Characterization of Nanocrystal Size Distribution using Raman Spectroscopy with a Multi-particle Phonon Confinement Model

Summary

We demonstrate how to determine the size distribution of semiconductor nanocrystals in a quantitative manner using Raman spectroscopy employing an analytically defined multi-particle phonon confinement model. Results obtained are in excellent agreement with the other size analysis techniques like transmission electron microscopy and photoluminescence spectroscopy.

Abstract

Analysis of the size distribution of nanocrystals is a critical requirement for the processing and optimization of their size-dependent properties. The common techniques used for the size analysis are transmission electron microscopy (TEM), X-ray diffraction (XRD) and photoluminescence spectroscopy (PL). These techniques, however, are not suitable for analyzing the nanocrystal size distribution in a fast, non-destructive and a reliable manner at the same time. Our aim in this work is to demonstrate that size distribution of semiconductor nanocrystals that are subject to size-dependent phonon confinement effects, can be quantitatively estimated in a non-destructive, fast and reliable manner using Raman spectroscopy. Moreover, mixed size distributions can be separately probed, and their respective volumetric ratios can be estimated using this technique. In order to analyze the size distribution, we have formulized an analytical expression of one-particle PCM and projected it onto a generic distribution function that will represent the size distribution of analyzed nanocrystal. As a model experiment, we have analyzed the size distribution of free-standing silicon nanocrystals (Si-NCs) with multi-modal size distributions. The estimated size distributions are in excellent agreement with TEM and PL results, revealing the reliability of our model.

Introduction

Semiconductor nanocrystals draw attention as their electronic and optical properties can be tuned by simply changing their size in the range compared to their respective exciton-Bohr radii.¹ These unique size-dependent features make these nanocrystals relevant for various technological applications. For instance, carrier multiplication effects, observed when a high energy photon is absorbed by the nanocrystals of CdSe, Si, and Ge, can be used in the concept of spectrum conversion in solar cell applications;^2–4 or size-dependent optical emission from PbS-NCs and Si-NCs can be used in the light emitting diode (LED) applications.^5,6 A precise knowledge and control on the nanocrystal size distribution will therefore play a determinant role on the reliability and the performance of these technological applications based on nanocrystals.

The commonly used techniques for the size distribution and morphology analysis of nanocrystals can be listed as X-ray diffraction (XRD), transmission electron microscopy (TEM), photoluminescence spectroscopy (PL), and Raman spectroscopy. XRD is a crystallographic technique that reveals morphological information of the analyzed material. From the broadening of the diffraction peak, estimation of the nanocrystal size is possible,⁷ however, obtaining a clear data is usually time consuming. Moreover, XRD can only enable the calculation of average of the nanocrystal size distribution. In the existence of multi-modal size distributions, size analysis with XRD can be misleading and result in wrong interpretations. TEM is a powerful technique that enables imaging of the nanocrystals.⁸ Although TEM is able to reveal the presence of individual distributions in a multi-modal size distribution, sample preparation issue is always an effort to be spent before the measurements. In addition, working on densely packed nanocrystal ensembles with different sizes is challenging because of the difficulty of individual nanocrystal imaging. Photoluminescence spectroscopy (PL) is an optical analysis technique, and optically active nanocrystals can be diagnosed. Nanocrystal size distribution is obtained from the size-dependent emission.⁹ Due to their poor optical properties of indirect band gap nanoparticles, large nanocrystals that are not subject to confinement effects, and defect-rich small nanocrystals cannot be detected by PL and the observed size distribution is only limited to nanocrystals with good optical properties. Although each of these abovementioned techniques has its own advantages, none of them have the capability of meeting the expectations (that is, being fast, non-destructive, and reliable) from and idealized size analysis technique.

Another means of size distribution analysis of nanocrystals is Raman spectroscopy. Raman spectroscopy is widely available in most of the labs, and it is a fast and non-destructive technique. In addition, in most cases, sample preparation is not required. Raman spectroscopy is a vibrational technique, which can be used to obtain information on different morphologies (crystalline or amorphous), and size-related information (from the size-dependent shift in the phonon modes that appear in the frequency spectrum) of the analyzed material.¹⁰ The unique feature of Raman spectroscopy is that, while size-dependent changes are observed as a shift in the frequency spectrum, the shape of the phonon peak (broadening, asymmetry) gives information on the shape of the nanocrystal size distribution. Therefore it is in principle possible to extract the necessary information, i.e., the mean size and the shape factor, from Raman spectrum to obtain the size distribution of nanocrystals analyzed. In the case of multi-modal size distributions sub-distributions can also be separately identified via deconvolution of the experimental Raman spectrum.

In the literature, two theories are commonly referred to model the effect of nanocrystal size distribution on the shape of the Raman spectrum. The bond polarizability model (BPM)¹¹ describes the polarizability of a nanocrystal from the contributions of all the bonds within that size. The one-particle phonon confinement model (PCM)¹⁰ uses size-dependent physical variables, i.e., crystal momentum, phonon frequency and dispersion, and the degree of confinement, to define the Raman spectrum of a nanocrystal with a specific size. Since these physical variables depend on the size, an analytical representation of the PCM that can be explicitly formulized as a function of nanocrystal size can be defined. Projecting this expression on a generic size distribution function will therefore be able to account for the effect of size distribution within the PCM, which can be used to determine the nanocrystal size distribution from the experimental Raman spectrum.¹²

Protocol

1. Planning of the Experiments

1. Synthesize or obtain the nanocrystals of interest¹³ (Figure 1a).
2. Avoid any confusion with the background signal by making sure that the substrate material does not have overlapping peaks in the Raman spectrum of the nanocrystals (Figure 1a).
3. Turn on the laser of the Raman spectroscopy setup. Wait enough time (approximately 15 min) for the laser intensity to stabilize.
4. Measure a bulk reference of the nanomaterial to be analyzed¹² (Figure 1b), following the measurement steps described in Step 2. From the peak position of the bulk material, estimate the relative shift¹².
5. Estimate the required laser power for Raman measurements using different powers on the nanocrystals going to be measured. Start a measurement with the lowest possible power to get enough signal (the ratio of the peak intensity to the background noise should be at least 50) , and increase the laser power if needed, as long as the position and shape of the nanocrystal Raman peak stays same^12,13.

2. Raman Spectroscopy of the Nanocrystal of Interest

1. Load the sample with nanocrystal powder deposited on the substrate into the measurement chamber.
   Note: The substrate dimensions are not critical (can be from millimeters to tens of centimeters) as long as it fits to the sample holder stage. The powder or thin film thickness should be at least tens of nanometers to have detectable signal from Raman spectroscope. For the planar substrate holder stage, simply lay the substrate under the optics (Figure 1b).
   1. Make sure the “Laser” and “Active” lights are off before opening the door in order to be safe from the unwanted illumination of the operating laser. If these lights are not off, perform the actions in steps 2.5 and 2.6. The “Interlock” sign always stays on.
   2. Press “Door Release” and open the door of the measurement chamber, and put the sample onto the sample holder stage (Figure 1b).
2. Adjust the focusing of the sample to be measured to get the highest possible signal.
   1. Select 50X objective and focus on the surface of the nanocrystal powder (Figure 1b).
   2. Bring the sample under focus using the z-direction manipulator of the sample holder. Check the clarity of the focused image from the live camera view on the computer screen.
   3. Close the door of the measurement chamber.
   4. Remove the shutter by clicking the “shutter-out” button from the Renishaw software, and let the laser light shine on the sample to be measured. Observe that the “Laser” and “Active” signs now flash green and blink red, respectively. In the live image from the screen, the laser will be visible (Figure 1c).
   5. From the live image, fine-tune the focusing of the sample using the wheel manipulator until the smallest laser spot, which is the best focus, is observed on the live image.
3. Set up a measurement from the Renishaw analysis software as described below (Figure 1d).
   1. From “Measurement” select new spectral acquisition option.
   2. From the pop-up window, set the measurement range from 150 to 700 cm^-1, set the time for the measurement as 30 sec, the total number of acquisition as 2x, and the percentage of the laser power as 0.5% (of a 25 mW laser) to be used during the measurement. Accept the parameters inserted, and the window will be closed.
   3. Start the measurement by clicking on the acquisition start button on the menu-bar. During the measurement the “Laser” and the “Active” lights will remain on.
4. Do not open the measurement chamber when these lights are on as the laser is in operation and measurement is being performed.
5. After the measurement is finished, put the shutter in by clicking the “shutter in” button from the Renishaw software. Observe that the lights of the “Laser” and the “Active” are turned off. Press “Door Release” and then open the door of the measurement chamber.
6. Before taking the sample out, lower the sample holder stage with the z-manipulator until there is a safe distance between the measured sample and the surface of the magnifying lens to remove the sample. Then, put the sample back to its container.
7. Turn off the laser.
8. Save the data in Renishaw software format, “.wxd”, and in the text file format, “.txt”. The latter will be used for the analysis of the experimental data.

3. Size Distribution Determination of the Nanocrystal of Interest

1. Open the text files of the measurements for the nanocrystal measurement, and the bulk reference.
2. Before plotting the data, smooth them using cubic spline, and normalize the data to 1 at their highest peak positions in order to have a good comparison of the relative peak shifts.
3. Plot the silicon nanocrystal and reference silicon data, determine the peak position of reference silicon, and estimate the amount of the shift, if any, from the actual peak position of 521 cm^-1.¹² Then save the processed silicon nanocrystal data as .txt file.
4. Start the fitting procedure.
   1. For the fitting procedure, type the fitting function shown at Figure 2f into an analysis program such as Mathematica.
   2. Import the normalized and corrected data as the input for the non-linear fitting model using the "Import" command.
   3. Ensure that the interval for skewness is between 0.1 and 1.0, and the mean size interval is between 2 nm and 20 nm.
   4. If necessary, insert additional peak(s) under the measured peak using the fitting function and repeat the steps 3.4.2 and 3.4.3 to fit the other sub-distribution(s).
   5. Press “Shift+Enter” to perform the fitting procedure.
   6. After that, insert the obtained values for the mean size and the skewness in the pre-defined generic distribution function shown at Figure 2b.
   7. After that, insert the obtained values for the mean size, D₀, and the skewness, σ, in the pre-defined generic distribution function shown in Figure 2b.
   8. Set the lower boundary of the of the integral as 1 nm. Set the upper limit of the integration to any size that does not exhibit any shift in the Raman spectrum (20 nm for Si-NCs)¹².
   9. Integrate the distribution function in Figure 2b as a function of nanocrystal size using the integral function definition a data analysis and plotting program by setting the lower and higher sizes as integral boundaries (1-20 nm for Si-NCs). Plot Φ(D) vs. D to give the size distribution. Alternatively, find a set of Φ(D) values for each value of D (for instance, from 1 to 20 nm for Si-NCs with an increment of 1 nm) and plot Φ(D) vs. D, which is the size distribution.
   10. If a multi-modal size distribution exists, first define the peaks to be fitted for other size distributions. Then, estimate their volume fractions of different size distributions with respect to each other by first finding the areas of each peaks obtained after deconvolution of the measurement data (with the size distribution determination procedure) and then calculating the areal ratio of each peak with respect to the total Raman peak.

Results

For using Raman spectroscopy as a size analysis tool, a model to extract the size-related information from a measured Raman spectrum is needed. Figure 2 summarizes the analytical multi-particle phonon confinement model.¹² All-size-dependent phonon confinement function (Figure 2c) is projected onto a generic size distribution function (Figure 2b), which is chosen as a lognormal...

Discussion

First discussion point is the critical steps within the protocol. In order not to have overlapping peaks with the material of interest, it is important to use another type of substrate material as mentioned in step 1.2. For instance, if Si-NCs are of interest, do not use silicon substrate for the Raman measurements. In Figure 1a, for instance, Si-NCs were synthesized on plexiglass substrates, which has completely flat signal roughly around the range of interest, i.e....

Materials

List of materials used in this article

Name	Company	Catalog Number	Comments
Raman Spectroscopy	Renishaw	In Via	Equipped with 514 nm Ar ion laser
Wire 3.0	Renishaw		Raman spectroscopy record tool
Mathematica	Wolfram		For fitting function and size determination
Substrate			Plexiglass (to avoid signal coincidence with Si-NCs)
Si wafer			Reference to Si-NC peak position
Photoluminescence Spectroscopy			334 nm Ar laser. For optical size distribution.
Transmission Electron Microscopy			Beam intensity 300 kV. For nanocrystal size and morphology determination.