Elsevier

Physics Letters A

Volume 401, 17 June 2021, 127346
Physics Letters A

Size effect on the bandgap change of quantum dots: Thermomechanical deformation

https://doi.org/10.1016/j.physleta.2021.127346Get rights and content

Highlights

  • Obtain an analytical relationship between bandgap change, strain and temperature change.

  • Increasing temperature leads to the decrease of bandgap.

  • The effect of surface energy exhibits similar size dependence to the effect of the Coulomb interaction.

Abstract

The size effect on the optoelectronic behavior of semiconductor crystals has attracted great interest for the applications of semiconductor nanocrystals in photonics, bioimaging, energy harvesting etc. In this work, we study the size effect on the bandgap change of spherical semiconductor nanocrystals under concurrent action of mechanical pressure and uniform temperature change and incorporate the theory of surface elasticity in the analysis. Using Taylor series expansion to the first order of approximation, we obtain an analytical relationship between the bandgap change of the spherical semiconductor nanocrystal, the volumetric strain, the maximum shear strain and the temperature change. The thermal expansion due to the temperature increase causes the decrease of bandgap. The effect of surface energy/elasticity on the bandgap change exhibits similar size dependence to the effect of the Coulomb interaction, and the effect of the volumetric strain on the bandgap change is equivalent to the pressure (hydrostatic stress) effect.

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]Ei=ni2π2ħ22mV2/3 where Ei 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. ni is the total quantum number for the energy level, Ei, of the particle, and the smallest value of ni2 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 2πħ/p (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]ΔEπ2h22χ1R2 in which the parameter, χ, is dependent on the extent of the confinement asχ={me+mhweak confinementmemh/(me+mh)strong confinementmevery small nanocrystal

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 asΔE=π2h22me+mhmemh1R21.786e2rR+0.284ER where e is the charge of electron, and r and ER 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 asΔE=3.73da1.39 with da 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 da can be replaced by the characteristic dimension of the crystal or V1/3 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 k=0, He(), can be expressed asHe(i)=υitr(ε)3bi[(Lx2L23)εxx+c.p.]23di[(LxLy)εxy+c.p.] 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, εxx and εxy are normal and shear components of strain tensor, ε, respectively, and Lx and Ly 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 beΔE=υ(S11+2S12)tr(σ) where S11 and S12 are the components of the compliance matrix, tr() 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 a<0, such as GaAs [12], the bandgap under hydrostatic pressure (tr(σ)<0) 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 asΔE(p)=α(p)p with a pressure-dependent coefficient of α(p). They demonstrated the dependence of the parameter of α(p) 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 a<0 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]ΔE={χT2/(T+β)Varshini [16]χB(2[exp(ΘB/T)1]1+1)Logothetidis et al. [17]S<ħω>[coth(<ħω>/2kT)1]O'Donnell et al. [18] Here, T is absolute temperature, χ and β are two constants for the semiconductor, χB(>0) represents average interaction between phonon and electron, ΘB is the average temperature of phonon, S and <ħω> 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 asΔE=χT2T+β+ΔE(0)+γ[TECfilmTECsub]T 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](σxx+χ1ΔTσyy+χ1ΔTσzz+χ3ΔT)=(C11C12C13C12C11C13C13C13C33)(εxxεyyεzz)(σxy,σxz,σyz)=2((C11C12)εxy,C44εxz,C44εyz) where σij and εij (i,j=x,y,z) are the components of stress tensor and stain tensor, respectively, Cij (i,j=1,2,,6) are the components of stiffness matrix, and χ1 and χ3 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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