First-Principles-Derived Effective Mass Approximation for the Improved Description of Quantum Nanostructures

The effective mass approximation (EMA) could be an efficient method for the computational study of semiconductor nanostructures with sizes too large to be handled by first-principles calculations, but the scheme to accurately and reliably introduce EMA parameters for given nanostructures still remains to be devised. Herein, we report on an EMA approach based on first-principles-derived data, which enables accurate predictions of the optoelectronic properties of quantum nanostructures. For the CdS/ZnS core/shell quantum rods, where relevant experimental data became recently available, we first carry out density functional theory (DFT) calculations for an infinite nanowire to obtain the nanoscopic dielectric constant, effective mass, and Kohn-Sham potential. The DFT-derived data are then transferred to the finite nanorod cases to set up the EMA equations, from which we estimate the photoluminescence (PL) and field-dependent switching properties. Compared with the corresponding method based on bulk EMA parameters, we confirm that our EMA approach more accurately describes the PL properties of nanorods. We find that, in agreement with the experimentally observed trends, the optical gap of nanorods is determined by nanorod diameter and the PL intensity is reduced with increasing nanorod length. On the other hand, the electric field-induced PL switching efficiency is shown to be enhanced in longer nanorods.


Introduction
With the advancement in nanofabrication and synthesis techniques, it is now possible to prepare semiconductor nanostructures with various sizes and shapes to tune their electronic and optical properties [2,3]. In these nanostructures that experience quantum and dielectric confinement effects, one can engineer the excitons to acquire features that are beneficial for various device applications such as light-emitting diode (LED), photosensor, solar cell, solar fuels production, and biological labelling [4]. Particularly, their optical properties can be further modulated by inducing various external stimuli such as the electric and magnetic fields, which particularly make them promising candidates for display applications.
In the characterization and design of the semiconductor nanostructures, computer simulations have been playing an important role. Here, in principle, first-principles schemes such as density functional theory (DFT) and many-body or quantum Monte Carlo simulations performed on top of DFT would be desirable [5]. In practice, however, DFT and DFT-based higher-level calculations that require very large computational costs are often too demanding or even impossible to be applied to nanostructures of realistic sizes. Accordingly, approxi-mate large-scale methods such as the effective mass approximation (EMA) and tight binding [6,7] techniques are still routinely employed for the study and design of large complex semiconductor nanostructures [8][9][10]. However, employing the effective mass and dielectric constant derived from the bulk crystal, they often fail to produce accurate and reliable results for the nanostructures experiencing quantum and dielectric confinement effects.
In this work, in view of the optoelectronic device applications based on semiconductor nanostructures, we extend the EMA simulator based on our grid-based Object-Oriented Real-space Electronic structure (OORE) framework [11][12][13][14][15][16] by employing the EMA parameters generated from first-principles calculations on reference nanostructures. Specifically, from the reference DFT calculations, we extract the nanoscopic dielectric constant, electron and hole effective masses, and additionally the Kohn-Sham (KS) potential. The envelope function of the atomistic Kohn-Sham potential then allows us to define a well-defined EMA potential. While our approach should be generally applicable to finite-size quantum nanostructures including zero-dimensional (0D) quantum dots, one-dimensional (1D) quantum rods, and two-dimensional (2D) nanoplatelets, we here focus on the optical properties of semiconducting nanorods [8,9,[17][18][19]. Specifically, we will study the type-I CdS/ZnS core/shell quantum rods, with which we have recently demonstrate the high blue photoluminescence (PL) and electric-field-induced PL switching [1]. We will show that our DFT-derived EMA approach produces the optical gaps in good agreement with the experimentally measured data [1]. It will be pointed out that the optical gap of nanorods is mainly determined by their diameter, indicating the important role of quantum confinement effects. We will further show that, while the PL intensity itself is reduced, the electric field-dependent PL on/off switching efficiency can be enhanced by increasing nanorod length.

DFT calculations
For the infinite CdS core-only and CdS/ZnS core/shell nanowires extended along the z-axis, we performed for their unit cell models DFT calculations within the local density approximation (LDA) exchange-correlation functional [20]. DFT calculations were performed with the VASP package, in which the core electrons are handled by the projector augmented wave method [21]. The plane-wave basis set with a kinetic energy cutoff of 400 eV and the self-consistency cycle energy criterion of 10 -4 eV were adopted. To avoid artificial interactions with the neighboring images within the periodic boundary condition, a vacuum space of more than 20 Å was inserted along the xy directions perpendicular to the nanowire axis. The Brillouin zone was sampled with a 1 × 1 × 10 Monkhorst-Pack grid. The edge states of the (1010) surfaces were passivated by pseudo-hydrogen atoms. Specifically, the Cd or Zn dangling bonds were passivated with the pseudo-hydrogen atom with nuclear charge Z = 1.5 electrons, and each S dangling bond was passivated by the pseudo-hydrogen atom with Z = 0.5 electrons [22]. The nanowire dielectric constants were calculated using the optical dielectric function calculation module available within VASP.

EMA calculations
To assess the electronic structures and PL intensities of nanorods with different lengths and diameters, we performed EMA calculations using our grid-based OORE code [11][12][13][14][15][16]. It utilizes the higher-order finite-difference expansions of the Laplacian operator [23,24], where ℎ is the grid spacing and are the finite-difference coefficients, and the multigrid iterative minimization schemes for the solutions of Schrödinger and Poisson equations. The OORE code includes a general framework to carry out grid-based first-principles DFT calculations [11], and by simply replacing pseudopotentials by EMA potentials one can perform large-scale 3D EMA calculations (OOREQD) [12,13]. The key features and further developments relevant for the calculation of optical properties of semiconductor nanostructures within the newly-developed DFT-based EMA scheme will be presented in Sec. 3. In view of the LED applications, the type-I band alignment will be assumed throughout this work. Figure 1 graphically summarize the strategy of the DFT-based EMA scheme proposed in this work. For the "ideal" low-dimensional nanostructures, we first carry out DFT calculations and obtain the electron/hole effective mass /ℎ * and dielectric constant , and Kohn-Sham (KS) potential . Note that the effective mass and dielectric constant together set the length scale as ⁄ . Here, we define the "ideal" systems as the nanostructures that are infinitely extended along the non-confined directions. For example, for the finite quasi-1D nanorods and quasi-2D nanoplatelets, we consider 1D nanowires and 2D slabs, respectively, with the periodic boundary condition (PBC) along the z-and xydirections, respectively. For the 0D quantum dots, we could adopt a reasonably-sized quantum dot and employ the nanostructure-derived dielectric constant and KS potential in combination with the bulk effective mass. We will below take the nanorod case as the representative example and discuss in more detail the procedure of systematically employing nanoscopic EMA parameters derived from DFT calculations.

The effective mass approximation formulation
We solve within the isotropic EMA framework the conduction band edge (electron) Schrödinger equation for the electron wavefunction and energy , and separately the valence band edge (hole) Schrödinger for the hole wavefunction ℎ and energy ℎ , where ℏ is the reduced Planck's constant, * is the electron effective mass, and ℎ * is the hole effective mass. We emphasize that the key development in this work is the adoption of the EMA parameters derived from first-principles calculations performed on the representative model nanostructures. Importantly, in addition to the dielectric constant and effective masses, we introduce the effective potential , /ℎ from the reference DFT calculations. At the fundamental level, it was argued that the "exact" DFT KS equations for Ν electrons can be characterized as the Dyson equation for Ν − 1 electrons, so unoccupied orbitals obtained in the KS calculations should physically describe number-conserving optical excitations of the Ν-electron system [11,[25][26][27][28].
While we adopt , = ,ℎ ≡ with this physical nature of the KS potential in mind, given that we start from LDA DFT calculations contaminated by self-interaction errors, we heuristically regard equations (2) and (3) as quasi-particle equations [5] and determine the expressions for quasiparticle and optical gaps of quantum nanostructures within EMA as described below.
Once the hole and electron Schrödinger equations are solved, we estimated the exciton transition energy or optical gap , by calculating the band edge transition energy or quasiparticle gap according to The CdS quasiparticle gap was obtained by adding the experimentally reported bulk optical bandgap value of 2.42 eV [29] to the bulk exciton binding energy of 0.026 eV calculated according to where and 0 are static dielectric constant and vacuum permittivity, and µ is the reduced effective mass, We calculated the exciton binding energy using the expression for a spherical quantum dot with the radius [30][31][32][33][34] or we assumed that the nature of exciton binding in a quantum rod with diameter 2 is essentially determined by the radial direction quantum confinement. Finally, the oscillator strength for the electron-hole band edge exciton transition was calculated according to [35,36] where ℎ presents the oscillator strength for the electron transition from at to ℎ at ℎ .

Effective mass and dielectric constant from DFT calculation
For the descriptions of the CdS/ZnS core/shell nanorods, for which we recently obtained relevant experimental data [1], we first carried out PBC DFT calculations on the corresponding 1D nanowires (see Ref. [1] for the details). Denoting the nanowire model with m CdS and n ZnS layers as (CdS)m(ZnS)n, we show in figure 2(a) the (CdS)3(ZnS)2 core/shell nanowires with diameter of ~ 3.5 nm (excluding passivating pseudo-hydrogen atoms) optimized within DFT. From these models, we derived the electron (hole) effective mass * ( ℎ * ) and dielectric constant . The DFT-calculated dispersions of the conduction (top panels) and valence (bottom panels) band edges of the (CdS)3(ZnS)2 nanowire are shown in figure 2(b). The procedure of extracting electron (hole) effective mass from the DFT-derived conduction (valence) band dispersion curve is also schematically described in figure 2(b), where the region of band dispersion used for effective mass fitting is marked with the shaded rectangle near the gamma (Γ) k-point. To obtain the electron and hole effective masses, we adopted the harmonic E-k dispersion relation near the Γ according to Where E0 is the energy eigenvalue of selected conduction or valence band used for the effective mass fitting. The effective masses fitted to the conduction band minimum (CBM) and valence band maximum (VBM) of the (CdS)3(ZnS)2 nanowire are presented in table 1. While the bulk CdS-derived electron and hole effective masses are 0.2 and 0.7, respectively [37,38], the corresponding values derived by fitting equation (10) to (CdS)3(ZnS)2 band edges are 0.2 and 0.51, respectively. Namely, we determined that while the bulk hole effective mass is translated into the nanowire hole effective mass ℎ * , the electron effective mass is reduced by ~ 30 % through the nanostructuring. The nature of the effective masses relatively insensitive to the nanostrucing can be taken as the justification of the isotropic EMA adopted in this work.
On the other hand, as summarized in table 1, we found that the dielectric constants of the CdS/ZnS nan-owires are significantly decreased from the bulk CdS dielectric constant value of 8.92 [39] due to the reduced electronic screening effect [40][41][42][43]. Quantitatively, the optical dielectric constants of the (CdS)3(ZnS)2 core/shell nanowire, the static dielectric constant values along axial and radial directions were = 2.3 and , = 2.2, respectively. Note the small differences between the dielectric constants along the radial and axial directions, which indicates the negligible anisotropy in the local dielectric screening environment. Accordingly, we also assume the isotropic dielectric constant within the EMA calculations.

The EMA potential from DFT calculation
As emphasized earlier, in addition to the effective mass and dielectric constant, the utilization of the KS potential information to construct the EMA effective potentials represents a key feature of our approach. Note that in general the confinement potential shape is a critical factor in determining the electronic and optical properties of quantum nanostructures. For example, it was theoretically suggested that the suppression of undesirable nonradiative Auger processes can be achieved by smoothing out the confinement potential [44]. Accordingly, much experimental efforts were recdently devoted to understand and optimize the material gradient at the core/shell interface [45,46].
In figures 2(c) and (d), we present the cylindricallyaveraged DFT KS potentials and the corresponding EMA effective potentials obtained for the (CdS)3(ZnS)2 and (CdS)3(ZnS)1 nanowire cases, respectively. The macroscopically smooth EMA potentials were generated by obtaining the envelope functions of the KS potentials that oscillate at the atomic scale using a double filtering process with the step function as the filter function [47], Here, we chose ≈ 7 Å, which is approximately the radial thickness of two CdS (or ZnS) layers. The radially smoothened 1D EMA potential profiles were then directly projected along the boundaries of quantum rods, for which we adopted the rectangular or cylindrical shapes as shown in figures 3(a) and 3(b), respectively.
To confirm the importance of the DFT-based EMA effective potential, we additionally adopted an abrupt potential with the potential depth fixed to the DFT-derived EMA potential as shown in figure 3(c). Based on the nature of the DFT KS equations mentioned above [11,[25][26][27][28], we used the same EMA potential profile (with the opposite sign) for both hole and electron wavefunctions.

Comparison of wavefunctions from DFT and EMA calculations
To check the quality of the EMA potentials employed in our scheme, we first analyzed the radial-direction EMA electron and hole wavefunctions obtained by solving equations (3) and (4), respectively, against their DFT counterparts. In figure 4(a), we first show the CBM (top) and VBM (bottom) wavefunctions obtained from the DFT calculations performed on the (CdS)3(ZnS)2 nanowire. We next show in figure 4(b) the corresponding wavefunctions obtained from the DFTbased EMA calculations performed for a 12 nm-long (CdS)3(ZnS)2 rectangular-shape nanorod. We note that as shown in Supplementary figure S1(c) very similar wavefunctions were obtained by adopting the cylindrical shape EMA potential. Additionally, in figure 4(c), we show the corresponding wavefunctions obtained with abrupt EMA potential [figure 3(c)].
Overall, as can be expected by the comparison of the DFT and EMA potentials, we observe that the atomic scale oscillations in the DFT-derived wavefunctions [figure 4(a)] are smoothend out in the EMA envelope wavefunctions [figures 4(b) and 4(c)]. Next, comparing the EMA calculations based on the DFT-derived and abrupt EMA potentials, we can notice that our DFTderived EMA scheme much more closely reproduces the envelope profiles of DFT wavefunctions: Both the electron and hole wavefunctions penetrate into the shell region, and particularly the electron wavefunctions exhibit more delocalized nature. On the other hand, the abrupt effective potential-based EMA method results in wavefunctions that are too strongly confined within the core region [figure 4(c)], leading us to conclude that our DFT-based EMA approach indeed represents an improvement in describing quantum nanostructures.

Optical properties from EMA calculations
We now consider the energy gaps of quantum rods computed in our EMA approach. In figure 5(a), we present the quasiparticle gap of the (CdS)3/(ZnS)2 nanorod with the lengths of 18 and 24 nm with the cylindrical (black triangle) and rectangular (blue square) confinement potential shapes shown in figures 3(a) and 3(b), respectively. In both cases, shows negligible changes (≲ 5 meV) with respect to the nanorod length, indicating that or electron/hole eigenvalues /− ℎ of nanorods is essentially determined by the smallerdimension or radial-direction quantum confinement.
Comparing the values obtained from the cylindrical and rectangular EMA potentials, we find that the former is about 0.1 eV larger than the latter because of the slightly smaller cross section in the cylinder (for a fixed radius r, πr 2 rather than 4r 2 in the rectangular rod shape). After all, due to the close correspondence between the wavefunctions and eigenvalues or values from the cylindrical and rectangular shape confinement potentials, we will from now on consider only the rectangular EMA potential case.
For comparison, we also present in figure 5(a) the values obtained from the EMA calculations using bulk effective masses and dielectric constant together with the abrupt confinement potential profile (red star) shown in figure 3(c). They are smaller than those obtained from DFT-based EMA calculations by about 0.6 eV, indicating the weaker quantum confinement effect. Because the abrupt potential is supposed to provide stronger quantum confinement effects than the smoother DFT-derived potential and the electron and hole effective masses are similar in the two schemes, this feature should have mainly resulted from the significantly reduced dielectric constant or electronic screening in nanostructures compared to the bulk case [40][41][42][43] (see section 3.3 and table 1).
In figure 5(b), we next show the optical gap values calculated according to equation (4) together with the experimentally measured values (black filled circles) [1]. The quantum confinement induces a bigger overlap between electron and hole wavefunctions, thus increases within nanostructures. For example, it was experimentally observed that the value of CdSe significantly increase from 15 meV [48] in the bulk limit to 130 ~ 230 meV in nanoplatelets, to about 240 meV in nanorods, and to about 400 meV in quantum dots [49][50][51]. The of CdS nanorods with diameters of 4 ~ 10 nm were reported to be about 220 ~ 300 meV [49], again an order of magnitude larger than the bulk CdS value of 28 meV [52]. Within our DFT-based EMA calculations of (CdS)3(ZnS)2 with a smaller (CdS)3 core diameter of about 2.3 nm, we obtained the E X value of 588 meV using equation (6). Then, the resulting E gap opt values obtained within our DFT-based EMA approach (blue open squares) are in excellent quantitative agreement with the experimental data.
On the other hand, using the abrupt confinement potential and bulk effective masses and dielectric constant, we obtained the value of 152 meV and the values smaller than the experimental ones by ~ 110 meV (red stars). This value is apparently an underestimate of the experimental values [49,52], demonstrating the shortcoming of the conventional EMA approach and the improvement achieved in our EMA scheme.
With the electron and hole wavefunctions obtained from DFT-based EMA calculations, we also evaluated the oscillator strength according to equation (9) and estimated the PL intensity for the electron-to hole-level transition. In figure 5(c), we show the PL intensities of the (CdS)3/(ZnS)2 nanorods in the cases of nanorod lengths 16 nm and 18 nm. For the 24 nm nanorod case, we obtained the PL peak position of 463 nm, which is in good agreement with experiment [1]. For the shorter 16 nm nanorod, due to the increased axial direction quantum confinement, we obtained a slight blueshift of the PL peak and an about 30% increase of the PL intensity. The trend of enhanced PL, or the higher probability of electron-hole recombination, in shorter nanorods is again in good agreement with experimental data [1].

Electric field-dependent switching properties
One of the main motivations to employ nanorods for optoelectronic applications is the distinctive electricfield-induced PL switching property [8,9,[17][18][19]. To test whether our DFT-based EMA scheme can reliably describe these features, we repeated calculations by including the external field effect. In figure 6(a), we schematically show the positions and shapes of the electron and hole wavefunctions before (top) and after (bottom) applying an electric field of 100 kV/cm. The external electric field causes the electron and hole swavefunctions shift toward the opposite directions, which should result in the reduction in their spatial overlap. Accordingly, as shown in figure 6(c), the PL intensity is gradually quenched as the intensity of the applied electric field increases. Regarding the dependence of the PL switching efficiency on the length of quantum rods, we find that it improves as the length of nanorods increases. This is easily understandable in that the space between separated electron and hole wavefunctions can be extended in longer nanorods.

Discussion and future directions
While the above trend of the field-induced PL quenching and its nanorod length dependence are overall in qualitative agreement with our earlier experimental results, we note that quantitatively the PL quenching efficiency is overestimated in our EMA calculations [1]. Specifically, whereas at ~ 100 kV/cm the PL was observed to be reduced by ~ 20 % in short nanorods and up to ~ 40 % in long nanorods [see figure 3(e) of Ref. [1], it was quenched almost completely in our computations [figure 6(b)].
As a potential direction to make further improvements, we implemented the iterative Hartree scheme, which has been successfully employed in the past decades for both EMA [8,9,[53][54][55][56] and pseudopotential calculations [34]. Here, one includes the electron-hole Coulomb interaction at the Hartree level and self-consistently solves the coupled electron-hole Schrödinger equations, and Unexpectedly, whereas the uncoupled Schrödinger equations (2) and (3) have been traditionally considered as the approximations to the coupled Schrödinger equations (11) and (12), we obtained overall deteriorated results. To show this behavior in detail, we present in figures 6(c) and 6(d) the results corresponding to 6(a) and 6(b), respectively. As shown in figure 6(c), the electron and hole wavefunctions are spatially too strongly localized. They then abruptly split at very high electric fields, resulting in the sudden and drastic PL quenching as summarized in figure 6(d). Namely, the electron and hole wavefunctions are too strongly bound by Coulomb attraction in the coupled Hartree scheme, and this also results in the overestimation of the exciton binding energy with the magnitude of > 1 eV. We consider this deficiency resulted from the omission of the exchange interaction term, and leave its inclusion for future study.

Conclusion
In summary, we developed a first-principles based EMA calculation approach for quantum nanostructures, and implemented the method within our grid-based OORE framework (OOREQD) [11][12][13][14][15][16]. The essential ingredient in the developed scheme is carrying out reference DFT calculations to extract the nanoscopic effective mass and dielectric constant information and to generate from the atomistic KS potential a realistic EMA confinement potential. Given that the size and shape of the confinement potential are important factors in determining the electronic and optical properties of nanostructures [44], the ability to accurately extract the confinement potential profile for the efficient yet approximate EMA calculation approach will have important implications for the computational design of semiconductor nanostructures.
Applying the method to study the optical properties of CdS/ZnS core/shell quantum rods [8,9,[17][18][19], for which we recently acquired relevant experimental data [1], we successfully reproduced their optical gaps and the dependence of the PL intensity on the physical dimensions of nanorods. We found that the optical gap or the PL peak position is essentially determined by the nanorod diameter or the radial direction quantum confinement. On the other hand, the length of nanorods or the axial direction quantum confinement was found to affect the overlap between electron and hole wavefunctions and accordingly the PL intensity. In this work, we mainly focused on solving the uncoupled electron and hole Schrödinger equations because extending our scheme to the coupled Hartree level deteriorated the agreements with experimental data. The additional inclusion of exchange and correlation effects will be addressed in the future.        Table 3. Parameters for CdS and CdS/ZnS nanomaterials utilized to perform EMA simulations in this work.