From Bulk Semiconductors to Quantum Dots

The semiconductor properties observed in many inorganic substances are not the proper-ties of the individual molecules or atoms themselves, but of an ordered crystal lattice of the constitutive elements in a specific array [Weller, 1993]. Therefore, the photophyiscs of semicondcutor quantum dots can be described by the band theory [Smyder and Krauss, 2011]. In general, an isolated atom consists of discrete energy levels where electrons orbit. When many atoms are in close proximity such as in crystalline solids, the atomic orbitals overlap leading to splitting of the original discrete levels to many separated lev-els [Johansen, 2005] [Weller, 1993]. They however may be considered as a continuous band of energy states because the levels are closely separated. The two highest energy levels are known as the electron rich valence band and the empty conduction band. These two levels are parted by a region known as the forbidden gap, or the bandgap, E_g and defined as energies that cannot be held by electrons in a solid [Johansen, 2005]. Bandgap can therefore be defined as the difference in energy that separates the highest occupied valence band energy and the lowest unoccupied conduction band energy.

If a semiconductor absorbs a photon with an energy that is eqaul to or above the bandgap energy, electrons (e^-) receive energy to move from valence band to the empty conduction band, which leave holes (h⁺) in the valence band [Smyder and Krauss, 2011]. A particle known as an exciton is formed by the combination of the two charge carriers, electrons and holes, while coulombic forces hold them together similar to that in a hydrogen atom [Smyder and Krauss, 2011] [Saris, 2017]. The formed exciton recombines either through fluorescence and re-emits a photon with an energy equal to E_g, or through non-radiative processes such as creating heat [Smyder and Krauss, 2011]. The distance that separates the electron and the hole is known as the exciton Bohr radius,a_B, which varies depending on the material and is given as,

a_B = h² e² ( 1

m_e∗+ 1

m_h∗) (1)

whereis known as the dielectric constant of the semiconductor,eis the electron charge, meis the effective electron mass andmhis the hole mass [Saris, 2017] [Bakkers, 2000].

Figure 2. (a) Energy scheme of a semiconductor [Weller, 1993] (b) The expression for the size dependent quantum confinement energy obtained with the spherical “quantum box” model.

[I. Klimov, 2003]

Bandgap and the associated energy is determined by the interconnection between the crys-tal size and the exciton Bohr radius, a defining feature of a semiconductor, which dictates the emission energy and the colour of light emitted in the visible region of the spec-tra [Mishra et al., 2012] [I. Klimov, 2003] [Chou and Dennis, 2015]. In semiconductors of macroscopic size, the bandgap is a established parameter depending on the identity of the material [I. Klimov, 2003] [Hollingsworth, 2006]. However, for semiconductor nanocrystals with sizes below about 25 nm, the particle size is identical or relatively smaller in comparison to the size of excitons observed in semiconductors of macroscopic size [Weller, 1993] [Hollingsworth, 2006] [I. Klimov, 2003]. This leads to an excited electron and a hole to be physically confined into a geometry smaller than their natural Bohr radius, in a regime of strong confinement, where a real materials system experiences the qauntum mechanical ’particle-in-a-box or quantum box’ potential energy function model [Weller, 1993] [Smyder and Krauss, 2011] [Norris and Bawendi, 1996]. Three dimensional quantum confinement is experienced by the exciton wave functions due to these geometrical constraints arising from the particle boundaries. As a result of this con-finement, continuous energy bands of a bulk material are quantized such that quantum

dots have discrete, atomic-like energy levels and electronic transitions that shift to higher energies with decreasing nanocrystal size [Norris and Bawendi, 1996] [I. Klimov, 2003].

The quantum box model envisages that for a spherical QD having a radius R, bandgap energy is proportional to 1/R² as shown by the expression in figure 2b), which implies that with decreasing QD size the gap increases [I. Klimov, 2003]. This phenomenon is recognized as the quantum-size effect as illustrated in figure 3, and it has a significant role in QDs [Kim et al., 2013] [Hollingsworth, 2006].

Figure 3. Quantum confinement of CdSe colloidal QDs(a)CdSe QDs with their core diameters ranging from 1.8 nm to 6.9 nm. (b) Schematic illustration of band structures and bandgaps in bulk semiconductors as well as QDs of different sizes. (c)Absorption and emission spectra with particle sizes from 1.8 nm to 20 nm. [Chou and Dennis, 2015]

As illustrated in figure 3(b), a bulk semiconductor of macroscopic size has a fixed energy gap,E_gthat separates its continuous conduction and valence energy bands. All states of a valence band up to the edge are typically completely occupied by the electrons, whereas those of a conduction band are empty. Figure 3(a) shows size dependent emission colour tunability of QDs [Chou and Dennis, 2015] [I. Klimov, 2003]. The decreasing QD size leads to a larger bandgap indicating a higher energy requirement to excite the electrons that produces higher frequency, shorter wavelengths corresponding to blue light in the spectra, whereas a small bandgap with lower energy corresponds to a red shift in the emission spectra with longer wavelengths as illustrated in figure 3(c) [Hollingsworth, 2006].

The unique quantum confinement of semiconductor nanocrystals thus facilitates new approaches for applications, particularly in electronics and optoelectronics due to their size-tunable emission, which makes it feasible to evaluate the electronic behaviour in a size regime transitional to molecular and bulk limits of matter [Norris and Bawendi, 1996] [Wang et al., 2017b] [Song and Jeong, 2017] [Yu et al., 2018] [Hollingsworth, 2006]. Optoelectronics have been developed based on QDs capable of functioning at wavelengths that were previously unfeasible or have access to, for a given semiconductor category [Bimberg and Pohl, 2011]. As fluorophores, high quality QDs are eminently bright with narrow emissions, whilst quantum yields (QYs) or fluorescence efficiencies have values close to unity, which is an indication of emitted fluorescence photons for each photon absorbed [Smyder and Krauss, 2011]. Photostability of QDs is remarkably longer, up to hours under continuous excitation in comparison to individual organic molecules that are prone to photobleach within seconds or minutes [Smyder and Krauss, 2011].

QDs further display minor Stokes shifts, which corresponds to the red shift observed in the emission spectra with regards to the absorption spectra, resulting due to vibrational re-laxation and solvent reorganization [Smyder and Krauss, 2011] [Britannica, 2018] [Bagga et al., 2007]. When a bulk semiconductor absorbs photons with energy greater than their bandgap, charge carriers having surplus kinetic energy are produced that dissipate via phonon emission, which are associated with lattice vibrations of a solid [Semonin et al., 2012]. However, the quantum confinement of semiconductor nanostructures has the dis-tinctive potential to provide novel routes that control the energy flow in optoelectronics when excited at energies far above their bandgap, that enhance the efficiency of the pri-mary photoconversion step [Smyder and Krauss, 2011] [Semonin et al., 2012]. Most significantly, a fixed excitation wavelength enables the collection of several QD emission wavelengths [Smyder and Krauss, 2011].

These outstanding properties of CQDs over bulk semiconductors make them much more versatile for new applications in electronics and optoelectronics.