# A Comparative Study of Theoretical Methods to Estimate Semiconductor Nanoparticles’ Size

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Materials

_{3})

_{2}·4H

_{2}O, 99.99%), thiophenol (C

_{6}H

_{5}-SH) (99%), sulphur powder (99.98%), toluene, methanol and dimethyl sulfoxide (DMSO) were purchased from Sigma-Aldrich (Darmstadt, Germany) and used without further purification.

#### 2.2. Characterization

_{exc}= 365 nm.

#### 2.3. Models

_{g})

^{n},

_{g}is the bandgap energy, n depends on the nature of the interband electronic transition, and A is an absorption constant [12]. Depending on the nature of the transition of the semiconductor, n can have the following values: For direct allowed transitions, n = 1/2; for direct forbidden transitions, n = 3/2; for indirect allowed transitions, n = 2; and for indirect forbidden transitions, n = 3. For this relation, Tauc proposed an extrapolation to locate the absorption edge in semiconductors. For this reason, (∝hν)

^{1/n}versus hν is plotted. By adjusting the variable n correctly, the absorption edge can be displayed.

#### 2.3.1. Brus Model

_{n}= E

_{b}+ (ħ

^{2}π

^{2}/2R

^{2}) × (1/m

_{e}*+1/m

_{h}*) − 1.8e

^{2}/(4πε

_{0}εR).

_{n}, and the nanoparticle radius, R. The constants associated with the material are: E

_{b}, the energy gap of bulk material; m

_{e}* and m

_{h}*, the effective masses of electrons and holes, respectively; and ε, the dielectric constant.

_{e}* and m

_{h}* is necessary to apply the Brus method. In this paper we will examine the possible effects that uncertainty of m

_{e}* and m

_{h}* could originate in the calculation of the size of some semiconducting monochalcogenides nanoparticles.

#### 2.3.2. Hyperbolic Band Model (HBM)

_{o}) outside the semiconductor [27].

_{n}

^{2}= E

_{b}

^{2}+ 2 ħ

^{2}× E

_{b}× (π/R)

^{2}/m*,

_{n}and E

_{b}are energy bandgap of nanoparticles and bulk semiconductor, respectively.

#### 2.3.3. Empirical Formula Suggested by Henglein et al.

_{a}) to the nanoparticles diameter (D) [15],

_{a}).

#### 2.3.4. Empirical Formula Obtained by Yu et al.

_{a}). For the calculation of the expression, CdS NPs were synthesized by their research group where nanocrystals sizes were determined by transmission electron microscopy (TEM) measurements.

^{−8}) λ

_{a}

^{3}+ (1.9557·10

^{−4}) λ

_{a}

^{2}+ (−9.2352·10

^{−2}) λ

_{a}+ 13.29.

#### 2.4. Synthesis of CdS Nanocrystals

_{3})

_{2}·4H

_{2}O was dissolved in distilled water and methanol, with ratio 1:1, and 0.2 M C

_{6}H

_{5}-SH was dissolved in methanol. Both solutions were agitated in different flasks until they became homogeneous. When solutes of both solutions were completely dissolved, solutions were mixed together and stirred during 15 min. The blend evolved and turned a whitish color. Once filtered and dried, the cadmium thiolate (Cd(C

_{6}H

_{5}S)

_{2}) was synthesized.

_{(S(1%))}. Three different nanoparticle solutions were prepared. In three different vials, Cd(C

_{6}H

_{5}S)

_{2}was dissolved in DMSO by keeping the relation, m

_{(Cd(C6H5-SH))}/V

_{(DMSO)}= 40 mg/mL, as shown in Table 1. These solutions were agitated and, when cadmium thiolate was dissolved, different volumes of V

_{(S(1%))}were added to each one. In the first one, 2.5 mL of V

_{(S(1%))}for each gram of Cd(C

_{6}H

_{5}-SH) were added. The mixture was stirred during 15 min and it became transparent and homogeneous. The output was the synthesis of CdS NPs with the following relation V

_{(S(1%))}/m

_{(Cd(C6H5-SH))}= 2.5 mL/g. For the synthesis of the other nanoparticles, two solutions with a relation V

_{(S(1%))}/m

_{(Cd(C6H5-SH))}= 5.0 and 7.5 mL/g, respectively, were selected. These nanoparticles are referenced as CdS(2.5), CdS(5.0), and CdS(7.5), respectively.

## 3. Results

#### 3.1. Optical Characterization

_{(S(1%))}in the synthesis led to higher wavelength in the PL emission of the nanocrystals.

#### 3.2. TEM Study

#### 3.3. Comparison of Theoretical Models to Estimate CdS Size

#### 3.3.1. Brus Model

_{e}* = 0.19·m

_{o}and m

_{h}* = 0.80·m

_{o}used by Brus et al. in their studies [33]. However, small variations of the value of the effective masses may lead to significant differences on the calculated sizes. The effective masses of the electrons and the holes had different values depending on to the consulted bibliography. For example, on the one hand, Praus et al. indicated the values m

_{e}* = 0.18·m

_{o}and m

_{h}* = 0.80·m

_{o}[23]. In the other hand, Dey et al. reported m

_{e}* = 0.42·m

_{o}and m

_{h}* = 0.61·m

_{o}[24].

_{h}* varied around the theoretical value, m

_{h}* = 0.80·m

_{o}, whereas the other variables remained constant. The variations of the radius when the hole effective mass parameter was changed are more evident for values of the bandgap energy close to the bulk. For example, for E

_{n}= 3.0 eV, radiuses oscillated between 1.82 nm for m

_{h}* = 0.70·m

_{o}to 1.62 nm for m

_{h}* = 0.90·m

_{o}. In contrast to this, for an energy value of 2.44 eV, the difference in the radius reached 0.85 nm. The ranges were based on the most usual values that are found in the bibliography for the variables studied. The interval was selected adding a variation of 0.1·m* to the common value.

_{e}*. As in the previous study, m

_{e}* varied around the bulk value, in this case, m

_{e}* = 0.19·m

_{o}. As detailed in Figure 6B, this variable was more critical than m

_{h}*. For E

_{n}= 3.0 eV, the radius varied between 1.88 and 1.64 nm for m

_{e}* = 0.10·m

_{o}and m

_{e}* = 0.30·m

_{o}, respectively. This difference increased for energies close to the bulk energy. It should also be noted that errors in m

_{e}* were more critical for energy gap close to CdS bulk. Small changes in m

_{e}* were determinant for energies close to the bulk energy (like in m

_{h}* variation study). Thus, accurate values for m

_{h}* and m

_{e}* should be used in order to avoid errors when using the Brus equation to estimate the size of quantum dots. Furthermore, m

_{e}* errors are more critical than m

_{h}* errors.

_{g}= 0.41 eV, m

_{e}* from 0.01·m

_{e}* to 0.20·m

_{e}*, m

_{h}* from 0.01·m

_{h}* to 0.20·m

_{h}*, and ε = 17.2 [34] and for ZnS, E

_{g}= 3.70 eV, m

_{e}* from 0.30·m

_{e}* to 0.50·m

_{e}*, m

_{h}* from 0.50·m

_{h}* to 0.70·m

_{h}*, and ε = 8.76 [23].

_{e}* variation where the other Brus equation variables remained constant. For the E

_{n}value kept constant, the increase in m

_{e}* value revealed a reduced PbS NPs diameter. It is noteworthy that for m

_{e}* values higher than m

_{e}* = 0.085·m

_{o}, size variations were not critical. In Figure 7A, we can observe results for the m

_{h}* simulation. This study obtained the same results as m

_{e}* because the m

_{h}* and m

_{e}* were identical for PbS NPs.

_{h}* and m

_{e}* variation on the estimated size is shown in Figure 8A and Figure 8B, respectively. Some results obtained were similar to results of CdS and PbS NPs tests. In Figure 8B, the m

_{e}* study showed that when m

_{e}* was increased nanoparticles’ radius decreased. Additionally, for low bandgap energy values, close to bulk, m

_{e}* variations were more critical than at higher energy values. Concerning Figure 8A, to m

_{h}* its influence was less than m

_{e}*, although it followed the same trend. For instance, for energy values close to 3.7 eV, the ZnS energy bulk m

_{h}* variations became more important because small changes produced estimable differences in nanoparticles’ size with respect to high energy values.

_{e}* and m

_{h}* values produced size estimation errors, a fact that became critical when getting close to bulk energy gap values. To avoid calculation errors in the Brus equation, a correct data collection is necessary for the m

_{e}* and m

_{h}* values.

#### 3.3.2. HBM Model

_{e}* and m

_{h}* may modify the results obtained when estimating NPs’ sizes. On the other hand, the HBM model simplified the effective masses of the electron and hole in a single mass, with the following equation:

_{e}* + 1/m

_{h}*,

_{e}* and m

_{h}* are the effective masses of electrons and holes, respectively. As in the Brus model, m

_{e}* and m

_{h}*variations were simulated and the effect on the HBM model was analysed.

_{e}* and m

_{h}* (Equation (6)) makes the study of both variables necessary. In the test of the effective mass (Figure 9), it can be noted that when m

_{e}* or m

_{h}* increased, m* increased. As in the previous studies, the NPs’ size decreased when m* increased. Additionally, for high bandgap energy values, m* variations were not critical for both m

_{e}* and m

_{h}*. For example, for E

_{n}= 4.5 eV, the diameter varied between 1.92 nm and 1.73 nm for variations of m

_{h}* and it varied between 1.97 nm and 1.78 nm for variations of m

_{e}*. In energies close to energy CdS bulk, the m* variations caused bigger differences in the size between extreme values (for E

_{n}= 2.6 eV, the difference is 0.75 nm for variations of m

_{h}*, and for variations of m

_{e}* the difference was 0.88 nm), but as the NP size was bigger, the relative error in the diameter estimation was negligible.

_{e}* and m

_{h}* variations for PbS (Figure 10) and for ZnS (Figure 11) NPs. For PbS and ZnS simulations, the results were similar to the CdS NPs’ simulations. An increase of m

_{e}* and m

_{h}* augmented the effective mass. This fact modified HBM model results, where the diameters were smaller due to the growth of m*. As in CdS, PbS, and ZnS NPs, variations towards low energies increased nanoparticles’ dimensions.

_{h}*, the m

_{e}*, and m* variables were critical when the energy gap was close to the bulk energy. Then, both the calculation of the absorption edge and an exact value of the effective mass became critical for a correct use of theoretical models to calculate the size of the nanoparticles. Attending to Figure 9 and Figure 11 versus Figure 6 and Figure 8, it is remarkable that HBM led to bigger nanoparticle diameter when energy was close to the bulk bandgap, contrasting with the Brus equation. This was observed for CdS and ZnS NPs, but not for PbS NPs. This result could be expected because the Brus model uses infinite square well-type quantum confinement.

#### 3.3.3. Henglein Model and Yu Equation

_{a}) and the diameter will be addressed.

## 4. Conclusions

_{n}due to sharp variation of size for energies closest to NPs’ E

_{b}, especially for PbS NPs. In our experiment with CdS NPs, the worse estimation of sizes was obtained by the Henglein formula.

_{e}*, m

_{h}*, and m* parameters must be well known, since errors in their values may lead to nonnegligible errors in the size estimation, especially for energies slightly higher than the bulk bandgap. These considerations must be taken into account in the theoretical models, regardless of the type of nanoparticles that are being studied.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Normalized photoluminescence spectra, pristine optical absorption spectra (

**A**) and optical absorption edge calculated using Tauc relation (

**B**) for CdS nanoparticles synthesized by different volumes of sulphur in toluene. The 2.5, 5.0, and 7.5 milliliters of sulphur solution per gram of cadmium thiolate were used for the synthesis of each nanoparticle, respectively.

**Figure 2.**Photoluminescence spectra (

**A**) and pristine optical absorption spectra (

**B**) for thiophenol (black) and cadmium thiolate (cyan) CdS nanoparticles synthesized with 7.5 milliliters of sulphur solution per gram of cadmium thiolate in toluene (blue) and DMSO (red).

**Figure 4.**Histograms of different nanocrystals located in TEM analysis for each solution of CdS nanoparticles, for CdS(2.5) (

**A**), for CdS(5.0) (

**B**), and for CdS(7.5) (

**C**).

**Figure 5.**Graphs of Tauc relation with different n values for CdS nanoparticles (

**A**), where n depends on the nature of interband electronic transition, varying between n = 1/2, 3/2, 2, and 3. Different simulation (

**B**) of nanoparticles’ (NPs’) sizes estimated with each model for CdS NPs as a function of bandgap energy. Horizontal lines set the average diameter of the nanoparticles calculated by the TEM images: CdS(2.5) (black line), CdS(5.0) (red line), and CdS(7.5) (blue line). The corresponding circle is located according to the optical gap calculated from absorbance spectra.

**Figure 6.**The m

_{h}* variation study (

**A**) for Brus equation of CdS nanoparticles for different energies, from high values of energy to CdS energy bulk. The m

_{h}* oscillated between m

_{h}* = 0.70·m

_{o}and m

_{h}* = 0.90·m

_{o}with the rest of the variables remaining constant: E

_{b}= 2.42 eV, ε = 5.7, and m

_{e}* = 0.19·m

_{o}. The m

_{e}* simulation (

**B**) for Brus equation of CdS nanoparticles for different energies, from high values of energy to CdS energy bulk. The m

_{e}* oscillated between m

_{e}* = 0.10·m

_{o}and m

_{e}* = 0.30·m

_{o}with the rest of the variables remaining constant: E

_{b}= 2.42 eV, ε = 5.7, and m

_{h}* = 0.80·m

_{o}.

**Figure 7.**The m

_{h}* variation study (

**A**) for Brus equation of PbS nanoparticles for different energies, from high values of energy to PbS energy bulk. The m

_{h}* oscillated between m

_{h}* = 0.01·m

_{o}and m

_{h}* = 0.20·m

_{o}with the rest of the variables remaining constant: E

_{b}= 0.41 eV, ε = 17.2, and m

_{e}* = 0.085·m

_{o}. The m

_{e}* simulation (

**B**) for Brus equation of PbS nanoparticles for different energies, from high values of energy to PbS energy bulk. The m

_{e}* oscillated between m

_{e}* = 0.01·m

_{o}and m

_{e}* = 0.20·m

_{o}and with the rest of the variables remaining constant: E

_{b}= 0.41 eV, ε = 17.2, and m

_{h}* = 0.085·m

_{o}.

**Figure 8.**The m

_{h}* variation study (

**A**) for Brus equation of ZnS nanoparticles for different energies, from high values of energy to ZnS energy bulk. The m

_{h}* oscillated between m

_{h}* = 0.50·m

_{o}and m

_{h}* = 0.70·m

_{o}and with the rest of the variables remaining constant: E

_{b}= 3.70 eV, ε = 8.76, and m

_{e}* = 0.42·m

_{o}. The m

_{e}* simulation (

**B**) for Brus equation of ZnS nanoparticles for different energies, from high values of energy to ZnS energy bulk. The m

_{e}* oscillated between m

_{e}* = 0.30·m

_{o}and m

_{e}* = 0.50·m

_{o}and with the rest of the variables remaining constant: E

_{b}= 3.70 eV, ε = 8.76, and m

_{h}* = 0.61·m

_{o}.

**Figure 9.**Estimated CdS NPs’ size by the hyperbolic band model (HBM) model as a function of nanoparticle bandgap energy with (

**A**) m

_{h}* and m

_{e}* (

**B**) as parameters. The m

_{h}* oscillated between m

_{h}* = 0.70·m

_{o}and m

_{h}* = 0.90·m

_{o}and m

_{e}* oscillated between m

_{e}* = 0.10·m

_{o}and m

_{h}* = 0.30·m

_{o}, with E

_{b}remaining constant, E

_{b}= 2.42 eV.

**Figure 10.**Estimated PbS NPs’ size by HBM model as a function of nanoparticle bandgap energy with (

**A**) m

_{h}* and m

_{e}* (

**B**) as parameters. The m

_{h}* oscillated between m

_{h}* = 0.01·m

_{o}and m

_{h}* = 0.20·m

_{o}and m

_{e}* oscillated between m

_{e}* = 0.01·m

_{o}and m

_{h}* = 0.20·m

_{o}, with E

_{b}remaining constant, E

_{b}= 0.41 eV.

**Figure 11.**Estimated ZnS NPs’ size by HBM model as a function of nanoparticle bandgap energy with (

**A**) m

_{h}* and m

_{e}* (

**B**) as parameters. The m

_{h}* oscillated between m

_{h}* = 0.50·m

_{o}and m

_{h}* = 0.70·m

_{o}and m

_{e}* oscillated between m

_{e}* = 0.30·m

_{o}and m

_{h}* = 0.50·m

_{o}, with E

_{b}remaining constant, E

_{b}= 3.70 eV.

**Figure 12.**Comparison among the nanoparticles’ size obtained with the theoretical methods studied in this paper and the experimental values obtained with TEM images (solid lines).

NPs | m_{[Cd(C6H5-SH)]} | V_{[DMSO]} | V_{[S(1%)]} |
---|---|---|---|

CdS(2.5) | 0.80 g | 20 mL | 2 mL |

CdS(5.0) | 0.80 g | 20 mL | 4 mL |

CdS(7.5) | 0.80 g | 20 mL | 6 mL |

**Table 2.**Absorption edge and photoluminescence peak of the CdS nanoparticles synthesized by 2.5, 5.0, and 7.5 milliliters of sulphur per gram of cadmium thiolate.

NPs | Absorption Edge | PL Peak | |
---|---|---|---|

CdS(2.5) | 3.54 eV | 350 nm | 456 nm |

CdS(5.0) | 3.33 eV | 372 nm | 480 nm |

CdS(7.5) | 3.21 eV | 386 nm | 495 nm |

**Table 3.**Results’ deviation between the different theoretical models estimation for the size and the real measurements obtained with TEM analysis.

NPs | Brus | HBM | Henglein | Yu |
---|---|---|---|---|

CdS(2.5) | −6% | −11% | −30% | −25% |

CdS(5.0) | −8% | −9% | −30% | −16% |

CdS(7.5) | −24% | −24% | −42% | −26% |

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## Share and Cite

**MDPI and ACS Style**

Rodríguez-Mas, F.; Ferrer, J.C.; Alonso, J.L.; Valiente, D.; Fernández de Ávila, S.
A Comparative Study of Theoretical Methods to Estimate Semiconductor Nanoparticles’ Size. *Crystals* **2020**, *10*, 226.
https://doi.org/10.3390/cryst10030226

**AMA Style**

Rodríguez-Mas F, Ferrer JC, Alonso JL, Valiente D, Fernández de Ávila S.
A Comparative Study of Theoretical Methods to Estimate Semiconductor Nanoparticles’ Size. *Crystals*. 2020; 10(3):226.
https://doi.org/10.3390/cryst10030226

**Chicago/Turabian Style**

Rodríguez-Mas, Fernando, Juan Carlos Ferrer, José Luis Alonso, David Valiente, and Susana Fernández de Ávila.
2020. "A Comparative Study of Theoretical Methods to Estimate Semiconductor Nanoparticles’ Size" *Crystals* 10, no. 3: 226.
https://doi.org/10.3390/cryst10030226