Semiconductor nanoparticles

Semiconductor NPs have attracted great attention due to the remarkable optical properties related to their absorption and emission features. A semiconductor material exposed to excitation shows a transition of electrons in the valence band into the unoccupied conduction band yielding an electron hole in the valence band. The electron-hole pair, which is attracted to each other by electrostatic Coulomb force, is termed as an exciton. The energy to overcome the Coulomb force of the exciton is equal to the bandgap of the semiconductor material.

Semiconductor NPs contain a finite number of atoms bonded together leading to the splitting of the energy levels into discrete states of different energies. The quantum confinement of the excitons within the NP and the dimensionality of nanoscale size lead to the terms of quantum dot (QD), quantum wire (QW) and quantum wells (QWE)⁷⁸. For a simple description, the electron–hole pair can be considered as an analog to a hydrogen atom, where the hole is a proton. Therefore, the classic Bohr model can be applied to estimate the average distance between electron and hole in a semiconductor NP also known as the exciton Bohr radius 𝑎_𝐵^𝑒𝑥 (Equ. 13)⁷⁹.

Chapter 2 – Fundamentals semiconductor material, 𝑚_𝑒, 𝑚_ℎ – effective mass of electron and hole.

Thus, an exciton Bohr radius of 3 nm and 2.5 nm can be determined for cadmium sulfide (CdS)⁸⁰ and zinc sulfide (ZnS)⁸¹ QDs. When the QD size is similar to the exciton Bohr radius of the semiconductor material, the properties of the exciton will strongly depend on the space available for the electron-hole pair. Therefore, 𝑎_𝐵^𝑒𝑥 becomes a threshold value and the QDs can show size-dependent optical properties below this value. The formation of the band structure with discrete energy states within this size range is also known as the ‘quantum size effect’.

To explain the exciton-related properties, one can use the linear combination of atomic orbitals (AO) approach, as shown in Fig. 10. In a diatomic molecule the atomic orbitals interact with each other forming highest occupied and lowest unoccupied molecular orbitals (HOMO and LUMO). With increasing number of atoms more atomic orbitals will be included in the linear combination leading to more molecular orbitals of discrete energy levels. If an infinite number of atoms is used to form an extended bulk material, the distances between the energy levels will ‘infinitely’ decrease, which yields continuous bands from the combination of the atomic orbitals. In the case of QDs the distance between the energy levels and the bandgap becomes larger compared to the solid material due to the small number of atoms. Therefore, QDs can be placed in between the molecule and bulk material. Anyhow, the band gap of QDs closely resembles the bandgap of the bulk semiconductor for sufficiently large QD size and the optical features such as exciton absorption, photoluminescence disappear indicating the transition to the continuous band structure.

Fig. 10: The evolution of the electronic structure from atoms and molecules to semiconductor QDs and bulk material indicates the change of energy levels and the decreasing bandgaps with increasing sizes.

HOMO LUMO

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The effective mass approximation (EMA) was firstly introduced by Efros⁸² and later extended by Brus⁸³ with including of the particle in the box model. This approach quantitatively describes the size-dependency of the QD band gaps. A particle of a mass m is placed in a box of length L and is surrounded by infinite potential energy of the box walls preventing the particle from escaping.

The particle in the box itself does not possess any potential energy. The solution of the

𝐸_{𝑔,𝐵𝑢𝑙𝑘} – bandgap of the semiconductor bulk material

The first term of Equ. 15 represents the energy from the quantum confinement effect, which shifts the Eg to higher value as R^-2, while the second term is the Coulomb interaction of the excitons shifting Eg to lower energy as R^-1. Thus, the bandgap of the semiconductor NP will increase with decreasing QD size. Fig. 11 A shows the size-dependency of bandgaps for ZnO-, CdS-, GaAs- and InSb-QDs of increasing sizes. In this thesis cadmium sulfide (CdS) and zinc sulfide (ZnS) QDs were synthesized and used for the experiments with the Hcp1 protein structures. CdS and ZnS belong to the II-VI class of semiconductor compounds and exist in the wurtzite and zinc blende crystal structure (Fig. 11 B-C). The increasing bandgaps from ZnS to CdS yield absorption and emission properties in the UV and visible range (Table 1).

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Table 1: Properties of ZnS and CdS bulk semiconductor material at room temperature⁸⁴. Material

Crystal structure Eg [eV] Lattice constant [Å]

ZnS Zinc blende 3.61 5.41

CdS Wurtzite 2.58 4.14/6.71 (a/c)

Fig. 11: A) Bandgaps of semiconductor nanoparticles with increasing sizes. The short black line represents the bandgap of the bulk semiconductor⁸⁰. B-C) Zinc blende and wurtzite structure of ZnS (zinc atoms in green and sulfur atoms in orange)⁸⁵.

The outstanding optical properties of QDs can be observed in UV-Vis and photoluminescence spectroscopies. The UV–Vis spectroscopy reveals the absorption characteristics, which are related to the electronic structure of QDs. For example, CdSe-QDs of radius smaller than the exciton Bohr radius (aBohr = 5.6 nm) feature a clearly resolved band structure in the UV-Vis spectrum in Fig. 12 A. The different peaks can be assigned to the electron interband transitions, where the 1S3/2–1Se transition corresponds to the band-edge exciton. This exciton of the lowest energy compared to other electron transitions is very useful to calculate the size of QDs⁸⁶. As the particle radius further grows, the band edge transition closely shifts to the value of bulk material indicating a decreasing bandgap (Fig. 12 B). The photoluminescence spectroscopy shows the emission characteristics. For QDs, a single size-dependent sharp emission peak is observed, which corresponds to the band-edge exciton recombination. The wavelength difference between the emission peak and the first exciton absorption peak is defined as the Stokes shift, which is found to decrease with QD size. The emission peak also shifts to higher wavelength, as the QD particle size increases (Fig. 12 A-B).

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Fig. 12: A) Normalized absorption (solid line) and full luminescence spectra (dashed line) for small CdSe-QDs between 12 and 56 Å in radius⁸⁷. B) Absorption and photoluminescence spectra of CdSe-QDs of a particle radius of 8 nm. Photoluminescence spectra are obtained under intermediate (dashed curve) and strong (short- and long-dashed curves excitation at wavelength of 355 nm⁸⁸.