Size effect on the bandgap change of quantum dots: Thermomechanical deformation
Introduction
Semiconductor nanostructures have stimulated great interest for potential applications in a variety of areas, including nanoelectronics, nanophotonics and bioimaging and for a better understanding of the fundamental mechanisms of optoelectronic characteristics of nanoscale materials. Various semiconductor nanostructures, such as nanoporous silicon and perovskite quantum dots, have been fabricated and exhibited photoluminescence (PL) under electric field and ultraviolet (UV) light in contrast to the corresponding bulk phase. In the heart of the field- and UV-induced photoluminescence is the size confinement to bandgap [1], [2], [3], [4], which results in the tunability of the bandgap.
The energy level of a particle (herein, an electron) in a cubic box (a confined system) can be calculated as [5] where and m are the energy and mass of the particle, respectively, ħ is the Planck constant divided by 2π, and V is the volume of the cubic box. is the total quantum number for the energy level, , of the particle, and the smallest value of is 3 for the ground state of the particle. Note that Eq. (1) is applicable to a crystal of any shape with dimensions larger than the de Broglie wavelength of (p is the momentum of the particle) [5]. Thus, the energy level is inversely proportional to the square of radius for a spherical box and the square of side-length for a cubic box. Decreasing the size of the system leads to the increase of the energy level of the particle.
The effect of quantum confinement is dependent on the size of a system and becomes significant for a system with a characteristic dimension of the order of or less than the de Broglie wavelength. In general, the increase in the lowest energy of a particle at a confined state in a spherical system of R in radius without the contribution of the Coulomb interaction can be expressed as [6], [7] in which the parameter, χ, is dependent on the extent of the confinement as
Including the contribution of the Coulomb and correlation energies, Brus [8] and Kayanuma [9] extended the work by Efros and Efros [6] and obtained the change of the bandgap as where e is the charge of electron, and and are the relative dielectric constant and the Rydberg energy of the corresponding bulk phase. It is evident that the contribution of the Coulomb and correlation energies reduces the effect of quantum confinement on the increase of bandgap for very small nanocrystals.
In the analysis of the PL characteristics of silicon nanocrystals, Ledoux et al. [10] suggested that the shift of the PL energy of silicon nanocrystals under a Nd:YAG (yttrium aluminum garnet) laser with the size of silicon nanocrystals followed a power-law relation as with being average size of silicon nanoparticles. The power index, −1.39, of the average size is significantly different from the power index, −2, in Eq. (2) for the dependence of the bandgap change on particle size, even though Ledoux et al. [10] stated that “the PL of the silicon nanocrystallites follows the quantum confinement model very closely.” Such a difference likely suggests the presence of the Coulomb interaction in the PL response of silicon nanocrystals. Note that Yorikawa and Muramatsu [11] obtained a power index of −1.25 from the PL analysis of nanoporous silicon.
As pointed out above, Eq. (1) is applicable to crystals of any shape with dimensions larger than the de Broglie wavelength. We expect that Eqs. (2), (4) and (5) are likely applicable to a crystal of any shape with characteristic dimension less than the de Broglie wavelength and the parameters of R and can be replaced by the characteristic dimension of the crystal or with different coefficients in the corresponding terms. Note that the results in Eqs. (2), (4) and (5) do not involve the effects of mechanical stress and temperature.
Deformation can induce the shift of the energy extrema of semiconductor [12]. The contribution of deformation to the change of the bandgap of semiconductor nanocrystals generally consists of two aspects; one involves the modification of the bandgap, and the other is associated with the size confinement. According to Pikus and Bir [13], the orbital-strain Hamiltonian of a bulk semiconductor for a given band at , , can be expressed as Here, the subscript i represents the index of a band, υ is the deformation potential from hydrostatic stress, b and d are the deformation potentials from the strain components associated with tetragonal and rhombohedral symmetries, respectively, and are normal and shear components of strain tensor, ε, respectively, and and are the components of angular momentum operator, L, respectively. Using the results given by Mintairov et al. [14] and Eq. (6), the change of the bandgap under the action of hydrostatic stress for a cubic system is found to be where and are the components of the compliance matrix, is the summation of the diagonal terms of a tensor, and σ is stress tensor. Equation (7) suggests the proportionality between the bandgap change and hydrostatic stress. For semiconductors with , such as GaAs [12], the bandgap under hydrostatic pressure () is less than that at stress-free state. Note that a nonlinear relation between the change of the bandgap and hydrostatic stress is expected for large hydrostatic stress.
Beimborn et al. [15] in the analysis of the pressure-dependent PL response of CsPbBr3 nanocrystals of different sizes suggested that the change of the PL peak energy as a function of pressure, p, can be phenomenologically described as with a pressure-dependent coefficient of . They demonstrated the dependence of the parameter of on pressure and the size of CsPbBr3 nanocrystals. For all the CsPbBr3 nanocrystals, the change of the PL peak energy decreases first with the increase of the pressure and then increases with the increase of the pressure. The decrease of the change of the PL peak energy initially with the increase of the pressure suggests in Eq. (7) for small pressure.
There are reports on the effect of temperature on the bandgap of semiconductor. The temperature dependence of the bandgap of semiconductor can be expressed in a variety of formulations as [16], [17], [18] Here, T is absolute temperature, χ and β are two constants for the semiconductor, represents average interaction between phonon and electron, is the average temperature of phonon, S and are a nondimensional constant and an average phonon energy, respectively, and k is the Boltzmann constant. Considering the concurrent action of strain and temperature, Lee et al. [19] suggested an empirical relation for the change of the bandgap of GaN films on sapphire as with the last two terms representing the strain contributions. γ is a strain-related coefficient, and TECfilm and TECsub are the thermal expansion coefficients of the film and substrate, respectively. It needs to be pointed out that the strain-related terms in Eq. (10) are similar to the pressure-related term in Eq. (7) and consist of the contribution of volumetric change.
These studies had been focused on the effects of the size confinement, temperature and pressure/strain. For nanocrystals, the large ratio of surface area to volume suggests that the nanocrystals likely experience different stress state from the corresponding bulk phase due to surface/interface energy (stress). One needs to incorporate thermomechanical interaction and surface/interface energy (stress) in the analysis of the bandgap change of semiconductor nanocrystals under concurrent action of thermal and mechanical loading.
In the following, we analyze the thermomechanical deformation of an elastic, semiconductor sphere in the framework of linear thermoelasticity, as shown in Fig. 1. Note that the elastic deformation of an isotropic sphere has been well studied. Here, the elastic sphere exhibits transverse isotropy and experiences a uniform change of temperature, ΔT, under the action of pressure, p. An analytical formulation is developed to take into account the effects of the thermomechanical deformation and surface stress/energy on the bandgap change of the semiconductor sphere.
Section snippets
Effect of thermomechanical deformation on the bandgap change of a spherical nanocrystal
For an elastic sphere of transverse isotropy, the stress-strain relations in a Cartesian coordinate system are [20] where and () are the components of stress tensor and stain tensor, respectively, () are the components of stiffness matrix, and and are thermal moduli in corresponding directions. With spherical symmetry of the problem under a uniform
Discussion
There are reports on the temperature dependence of photoluminescence of semiconductor quantum dots [28], [29], [30], [31] and thin films [32]. All the results reveal that the change of bandgap is an approximately decreasing function of temperature, which support the relationship of (44).
Hartel et al. [33] studied the temperature effect on the bandgap of SiO2-capped silicon nanocrystals with sizes of 1.5 to 4.5 nm in a temperature range of 4 to 350 K for the laser power density of 0.4, 8.8, 88.4
Summary
In summary, we have investigated the size effect on the bandgap of spherical semiconductor nanocrystals under concurrent action of pressure and uniform temperature change in the framework of linear thermoelasticity and simplified surface elasticity. Closed-form solutions have been obtained for the volumetric strain, maximum shear strain and hydrostatic stress. Using the solutions and the Taylor series expansion to the first order of the temperature change, volumetric strain, maximum shear
CRediT authorship contribution statement
FY: Conceptualization, Methodology, Derivation, Writing-Editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
FY is grateful for the support by the National Science Foundation through the grant CBET-2018411 monitored by Dr. Nora F Savage.
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