Band-structure-corrected local density approximation study of semiconductor quantum dots and wires

This paper presents results of ab initio accuracy thousand atom calculations of colloidal quantum dots and wires using the charge patching method. We have used density functional theory under local density approximation (LDA), and we have corrected the LDA bulk band structures by modifying the nonlocal pseudopotentials, so that their effective masses agree with experimental values. We have systematically studied the electronic states of group III-V (GaAs, InAs, InP, GaN, AlN, and InN) and group II-VI (CdSe, CdS, CdTe, ZnSe, ZnS, ZnTe, and ZnO) systems. We have also calculated the electron-hole Coulomb interactions in these systems. We report the exciton energies as functions of the quantum dot sizes and quantum wire diameters for all the above materials. We found generally good agreements between our calculated results and experimental measurements. For CdSe and InP, the currently calculated results agree well with the previously calculated results using semiempirical pseudopotentials. The ratios of band-gap-increases between quantum wires and dots are material-dependent, but a majority of them are close to 0.586, as predicted by the simple effective-mass model. Finally, the size dependence of $1S_e\text{−}1P_e$ transition energies of CdSe quantum dots agrees well with the experiment. Our results can be used as benchmarks for future experiments and calculations.

Figure 1

Schematic configuration for Ga-terminated and N-terminated surface. Dangling bonds passivated by H contain 1.25 and 0.75 electrons on the Ga-terminated and As-terminated ideal surface, respectively. The surface charge-density motifs are generated for the group of atoms inside the dashed line square box. (a) Ga-terminated surface passivated by one H atom. (b) N-terminated surface passivated by one H atom. (c) Ga-terminated surface passivated by two H atoms. (d) N-terminated surface passivated by two H atoms.

Figure 2

Pseudopotentials of (a) Ga and (b) As atoms in real space. The nonlocal pseudopotentials are angular momentum dependent. $V_s$ and $V_p$ are the $s$ and $p$ states of the valence wave functions. The modified pseudopotentials are $V_s+\text{correct}$ and $V_p+\text{correct}$, respectively. $Z$ is the pseudo core charge.

Figure 3

Band structures of bulk GaAs of LDA self-consistent calculation (solid curves) and of the modified nonlocal pseudopotentials (dotted curves).

Figure 4

(Color online) The charge motifs used to generate the charge densities of GaNP systems and GaPH quantum dots. One charge density isosurface is plotted.

Figure 5

Planar averaged atomic potentials of CdSe QDs with zinc blende structure. The diameter of QDs is $1.99\mathrm{nm}$. The horizontal position axis is along the (100) direction. Potentials from self-consistent LDA calculation and charge patching method are shown by the solid curve and dotted curve, respectively.

Figure 6

The single particle eigen energies of a 149 Si atom quantum dot, comparison between the charge patching results, and the direct self-consistent LDA results. Each verticle bar is an eigenenergy.

Figure 7

Wave function isosurfaces of CdSe QDs with $1.32\mathrm{nm}$ diameter for (a) CBM and (b) VBM states. The bonding geometry of CdSe QDs is a wurtzite structure. Isosurfaces are drawn at 20% of maximum. The small isolated black spheres are the surface passivation pseudo-H atoms.

Figure 8

Same as Fig. 7, but the diameter of the QDs is $4.33\mathrm{nm}$.

Figure 9

Same as Fig. 7, but for CdSe QWs with $1.39\mathrm{nm}$ diameter. We are looking down the QW in the wurtzite (0001) direction.

Figure 10

Same as Fig. 9, but for CdSe QWs with $4.37\mathrm{nm}$ diameter.

Figure 11

Comparison of quantum confinement energies by present “$\mathrm{LDA}+\mathrm{C}$” calculations (bulk effective-mass is corrected to experiment) and previous “LDA” calculation (no correction on LDA band structure) in CdSe QDs and QWs. The bond geometry of QDs and QWs is zinc blende structure. No Coulomb energy is considered in this figure.

Figure 12

Comparison of the exciton energy shift (from its bulk value) of CdSe QDs between experiment, “$\mathrm{LDA}+\mathrm{C}$” (present work), and SEPM calculations. Coulomb energies are considered in this calculation. Experimental data is from Ref. 5.

Figure 13

Comparison of the quantum confinement energy gap of CdSe QWs between experiment, “$\mathrm{LDA}+\mathrm{C}$” (present work), and SEPM calculations. Experimental data are from Ref. 65.

Figure 14

Size dependence of exciton energies of CdS QDs. Experimental data are from Ref. 83.

Figure 15

Size dependence of exciton energies of CdTe QDs. Experimental data are from Ref. 86.

Figure 16

Comparison of the exciton energy shift (from its bulk value) of InP QDs between experiment, “$\mathrm{LDA}+\mathrm{C}$” (present work), and SEPM calculations. Coulomb interactions are considered in this calculations. Experimental data are from Refs. 87, 88.

Figure 17

Comparison of the quantum confinement energy gap of InP QWs between experiment, “$\mathrm{LDA}+\mathrm{C}$” (present work), and SEPM calculations. Experimental data are from Refs. 12, 64.

Figure 18

Size dependence of exciton energies of InAs QDs. Experimental data are from Refs. 96, 97.

Figure 19

Size dependence of exciton energies of ZnS QDs. Experimental data are from Refs. 100, 101.

Figure 20

Size dependence of exciton energies of ZnSe QDs. Experimental data are from Ref. 104.

Figure 21

Size dependence of exciton energies of ZnO QDs (a) and QWs (b).

Figure 22

Size dependence of exciton energies of GaN QDs (a) and QWs (b).

Figure 23

Size dependence of exciton energies of InN QDs (a) and QWs (b).

Figure 24

Size dependence of energy gaps for CdSe QDs (a) and QWs (b). Note, the electron-hole Coulomb interaction is not included, thus (a) is different from Fig. 12.

Figure 25

Theoretical optical absorption spectra of wurtzite structure CdS QDs. Coulomb interaction is taken into account in the calculation. $E_g$ is the ground exciton energy. The four quantum dots from the top curve to the bottom curve are: ${\mathrm{Cd}}_{87}{\mathrm{S}}_{96}$, ${\mathrm{Cd}}_{217}{\mathrm{S}}_{220}$, ${\mathrm{Cd}}_{443}{\mathrm{S}}_{432}$, and ${\mathrm{Cd}}_{750}{\mathrm{S}}_{765}$, respectively. The peak a corresponds to VB1,2,3-CB1 transitions; peak b corresponds to VB7,8,9-CB1 transitions; peak c→ VB1,2,3-CB2,3,4; peak d→ VB4,5-CB2,3,4; peak e→ VB10,11-CB2,3,4; peak g→ VB8,9-CB5,6,7,8,9,10.

Figure 26

Theoretical optical absorption peaks of CdS QDs extracted from Fig. 25.

Figure 27

Size dependence of $1S_{e}1P_e$ transition energy of $n$-type CdSe QDs.