We theoretically compare the energies and wave functions of the electron/hole states between InP- and CdSe-based core/shell/shell colloidal quantum dots (QDs) and investigate how the bandgap energy of the core material affects the light emission characteristics such as the photoluminescence quantum yield and linewidth. The band diagrams and electron/hole energies of InP/ZnSe/ZnS and CdSe/ZnSe/ZnS QDs, having the same emission wavelength, are calculated on the basis of strain-modified effective mass approximation (EMA). The QD strain distribution, caused by the lattice mismatch, is considered based on the continuum elasticity theory. The energies and wave functions of all the electron and hole states in the InP- and CdSe-based core/shell/shell QDs are obtained through the analytical solution of the Schrödinger equation under the EMA. Then, the emission spectra of the two QDs are calculated while considering the homogeneous and inhomogeneous broadening. Finally, we elucidate why the emission characteristics of InP-based QDs, such as the quantum efficiency and emission linewidth, are inferior to those of CdSe-based QDs, and how these can be improved by using the III-V ternary core materials with a bandgap energy comparable to or larger than that of CdSe.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Colloidal semiconductor quantum dots (QDs), owing to the advantages of their size-tunable emission wavelengths and narrow emission linewidths, have been investigated intensively as one of the promising display technologies . While the QD-based down-conversion light-emitting applications have already been commercialized , their electroluminescence applications have been studied to improve the performance of organic light-emitting diodes (OLEDs) in the framework of QD-OLEDs [3,4]. There has been a rapid performance improvement in Cd-based II-VI compound QDs, which shows a very high photoluminescence quantum yield (PLQY) of more than 90% and a narrow emission linewidth of 25 nm . In spite of their outstanding characteristics, the commercialization of Cd-based QDs is strictly restricted because of the use of hazardous Cd substances.
Considerable research efforts to develop a nontoxic alternative to the Cd-based QDs, which include nontoxic I-III-VI ternary QDs such as CuInS2 QDs [6,7] or carbon QDs [8,9], have already been made. Among them, InP-based QDs have been investigated intensively as a promising nontoxic alternative to the Cd-based QDs and currently commercialized as down-conversion materials in white light-emitting diodes . However, the output performance of InP-based QDs is worse than Cd-based QDs because of their lower PLQY and broader emission linewidth . Furthermore, there has been no theoretical or experimental research to elucidate why the emission characteristics of Cd-based QDs are superior to those of InP-based QDs, which could provide a possible solution to improve the emission characteristics of InP-based QDs.
The ab initio calculation of colloidal semiconductor QDs, such as the density functional theory  and atomistic pseudopotential approach , has been performed to predict the emission wavelength with respect to various structure parameters such as the size and material composition of QDs. However, these first-principle calculation methods involve a significant computation complexity and cost. Therefore, they are not applicable to colloidal QDs thicker than 5 nm. For 3–10 nm thick colloidal semiconductor QDs, the strain-modified effective mass approximation (EMA) provides reliable calculation results of the energies and wave functions of the electron/hole states without heavy mathematical and computational complexity [13–16]. In the strain-modified EMA, the strain distribution of the colloidal QD is calculated on the basis of the continuum elasticity theory . The analytical solution of the Schrödinger equation under the strain-modified band diagram provides the energies and wave functions of the QD electron and hole states . The band diagram and electron/hole wave functions of Cd-based core/shell/shell QDs were calculated on the basis of the strain-modified EMA . Recently, the band diagram and electron/hole wave functions of InP-based core/shell/shell QDs were calculated without considering the strain effect . However, there has not yet been any direct comparison of the strain-modified band diagram and electron/hole wave functions between Cd- and InP-based QDs of the same emission wavelength.
In this paper, we theoretically compare the energies and wave functions of the electron/hole states in Cd- and InP-based core/shell/shell QDs of the same emission wavelength. Furthermore, we elucidate how the emission characteristics of InP-based QDs are different from those of Cd-based QDs in respect of the bandgap energy of the core material. In the calculation, after the natural band offsets of the conduction and valence bands are determined, the strain distribution caused by different material compositions of the core/shell/shell structure is calculated on the basis of the continuum elasticity theory. The energies and wave functions of the electron and hole states confined to the QD are obtained through an analytical solution of the Schrödinger equation under the EMA. Then, the emission spectra of the two QDs are calculated when the homogeneous and inhomogeneous broadenings are taken into account. Finally, we explain why the emission characteristics of InP-based QDs are inferior to those of Cd-based QDs, and how these can be improved by using III-V ternary core materials, whose bandgap energy is comparable to or larger than that of CdSe.
2.1 Strain-modified band diagram
Figure 1 shows the layer structures of InP/ZnSe/ZnS and CdSe/ZnSe/ZnS QDs along with the schematics of their band diagrams. The shell materials of ZnSe and ZnS are chosen to be the same so that any difference between the energies and wave functions of the electron/hole states is ascribed to the effect of the different core materials alone. The thicknesses of all layers are determined to make the emission wavelength of the two QDs equal. When both QDs have the same emission wavelength of 563 nm, the thicknesses of layers are InP(1.05 nm)/ZnSe(0.25 nm)/ZnS(0.35 nm) and CdSe(3 nm)/ZnSe(0.5 nm)/ZnS(0.7 nm). As shown in Fig. 1, the bandgap energy of InP (1.35 eV)  is smaller than that of CdSe (1.74 eV) . To obtain the same emission wavelength as that of the CdSe QD, the InP QD should have a smaller layer thickness, which enhances the energies of the QD electron and hole states caused by the quantum confinement of the three-dimensional QD electric potential. This fact plays an important role in determining the reasons for the difference between the emission characteristics of InP and CdSe QDs. A detailed explanation will be given later.
For a spherical core/shell/shell QD, the strain distribution caused by the lattice mismatch at the core/shell and shell/shell interfaces plays an important role in the band diagram of the QD because the applied strain produces a significant change in the band-edge energy of conduction and valence bands [13,14,17]. In a spherical QD described by the spherical coordinate system (r, θ, ϕ), the hydrostatic strain (εhyd) causes changes in the band-edge energy of conduction () and valence () bands, which are given as 
The continuum elasticity theory has been widely used to calculate the strain distribution of the colloidal core/shell [13,14] or core/shell/shell QDs . Material parameters such as the lattice constant (a), Young′s modulus (EY), and Poisson′s ratio (μ) are used to obtain the strain distribution of εrr, εθθ, and εϕϕ. The detailed theory regarding the use the strain calculation in this paper can be found in . In the case of the colloidal InP QD with the core region of around 1 nm, the strain distribution calculated by using the valence-force-field model  will be more accurate than that obtained by using the continuum elasticity theory in this scale. However, the QD strain distribution calculated through the continuum elasticity theory has shown a good agreement with that obtained by using the valence-force-field model when the shape of the QDs, made of homogeneous and isotropic materials, is assumed to have a spherical geometry and with no sharp edge effect [22,23]. In addition, the valence-force-field model is mathematically complex and requires a priori parameters related to the interface structure and surface passivation [24,25]. Thus, we choose to use the continuum elasticity theory, which provides a relatively simple analytical expression for the strain distribution of colloidal QDs with an acceptable accuracy.
Figure 2 shows the calculation results of the strain distribution of InP/ZnSe/ZnS and CdSe/ZnSe/ZnS QDs on the basis of the continuum elasticity theory, where three-dimensional strain distributions of the spherical QD are obtained in the three-dimensional calculation and can be separately expressed in the r, θ, and ϕ directions . The material parameters used in the strain calculation are shown in Table 1 [17,19]. In Fig. 2, the positive and negative values of the strain represent the tensile and compressive strains, respectively. Because the lattice constant of the core material in both QDs is smaller than that of the shell materials, the core regions experience isotropic, compressive strain caused by the compression from the shell regions. In the shell regions, the tensile strain is applied in tangential (εθθ, εϕϕ) directions, and the compressive stain is induced in the radial (εrr) direction. The strain distribution in the shell regions decays with the increasing distance from the core/shell and shell/shell interfaces. However, the hydrostatic strains in the ZnSe and ZnS shell regions are uniformly compressive and tensile, respectively.
Figure 3 shows the band diagrams without and with the consideration of the strain-induced band-edge energy change calculated by using (1) and (2). First, the band diagrams of unstrained QDs are determined on the basis of the natural band offsets adopted from the literature, where the revised ab initio method is used . Then, the change in the band-edge energy induced by the strain distribution is added for conduction (ΔEc) and valence (ΔEv) bands. The strain-modified bandgaps of the core and ZnSe shell increase owing to the applied compressive strain. On the other hand, the tensile strain is applied in the ZnS shell, where the strain-modified bandgap decreases. The amount of the increased bandgap in the core of the InP/ZnSe/ZnS QD is 0.297 eV and that in the core of the CdSe/ZnSe/ZnS QD is 0.356 eV. This large amount of bandgap shift caused by the strain indicates that the consideration of the hydrostatic strain is essential in the electronic band calculation of colloidal core/shell/shell QDs.
2.2 Wave function and energy of a single colloidal QD
The wave functions and confined energies of the QD electron and hole states are obtained by the solution of the stationary Schrödinger equation, which is given byFig. 3 is assumed to be spherically symmetric so that the three-dimensional wave function can be written as18]
The coefficients of the wave functions in (5) can be determined by means of the boundary conditions, which are expressed as
The exciton binding energy between the 1s-1s ground states of the electron and hole is calculated by 
2.3 Optical emission spectrum of the colloidal QD ensemble
After the wave functions and confined energies of the QD electron and hole states are determined, the emission spectrum of an ensemble of colloidal QDs is given by 17,20]27,28]. Finally, the size or material composition of a QD ensemble are not identical and give rise to the inhomogeneous broadening, which is expressed as [29,30]
3. Calculation results
Figure 4 shows the calculated energy levels of the electron and hole states for InP/ZnSe/ZnS and CdSe/ZnSe/ZnS QDs by solving the Schrödinger equation in (3). The radial electric potentials are obtained by the calculated strain-modified band diagram in Fig. 3. Because we focus on the emission spectrum of the ground-state exciton in this calculation, we only consider the wave function (l = 0, m = 0), which includes the ground state of the colloidal QD. According to the rigorous band structure calculation near the band edges of the direct bandgap, the valence band comprises heavy hole (HH), light hole (LH), and spin-orbit split-off (SO) bands . Because we concentrate on the calculation of the emission spectrum generated by the ground-state exciton, we only consider the energies and wave functions of the HH states, which include the ground state in the valence band. The material parameters used in the calculation of the energies and wave functions are summarized in Table 1 [17,19]. In addition, we assume that the InP/ZnSe/ZnS and CdSe/ZnSe/ZnS QDs are surrounded by the same matrix material, whose material parameters are assumed to be = = 1 and εr = 4.0 with the conduction and valence band edges of 3.35 and −2.0 eV. In Fig. 4, the InP/ZnSe/ZnS QD has one electron and three HH states, whereas the CdSe/ZnSe/ZnS QD has two electron and six HH states. Because of the deep potential barrier and large radius, the ground states (C1 and HH1) of the CdSe/ZnSe/ZnS QD are closer to the band edge and more confined to the QD potential than those of the InP/ZnSe/ZnS QD.
Figure 5 shows the calculated radial probabilities of the electron and HH ground-state wave functions in the InP/ZnSe/ZnS and CdSe/ZnSe/ZnS QDs. The wave functions of the electron and HH states in the CdSe QD are more confined to the core region than those in the InP QD. Thus, the CdSe QD has larger overlap of the wave functions between the electron and HH states than that of the InP QD. Because of the larger overlap of the wave functions, the CdSe QD has stronger emission efficiency than the InP QD. In addition, the ground-state electron and HH in the CdSe QD is less scattered owing to the strong confinement of the wave functions to the ground state. This results in less carrier scattering and a smaller broadening linewidth of the CdSe QD than that of the InP QD.
Figure 6 shows the comparison of the calculated emission spectra of the InP/ZnSe/ZnS and CdSe/ZnSe/ZnS QDs. The standard deviation of the inhomogeneous broadening is assumed to be σ = 40 meV for both QD ensembles because the degree of the fluctuation in the size or material composition of InP and CdSe QDs is assumed to be the same. The linewidth of the homogeneous broadening is assumed to be γ = 30 meV for the CdSe QD and γ = 40 meV for the InP QD on the basis that the ground-state carrier of the CdSe QD is more strongly confined than that of the InP QD. The values of the homogeneous and inhomogeneous broadening of the emission spectra in two cases are chosen to make the full width at half maximum (FWHM) of the calculated emission spectra agree with the FWHM of the measured emission spectra in the literature. In Fig. 6, the FWHM of the emission spectrum is 40 nm for the InP/ZnSe/ZnS QD and 30 nm for the CdSe/ZnSe/ZnS QD; these values match with the measured values . Because of the larger wave function overlap and the smaller homogeneous broadening linewidth, the CdSe QD shows larger emission intensity than that of the InP QD.
As shown in Fig. 6, the reason the emission spectrum of the InP QD has low peak intensity and broader linewidth is that the bandgap energy of InP is smaller than that of CdSe. To compensate the smaller bandgap energy, the quantum confinement energy of the InP QD should be increased by means of a smaller QD radius, where the wave functions of carriers are less confined to the QD core potential, as shown in Fig. 5. Then, the InP QD has a smaller wave function overlap and larger homogeneous linewidth, which deteriorates the emission characteristic of the InP QD. Therefore, by using another InP-based or III-V material with larger bandgap energy as the core material, we can improve the output emission characteristics of Cd-free InP-based QDs. One example is to use the InGaP material as shown in Fig. 7. The bandgap energy of InGaP increases with respect to the fraction of GaP and becomes comparable to or larger than the bandgap energy of CdSe. Another way is to use the GaAlAs as the core material of a core/shell/shell QD, whose bandgap energy can be tuned to become comparable to or larger than the bandgap energy of CdSe. In this case, we can expect the optical characteristics of the InGaP- or GaAlAs-based QD to be comparable to or better than those of the CdSe-based QD.
To demonstrate the feasibility of the proposed idea, we calculate the emission spectrum of the In1-xGaxP/ZnSe/ZnS QD, whose peak intensity with the same wavelength is comparable to that of the CdSe/ZnSe/ZnS QD shown in Fig. 6. When the fraction of GaP in In1-xGaxP is chosen as x = 0.57, the bandgap energy of In1-xGaxP is determined to be 1.977 eV based on the interpolation formula of 1.35 + 0.668x + 0.758x2 . According to Fig. 7, the In0.43Ga0.57P core has the direct bandgap when In1-xGaxP is an alloy of the direct-gap InP and the indirect-gap GaP. The other simulation parameters of In0.43Ga0.57P are determined on the basis of the linear interpolation of the material parameters between InP and GaP. The simulation parameters of InP are tabulated in Table 1. The material parameters of GaP are a = 5.45 Å, ac = −7.14 eV, av = 1.70 eV, EY = 103 Gpa, μ = 0.31, = 0.25, = 0.326, and εr = 11.11 [20,32]. The respective layer thickness of InGaP QD is set to In0.43Ga0.57P(9.0 nm)/ZnSe(0.5 nm)/ZnS(1.7 nm) to obtain the peak emission wavelength of 565 nm, which is nearly the same as the peak emission wavelength of 563 nm in the CdSe QD.
Figure 8 shows the calculation results of the strain distribution and the band diagram of the In0.43Ga0.57P/ZnSe/ZnS QD. Because the lattice constant of the In0.43Ga0.57P core (5.631 Å) becomes close to that of the ZnSe shell (5.66 Å), the lattice mismatch between the In0.43Ga0.57P core and ZnSe shell is significantly reduced. Correspondingly, the amount of the strain applied to the In0.43Ga0.57P QDs shown in Fig. 8(a) is reduced compared with the strain applied to the InP or CdSe QD in Fig. 2. Regarding the calculated band diagram in Fig. 8(b), the strain-modified band-edge energies of the In0.43Ga0.57P core and ZnSe inner shell increase whereas that of the ZnS outer shell decrease, which shows the same behavior for InP- or CdSe-based QDs shown in Fig. 3.
In Fig. 9(a), the energy levels of the electron and HH states are calculated in the In0.43Ga0.57P/ZnSe/ZnS QD, which have two electron and eight HH states. Figure 9(b) shows the calculated radial probabilities of the electron and HH ground-state wave functions in the In0.43Ga0.57P/ZnSe/ZnS QD. The confinement of the wave functions in the In0.43Ga0.57P-based QD is similar to that in the CdSe-based QD shown in Fig. 5(b), which results in the wave-function overlap value of 0.83 in the In0.43Ga0.57P/ZnSe/ZnS QD being very close to that of 0.861 in the CdSe/ZnSe/ZnS QD. Compared with the wave-function overlap value of 0.585 in the InP/ZnSe/ZnS QD, this improvement in the wave-function overlap in the In0.43Ga0.57P/ZnSe/ZnS QD is ascribed to the increase in the bandgap energy by using the In0.43Ga0.57P core material.
Figure 10 shows the comparison of the calculated emission spectra of the InP/ZnSe/ZnS, CdSe/ZnSe/ZnS, and In0.43Ga0.57P/ZnSe/ZnS QDs, where the standard deviation of the inhomogeneous broadening is assumed to be σ = 40 meV for all the QD ensembles. The linewidth of the homogeneous broadening for InGaP QD is set to be γ = 30 meV because the degree of the ground-state carrier confinement in the InGaP, related to the wave-function overlap, is close to that in the CdSe QD. The peak intensity of the In0.43Ga0.57P QD becomes larger than that of InP/ZnSe/ZnS QD and is comparable to that of CdSe/ZnSe/ZnS QD. This calculation result demonstrates that the output emission characteristics of Cd-free InP/ZnSe/ZnS QDs can be improved by using the InGaP core material, whose bandgap energy is comparable to or larger than the bandgap energy of CdSe.
Finally, the EMA used in this calculation of the energies and wave functions of the electron/hole states in the colloidal QDs has the limitation that it cannot consider the complicated valence band structure and/or the non-parabolic dispersion of the conduction band, which has been analytically calculated by using the k⋅p method [33–36]. In the k⋅p method applied to the spherical QD, the envelope function was expanded by spherical wave functions for mathematical and computational simplicity . The complicated valence band structure and conduction band non-parabolic dispersion were calculated on the basis of the k⋅p method in the CdSe core , InP core , and ZnTe/ZnSe core/shell  colloidal QDs. However, the k⋅p method has not been applied to the core/shell/shell QDs. According to the calculation results in , the electron and hole energies of the InP QD, calculated by using the EMA-based parabolic dispersion, were larger than those obtained through the k⋅p method because the EMA ignores the coupling effect among the conduction, HH, LH, and SO bands and overestimates the effective masses of the electron and hole. If the complicated valence band structure and non-parabolic dispersion of conduction band are considered, our calculation results obtained through the strain-modified EMA will be modified accordingly. The electron and hole energies of the InP- and CeSe-based core/shell/shell QDs become smaller, which reduces the emission wavelength. The calculation of the electron/hole energies and wave functions in the colloidal core/shell/shell QDs based on the k⋅p method is under investigation.
We theoretically compared the energies and wave functions of the electron/hole states in InP- and Cd-based core/shell/shell QDs of the same emission wavelength by using the strain-modified EMA. To compensate for the smaller bandgap energy, the InP/ZnSe/ZnS QD had to consist of thinner layers than the CdSe/ZnSe/ZnS QD, which enhanced the confinement energy of the three-dimensional QD potential. The strain distribution inside the QD was calculated on the basis of the continuum elasticity theory. The increases in the bandgap, caused by the compressive strain in the core, were 0.297 eV for the InP-based QD and 0.356 eV for the CdSe-based QD, which were significant values in the band diagram of colloidal core/shell/shell QDs.
The analytical solution of the Schrödinger equation gave the energies and wave functions of the QD electron and hole states. Because of the shallow potential barrier and small QD radius, the ground state of the InP/ZnSe/ZnS QD was less confined to the QD potential, which resulted in a smaller wave function overlap and a larger homogeneous linewidth. Consequently, the peak intensity of the calculated emission spectrum of the InP/ZnSe/ZnS QD was smaller than that of the CdSe/ZnSe/ZnS QD. The reason for the emission characteristics of InP-based QD being inferior to those of CdSe-based QD is the smaller bandgap energy of InP. If other large bandgap materials such as InGaP or GaAlAs are used as the core material, the output performance of the InGaP- or GaAlAs-based QD is expected to be comparable to or even better than that of the CdSe-based QD. To verify these results, the emission spectrum of the In0.43Ga0.57P/ZnSe/ZnS QD was calculated, where the bandgap of the In0.43Ga0.57P core was larger than that of the CdSe core. The peak intensity of the emission spectrum in the In0.43Ga0.57P/ZnSe/ZnS QD was comparable to that in the CdSe/ZnSe/ZnS QD when the emission wavelengths of both QDs were nearly the same.
National Research Foundation of Korea (NRF) (No. 2018R1D1A1B07047249).
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