0 nm. (a, c) Only the data used by Owen et al. in their sizing curve fit are shown. The bulk optical band gap is indicated with gray dashed lines. Figure 2. UV absorption spectra of the two largest samples used by Owen et al. (sample 13, estimated diameter 3.81 nm; sample 17, estimated diameter 4.47 nm) in their fit, and the position of their extracted first optical transition (reproduced from the Supporting Information of ref (1)). This article references 3 other publications.
京公网安备 11010802027423号