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Essay: Assignment: The Brus equation

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1.Review of J. Phys. Chem., 1986, 90, 2555-2560

This paper mainly focuses on the experimental and theoretical background of properties, which differ in nanoparticles from the bulk material. This change depends on the size in semi conductor crystallites of ~15Å to several hundred angstroms.

The “crystallites” are named as clusters. Generally, the word ‘nanocluster’ is coined for particles, which have at least one dimension in 1-10 nm in size and a narrow size distribution.

This paper discusses three main aspects of the “cluster problem” and provides a theoretical and an experimental approach to each.

  • Solid-state properties as a function of size
  • Lattice and electronic properties as a function of size
  • Size-dependent interaction of the cluster with the electromagnetic field

Solid-state properties as a function of size

Properties of bulk materials mainly depend on the electron delocalization.

In a graphite sheet, large number of bonding and anti-bonding orbitals merge to form conduction and the valence band.

A single graphite sheet is known as a “semi metal” since in the limit that N (ring fragment of graphite) reaches infinity, the HOMO to LUMO transition energy goes to zero.

The high atomization energy of a diamond crystal (7.4 eV/ atom), extreme hardness, large band gap (5.4 eV) of diamond is due to its unique crystal structure.

Carbon atoms in diamond are sp3 hybridized leading to a tetrahedral crystal structure. Thus, extensive electron delocalization occurs in three dimensions leading to unique properties in diamond.

Silicon is a semi conductor with a band gap of about 1 eV. Si also has a tetrahedral crystal structure similar to diamond.

Lattice and the electronic properties as a function of size

Depending on the number of atoms (n) in the relatively open diamond and zinc-blend lattices, clusters can be categorized into two cluster size regimes.
n≤30- 〖10〗^3

In this regime, bulk lattice structure is absent and molecular like properties are observed
30-〖10〗^3≤n < 〖10〗^5

In this regime, bulk lattice structure is observed but bulk electronic properties are absent. This paper focuses on this regime.

Semi conductor particles which are ~ 100Å or larger in size behave as bulk material.However, small particles of ~15-50 Å which were synthesized from homogenous precipitation reactions had same unit cell and bond length as the bulk material but differed in their electronic spectra. Since clusters have several hundreds of atoms, diffraction structural methods like X-ray or electrons can be used for characterization of interfaces and surfaces of clusters.

Different synthetic routes are used for the synthesis of these semi-conductor cluster materials. These include colloidal preparations, arrested precipitation and solid-state precipitation.

In order to circumvent the aggregation and physical manipulation of small crystallites in liquid medium, synthesis can be carried out at low temperatures and in nonaqueous solvents followed by adsorption of foreign stabilizing molecules.

During this procedure thermal coagulation and Ostwald ripening is controlled by the double-layer repulsion of individual crystallites.

Solvents such as 2-propanol and ethanol, methanol mixtures stabilize small crystallites. Since these solvents can form clear organic glasses when cooled, crystallites can be easily characterized using Raman and luminescence measurements.

Size-dependent interaction of the cluster with the electromagnetic field

The relationship between size and optical spectra was first resolved for ZnS and CdS.

Table 1. Change of band gap with the size of the crystal

Name Size / Å Band gap / eV Absorption threshold / nm
Bulk material 100Åand larger crystals 2.8 430
Medium 40-50 Blue shift –
Small 20-30 Blue shift more than 1 eV –

Clear peaks in spectra represent the discrete excited electronic states. This phenomenon was observed in a variety of semiconductors.

Size dependence of optical absorption and emission is due to the quantum confinement effect. In order to explain this behavior, a model was put forward.

The model is based on several assumptions,

The nanocrystal is spherical and the radius is R

There is a uniform medium inside the nanocrystal without any point charges or occupied spaces other than excited electron and hole.

Potential energy is infinite outside the nanocrystal

When an electron is promoted to the conduction band from the valence band by absorption of a photon, it leaves behind a hole in the valence band. Thus, the formed exciton can be treated as a particle in a box by above assumptions.

The Hamiltonian for this electron-hole pair consists of the kinetic and potential energy terms. Potential energy in turn has two components.

Coulomb attraction between the negatively charged electron and the positively charged hole

Polarization energy which arises due to the polarization of the nanocrystal by the point charge inside

By solving the Schrodinger equation for this system, the energy of the first excited electronic state Eex can be calculated as follows.

As the size of the nanoparticle increases, the physical properties approach slowly to that of the bulk. This slow approach in the region of fundamental gap is due to strong chemical bonding.

Above results are obtained when there is a uniform medium in the interior of the nanocrystal. The presence of defects in the crystal lattice may result in deep and shallow traps in the nanocrystal. These trapped carriers of the nanocrystal may result in absorption of energy below the exciton absorption peak in the absorption spectra.

The major difference between the metal and semi conductor cluster is the luminescence. Transitional metal clusters, since they have an extremely high density of electronic states above HOMO, do not show any electronic transitions emitting radiations. On the other hand, semi conductor clusters show luminescence.

2. Calculation of the band gap energy with size of the CdS nanocrystal.

Table 2. Calculated band gap energy with the size of CdS quantum dot
Radius of the particle/ nm Calculated band gap energy/ eV
3 2.605
4 2.518
5 2.483
6 2.466
7 2.458
8 2.454
9 2.452
10 2.451

Following figure 3 shows the variation of the band gap energy with the size of the CdS quantum dot. It can be observed that the change in band gap energy is drastic in the range of 3 – 6 nm range where as the change is much slow when it is greater than 7 nm. Bulk band gap energy of CdS is 2.42 eV. Thus we can observe clearly that the band gap energy of the quantum dot approaches the bulk band gap as the size increases.

For a semi-conductor nanocrystal of diameter 60 Å, using Brus equations,

Bulk band gap of CdS = 2.42 eV
Kinetic energy of the exciton =4.57×〖10〗^(-20) J
Coulombic energy =2.41×〖10〗^(-20) J
Polarization term =8.01×〖10〗^(-21) J
Energy of excitation =4.17×〖10〗^(-19) J
Energy of excitation in eV =2.61 eV
Shift in energy =2.61 eV- 2.42 eV = 0.19 eV

3. Synthetic procedures for preparing colloidal CdSe quantum dots and colloidal gold nanoparticles.

Nanoparticles act as the bridge between the bulk material and the atomic or molecular structures. Although the physical properties of the bulk material do not change with the size, physical properties of nanoparticles change with size. Thus, it is of paramount importance to prepare nearly monodisperse samples of nanoparticles to distinguish truly novel properties inherent with size. Monodisperse means that the particles are identical or indistinguishable but in nanoscience, monodisperse requires the standard deviation of diameter to be ≤5%.

For example, the nanometer size crystals known as nanocrystals should be monodisperse in size, shape, internal structure and in surface chemistry to reveal the size dependent properties like optical, electrical and magnetic properties.(2)

A variety of methods are employed for the fabrication of semiconductor nanocrystals(1). These include vapor deposition, ion implantation, micelle methods, sol-gel methods, and organometallic synthesis. Synthesis of cadmium selenide quantum dot is carried out by the pyrolysis of organometallic reagents. (1) This method is also called as the hot injection method.

Synthesis of monodisperse colloids occurs through nucleation followed by slow, controlled growth of the existing nuclei. First, metal-organic precursor is rapidly injected into a vigorously stirred flask containing a hot (~150-350 oC) coordinating solvent as shown in figure 5. This triggers the nanocrystal nucleation process. Mixtures of solvents like alkylphosphines, alkylphosphine oxides are used in these reactions along with metal alkyls and organophosphine chalcogenides as precursors. Then, the temperature of the reaction mixture is controlled in order to control the growth process. Growth occurs due to Ostwald ripening over 1-6 days to reach the required NC size. Since the growth occurs with time, NCs of different size can be isolated by periodic removal of aliquots from the reaction mixture.

Cd(CH3)2 + Se-TBP (tributyl phosphine) CdSe

Ref : (1)
Molecules of the coordinating solvent bind to the surface atoms of the NC giving rise to an organic shell. This stabilizes the surface of NCs, provide solublility in organic solvents and polymers.

In general, gold nanoparticles are synthesized by liquid chemical methods using chloroauric acid (H[AuCl4]) as the precursor. A solution of H[AuCl4] is stirred rapidly while a reducing agent is added to the solution. Thus, Au3+ ions are reduced to form neutral Au atoms. With time, the solution becomes supersaturated with Au atoms and gold starts to precipitate in the form of sub-nanometer particles that is nucleation occurs. Then, these particles will grow with time. A monodisperse NPs can be obtained through vigorous stirring of the reaction mixture. (5) Several synthetic routes are available for the synthesis of colloidal metal nanoparticles but only a few methods have the potential to yield monodisperse metal NP samples. Colloidal gold nanoparticles are synthesized mainly by aqueous reduction of metal salts with citrate ions (Turkevich method (2)) and two phase reduction method (Brust method (3)).
The simplest method to synthesize AuNPs is the Turkevich method. Modestly monodisperse spherical nanoparticles of around 10-20nm, suspended in water can be produced. In this process small amounts of hot chloroauric acid will be reacted with small amounts of sodium citrate solution. Citrate ions act as the reducing agent and also as a capping agent, which prevent aggregation of particles. Moreover it stabilizes the NPs. (4)

The other most common method to synthesize AuNPs is the two-phase reduction method described by Brust, Schiffrin and co-workers. The obtained AuNPs will be monodisperse in an organic solvent and 2-6 nm in diameter. In this method, aqueous chloroauric acid will be mixed in a toluene solution containing long-chain alkylammonium surfactants like tetraoctylammonium bromide (TOAB) to form a two-phase system. By vigorous stirring of the solution, metal ions can be transferred to the organic phase. Then a capping agent, typically a long-chain thiol, is added to the solution while stirring followed by a reducing agent like sodium borohydride. This triggers the nucleation of NPs.

4. Size dependent optical properties of gold nanoparticles and quantum dots.

Gold nanoparticles are nanometer size particles of gold metal where as quantum dots are nanometer size crystals of semi-conductor material in the range of 2-10 nm. Both quantum dots and gold nanoparticles exhibit different colors with different sizes. However, the phenomena behind these size dependent optical properties are different between gold nanoparticle and quantum dot.

Size dependent optical property of gold nanoparticles is due to a phenomenon known as surface plasmon resonance, which arises when the size of the nanoparticle is smaller than the wavelength of incident radiation. For noble metals this lies in the visible region of the electromagnetic spectrum. When the electric field of a light ray interacts with free electrons (electron in the conduction band) of a colloidal gold nanoparticle, the electrons start to oscillate in a concerted manner, in resonance to the frequency of the visible light as shown in figure 7. These resonating oscillations are known as the surface plasmons. In monodisperse gold nanoparticles, this surface plasmon resonance leads to absorption of light. As the particle size increases, the surface plasmon resonance wavelengths move to longer wavelengths. Thus, small particles (~30nm) with shorter plasmon resonance wavelength absorb blue-green light while red light is reflected. Surface plasmon resonance of larger particles move into the IR region leading to clear or translucent color. The wavelength strongly depends on the size, shape, surface and the agglomeration state of the gold nanoparticle.

In quantum dots, the optical properties arise due to fluorescence. Since quantum dots are in the range of 2-10nm (10-50 atoms), their electrons are confined to a small space. According to Pauli’s exclusion principle, when the radii of the semiconductor nanocrystal is smaller than the exciton Bohr radius, quantization of energy levels occur. The energy difference between the highest energy band and the lowest conduction band increases with the decreasing size of the quantum dot as shown in figure 5. As a result, the amount of energy needed to excite an electron from the valence band to the conduction band increases, which in turn results in release of more energy when the electron returns to the ground state. Thus, with the decrease of the size of the quantum dot, color shifts from red to blue in the emitted light.

5. Comparison of experimental and calculated first excited electronic state energy of CdSe nanocrystals

The energy associated with the first excited electronic state can be calculated by using the Brus equations. The calculated values and experimental values are as shown in table 3.

Table 3. Calculated and experimental values of, first excited electronic state energy for CdSe quantum dots.
Size/ nm Calculated value/ nm Experimental value(1)/ nm
1.7 188.6 410.0
2.9 375.0 500.0
3.3 424.3 540.0
5.5 589.1 600.0

Calculated and experimental values approach each other as the size increases, but they diverge significantly as size decreases. This can be due to several reasons.

Energy of the first excited electronic state is calculated using the Brus equation, which is derived based on several assumptions. (Polarization term is ignored during the calculation) As the size of the quantum dot decreases, the probability to satisfy these assumptions may decrease. In order to obtain calculated values in good agreement, prolate elliptical shape of the nanocrystal, quantum mechanical exchange interaction between the electron and hole, the effect of wurtzite crystal structure on the energy of the valence band should be taken into account.(1)

6. Applications of quantum dots.

Quantum dots were discovered in early 1980s but commercialization of them peaked in last few years due to the commercialization of first quantum dot LCD tv introduced by Sony (9). Since then quantum dots are emerging as promising material for photoluminescence, electroluminescence, optical, infrared, and biological labeling. After a laborious time period, commercialization of quantum dots turns into a success story.

Thermo Fisher Scientific company harness optical properties of quantum dots to synthesize “Qdot Probes”, which act as fluorophores.
Qdot naocrystal is a core/shell structure, which is about 10-20 nm in size. The core is made with either CdSe or CdTe where as the shell is made with a semi-conductor material like ZnS. The shell enhances optical properties of the material. In addition to the core/shell structure, Qdot may have a polymer coating and an outer biomolecule layer like a protein or on oligonucleotide depending on its function.

Fluorescence of Qdot is due to the formation of excitons at the core of the Qdot nanocrystal, which is analogous to an excited state of a fluorophore. The energy of the exciton depends on the physical size of the quantum dot owing to the quantum confinement effect. This property is known as “tuneability” and can be exploited for the development of multicolor assays.

Owing to the high efficiency in generating fluorescence, Q dot nanocrystals have many advantages over conventional fluorophores. These include brightness and photostability, which are many orders of magnitude higher than traditional fluorophores. Thus, Q dot nanocrystals allow long-term imaging and more time for image optimization. Moreover, the emission signal from Q dot is a narrow and symmetric. Thus the probability for the overlap of colors is minimal and this allows the use of different colors simultaneously.

Q dot nanocrystals have the potential for different applications in biological labeling. One potential application of this Qdot nanocrystal is in flow cytometry. Flow cytometry is a key technique used in clinical chemistry for the analysis of cells. It allows quantitative and qualitative analysis of cells as they flow in a single file in a flow through a laser based on their chemical and physical characteristics. Light scattered by cells is used to distinguish differences by size and internal complexity where as the light emitted by the fluorescently labeled antibodies can identify a variety of cell surface and cytoplasmic antigens. Thus, flow cytometry is versatile in analyzing a complex population of cells within a short time period.

Q dot bioconjugate, that is Q dot with a conjugated biomolecule like protein A, streptavidin, and biotin, can bind to specific proteins on cell membranes or inside cells. Thus it can be used to visualize and analyze cells. Since the nanocrystal fluorophore has a larger surface area, many biomolecules can be conjugated to one nanocrystal. Advantages of this approach include enhanced affinity for targets, cooperative binding and potential to use efficient signal amplification methods, multicolor, multiplexed assays.

Q dot nanocrystals are also used for the staining of tissues and cells. Photobleaching is a challenge when using conventional dye conjugates to detect low abundance antigens. This can be circumvent by the use of Q dot nanocrystal conjugates owing to their narrow, symmetric emission spectra and the ability to use a single excitation source. Moreover, these specialized quantum dots have the potential to be used for cell movement and location studies.
When considering all above factors, quantum dots find themselves as a versatile tool for the biological labeling. Tunable optical properties, narrow and symmetric emission spectra, high photostability are some of the desired physical properties in quantum dots that have led to the commercialization of quantum dots as biological labeling agents.


  • Brus, L., 1986. Electronic wave functions in semiconductor clusters: experiment and theory. The Journal of Physical Chemistry, 90(12), pp.2555-2560.
  • Brust, M., Walker, M., Bethell, D., Schiffrin, D.J. and Whyman, R., 1994. Synthesis of thiol-derivatised gold nanoparticles in a two-phase liquid–liquid system. Journal of the Chemical Society, Chemical Communications, (7), pp.801-802.
  • Turkevich, J., Stevenson, P.C. and Hillier, J., 1951. A study of the nucleation and growth processes in the synthesis of colloidal gold. Discussions of the Faraday Society, 11, pp.55-75.
  • Kimling, J., Maier, M., Okenve, B., Kotaidis, V., Ballot, H. and Plech, A., 2006. Turkevich method for gold nanoparticle synthesis revisited. The Journal of Physical Chemistry B, 110(32), pp.15700-15707.
  • Cytodiagnostics.com. (2017). Gold Nanoparticle Properties – Cytodiagnostics. [online] Available at: http://www.cytodiagnostics.com/store/pc/Gold-Nanoparticle-Properties-d2.htm [Accessed 6 Mar. 2017].
  • Sigma-Aldrich. (2017). Quantum Dots. [online] Available at: http://www.sigmaaldrich.com/technical-documents/articles/materials-science/nanomaterials/quantum-dots.html [Accessed 6 Mar. 2017].
  • Idtechex.com. (2017). Quantum Dots 2016-2026: Applications, Markets, Manufacturers: IDTechEx. [online] Available at: http://www.idtechex.com/research/reports/quantum-dots-2016-2026-applications-markets-manufacturers-000452.asp [Accessed 6 Mar. 2017].
  • Brown, M. and Wittwer, C., 2000. Flow cytometry: principles and clinical applications in hematology. Clinical chemistry, 46(8), 1221-1229.


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