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Quantum Dots : a True “Particle in a Box” System

quantum dot (QD) is a crystal of semiconductor material whose diameter is on the order of several nanometers – a size which results in its free charge carriers experiencing “quantum confinement” in all three spatial dimensions. The electronic properties of quantum dots are intermediate between those of bulk semiconductors and of discrete molecules and closely related to their size and shape. This allows properties such as the band gap, emission color, and absorption spectrum to be highly tuneable, as the size distribution of quantum dots can be controlled during fabrication. For example, the band gap in a quantum dot, which determines the frequency range of emitted light, is inversely related to its size. In fluorescent dye applications, the frequency of emitted light increases as the size of the quantum dot decreases, shifting the color of emitted light from red to violet.

The small quantum dots, such as nanocrystalline semiconductors in a colloidal solution, have dimensions between 2 and 10 nanometers, corresponding to about 10-50 atoms in diameter, and may reach a total of 100-100000 atoms for each quantum dot .

The self-assembled quantum dots have a size of 10-50 nanometers; while those defined by means of electronic lithography have larger sizes around 100 nm.

Besides confinement in all three dimensions (i.e., a quantum dot), other quantum confined semiconductors include:

  • Quantum wires, which confine electrons or holes in two spatial dimensions and allow free propagation in the third.
  • Quantum wells, which confine electrons or holes in one dimension and allow free propagation in two dimensions.

The Quantum Dot which contain electrons can also be compared to atoms: both have discrete energy levels and contain a small number of electrons, but unlike the atoms, the confinement potential of Quantum Dot not necessarily show spherical symmetry. Moreover the electrons do not move in the limited space, but inside the semiconductor crystal that hosts them.

Researchers have studied applications for quantum dots in transistors, solar cells, LEDs, and diode lasers. They have also investigated quantum dots as agents for medical imaging and as possible qubits in quantum computing. The small size of quantum dots allows them to be suspended in various solvents and thus compatible with solution processing techniques such as spin coating and inkjet printing.

quantumDot
“Quantum Dot” scheme

Quantum Dots Fluorescence experiment

In our experiment we want to measure the fluorescence produced from a colloidal solution of quantum dots, excited by a UV laser at 405nm.
In particular we intend to show that the wavelength of the emission is related to the size of the Quantum Dot : the larger the size, the greater the wavelength of the fluorescent radiation.

qdsizes-spectrum

For this experiment we used colloidal solutions of Quantum Dots type CdTe (cadmium – tellurium) produced by PlasmaChem declared with the following characteristics :

CdTe hydrophilic quantum dots Declared Emission max (nm) CdTe radius (nm) Average molar weight (Da)
PL-QDN-520 520 2,04 16’000
PL-QDN-570 570 3,12 59’000
PL-QDN-600 600 3,39 76’000
PL-QDN-640 640 3,66 96’000
PL-QDN-680 680 4,22 146’000
DSC_0221_1
Colloidal Solutions of PlasmaChem Quantum Dots

Fluorescence Spectra

Below are shown the fluorescence spectra of colloidal solutions obtained with a self-built webcam spectrometer using the software Theremino Spectrometer. The emission peak around 405nm is the excitation source.

Fluorescence Spectrum of CdTe Hydrophilic Quantum Dot – Green Color – excited by 405nm UV laser
Fluorescence Spectrum of CdTe Hydrophilic Quantum Dot – Yellow Color – excited by 405nm UV laser
Spettro di Fluorescenza di CdTe Hydrophilic Quantum Dot – Colore Arancione - eccitato da laser UV 405nm
Fluorescence Spectrum of CdTe Hydrophilic Quantum Dot – Orange Color – excited by 405nm UV laser
Spettro di Fluorescenza di CdTe Hydrophilic Quantum Dot – Colore Rosso - eccitato da laser UV 405nm
Fluorescence Spectrum of CdTe Hydrophilic Quantum Dot – Red Color – excited by 405nm UV laser
Spettro di Fluorescenza di CdTe Hydrophilic Quantum Dot – Colore Rosso Scuro- eccitato da laser UV 405nm
Fluorescence Spectrum of CdTe Hydrophilic Quantum Dot – Dark Red Color – excited by 405nm UV laser

Theoretical Considerations

In semiconductors, light absorption generally leads to an electron being excited from the valence to the conduction band, leaving behind a hole. The electron and the hole can bind to each other to form an exciton. When this exciton recombines (i.e. the electron resumes its ground state), the exciton’s energy can be emitted as light. This is called Fluorescence. In a simplified model, the energy of the emitted photon can be understood as the sum of the band gap energy between the highest occupied level and the lowest unoccupied energy level, the confinement energies of the hole and the excited electron, and the bound energy of the exciton (the electron-hole pair):

teorico1

As the confinement energy depends on the quantum dots size, both absorption onset and fluorescence emission can be tuned by changing the size of the quantum dot during its synthesis. The larger the dot, the redder (lower energy) its absorption onset and fluorescence spectrum. Conversely , smaller dots absorb and emit bluer (higher energy) light. Recent articles in Nanotechnology and in other journals have begun to suggest that the shape of the quantum dot may be a factor in the coloration as well, but as yet not enough information is available. Furthermore, it was shown that the lifetime of fluorescence is determined by the size of the quantum dot. Larger dots have more closely spaced energy levels in which the electron-hole pair can be trapped. Therefore, electron-hole pairs in larger dots live longer causing larger dots to show a longer lifetime.

To improve fluorescence quantum yield, quantum dots can be made with “shells” of a larger bandgap semiconductor material around them. The improvement is suggested to be due to the reduced access of electron and hole to non-radiative surface recombination pathways in some cases, but also due to reduced auger recombination in others.

Quantum Dots as a “Particle in a Box”

The problem of quantum mechanics, which corresponds to the ” Particle in a Box ” is a rather difficult thing to display. This is because until now there was not a real good example of such a system. With Quantum Dots the situation has changed. Inside the semiconductor used in microelectronics there are small particles of semiconductors. These may contain an electron and a “hole” (absence of an electron). It is a real world ” Particle in a Box” system : Indeed, the electrons will never go out to the outside of the particle of the semiconductor. In Quantum Dots the effects of changing the size on the energy levels of the system can be easily viewed.

teorico2

The simplest form of the particle in a box model considers a one-dimensional system. Here, the particle may only move backwards and forwards along a straight line with impenetrable barriers at either end. The walls of a one-dimensional box may be visualized as regions of space with an infinitely large potential energy. Conversely, the interior of the box has a constant, zero potential energy. This means that no forces act upon the particle inside the box and it can move freely in that region. However, infinitely large forces repel the particle if it touches the walls of the box, preventing it from escaping. The potential energy in this model is given as

teorico3

Where L is the length and x  is the position within the box.

In quantum mechanics, the wavefunction gives the most fundamental description of the behavior of a particle; the measurable properties of the particle (such as its position, momentum and energy) may all be derived from the wavefunction.
The wavefunction can be found by solving the Schrödinger equation for the system. Inside the box, no forces act upon the particle, which means that the part of the wavefunction inside the box oscillates through space and time with the same form as a free particle:

teorico4

The energies which correspond with each of the permitted wavenumbers may be written as :

teorico5

This problem is not the same as the Quantum Dot since in this last case the box is three-dimensional and spherical in shape. However, the equation of the problem Quantum Dot has a similar expression and is known as the equation of Brus and can be used to describe the emission of energy of Quantum Dot in terms of energy gap of the band Egap, Planck’s constant h, radius of the Quantum Dot r, as well as the effective mass of the electron me* and mh* of the hole.

The radius of the Quantum Dot influence the wavelength of the emitted light due to quantum confinement, this equation describes the effect of a change of the radius of the quantum dot on the wavelength λ of the emitted light (and hence on the emission energy E = hc / λ, where c is the speed of light). This is useful for calculating the radius of a quantum dot with the parameters experimentally determined.

The general equation is :

teorico6

In the above equation, the two m are respectively the effective mass of the electron and the effective mass of the hole, and R is the radius of the quantum dot. Egap is the energy of the bandgap of the semiconductor.

For CdTe semiconductor the values of me*, mh*, ed Eg  are the following :

me* = 0.096 me  = 8.75×10-32kg

mh* =0,84 me  =7.65×10-31kg

Eg = 2.40×10-19J = 1,5 eV

Measurement Data

The data obtained from the measurements of fluorescence can be summarized in the following table and in the graphs below :

Color Measured Emission (nm) Calculated Radius (nm)
Green 534 1,99
Yellow 586 2,23
Orange 629 2,49
Red 678 2,82
Dark Red 703 3,01

excelImage1

excelImage2

The experimental data are in substantial agreement with the equation of Brus and with the data declared by the manufacturer of Quantum Dot.

Pdf document with complete description of the experimentQuantumDots_ENG.pdf

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