Optical transitions and nature of Stokes shift in spherical CdSquantum dots Page: 2 of 5
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of Cd and S atoms), which have effective radii 6.32 A,
8.15 A, 10.24 A, 13.70 A, 17.26 A, and 30 A respec-
tively. The calculations were performed with the plane-
wave pseudopotential method, using local density ap-
proximation (LDA) and norm-conserving pseudopoten-
tials. The plane wave energy cut-off of 35 Ry was used
in all calculations. A well known LDA shortcoming of
underestimating the bandgaps and the electron effective
mass was corrected by modifying the nonlocal pseudopo-
tentials (after Ref. 19), such that the electron effective
mass in the bulk is in a very good agreement with ex-
periment, i.e. me 0.213mo calculated here, versus
me = 0.210mo measured,20 and the bandgap is partially
corrected (it is not possible to correct both simultane-
ously), i.e. Eg 2.23 eV calculated versus Eg 2.58 eV
measured20 (uncorrected LDA yields19 me 0.127mo
and Eg 1.315 eV). Spin-orbit coupling is included and
adjusted to yield Aso=0.068 eV corresponding to the
experimental value.20 In order to eliminate the quan-
tum dot surface dangling bond states and keep the sys-
tem charge neutral we passivate the surface with pseudo-
hydrogen following Ref. 21. A surface atom of valency
m is passivated with a pseudo-atom with Z (8-m)/4,
therefore Cd and S are passivated with H(Z 1.5) and
H(Z 0.5), respectively.
The charge patching method22 is used in order to ob-
tain the self-consistent quality real space charge distribu-
tion in the quantum dot, without having to perform di-
rect LDA calculations, which for the sizes of the systems
considered here are prohibitively expensive. Here, only
small prototype systems are computed self-consistently
for different atoms and their local environments to gen-
erate the motif charge densities . These charge motifs
are then used to assemble the total charge density for
the entire quantum dot. Knowing the charge density the
corresponding LDA potential is generated and the Hamil-
tonian of a given quantum dot is constructed. The band
edge eigenstates of this single particle Hamiltonian are
then solved using the folded spectrum method.23
Our calculated single particle eigenenergies for the va-
lence band are shown in Fig.1. Our ab initio calculation
produces a 1S3/2 state as the top of the valence band, and
the energies in Fig.1 are plotted relative to this state.
For quantum dots in the region of Reff > 17A the
1P3/2 is the second hole state. For smaller quantum
dots, there is another S3/2 state above the 1P3/2 state.
Even for the largest QD calculated here, i.e. (CdS)4586
with effective radius of Reff 30A, we do not observe
a S-to-P hole groundstate transition, and 1S3/2 is still a
groundstate. On the contrary, the 6 x 6 k-p theory in
the spherical approximation, using the effective mass pa-
rameters derived from our ab initio bulk band structure
(-1 2.31, 12 0.79, and Aso 0.068 eV), yields the
1P3/2 groundstate, rather than the 1S3/2 state, as shown
in the inset to Fig. 1. For this set of parameters the P-to-
S hole groundstate transition occurs at the QD effective
radius of ~100A. This is similar to other k-p calculations
even though k-p parameters used might be somewhat-0.051
0
0
Oh
x)-0.1
-0.15
-0.2
-0.2510 15 20
Radius [A]25 30
FIG. 1: Size dependence of the valence band energies without
electron-hole Coulomb interaction in CdS quantum dots, plot-
ted relative to the top of the valence band, 153/2 state. The
inset shows the hole energies of the 1P3/2 and 153/2 states as a
function of the quantum dot size, relative to the bulk valence
band maximum, calculated with the spherical k-p method.
The effective mass parameters used are, >1 = 2.31, y2 = 0.79,
and Aso = 0.068 eV.
different.7 (Note, the caption of Fig. 5 in Ref. 7 is in er-
ror. The energy is given in the units of o 11/(2R2) not
eo -11/(2R)2, and the x-axis is 0.529177 x R(nm), not
R(nm).) Note, that the 8x8 k-p formula should give qual-
itatively similar results due to the large bandgap. Since
the k-p predicts 1P3/2 hole groundstate for Reff < 100A,
our ab initio calculations indicate that it is possible that
the size range of ~40A < Reff < 100A, the 1P3/2 is in-
deed the hole groundstate. However, the possible spatial
symmetry induced dark exciton in that size range cannot
be used to explain the experimentally observed Stokes
shift, which is in the range of 15 - 70 meV, while the
spatial dark exciton in that size range will have a Stokes
shift of 0.6 - 6 meV, as predicted by the k-p model.
To compare our results with experimental optical mea-
surements, in addition to the single particle energies, we
need to calculate the exciton energies. Exciton energies
of optical transitions in the strong confinement regime,
where correlation effects are negligible, can be calculated
from the:Ee -EC - E C -E ,
(1)
where, ec and eg are the single-particle conduction and
valence states energies, respectively, and EC is the
electron-hole Coulomb energy, obtained as242 JJKC(x1V2bv(x2)2dd
ECV . 6. (r1 r2)Jr, r2 d id 2(2)
where, x - (r, o) includes both spatial r and spin o T, I
variables, c(ri - r2) is a position-dependent dielectric
function (described below), and vc(x) and bv(x) are theeA' --e- -------------------0
9 '-o..
-03 1--- IaS
G--2S32 04 1
P 20 Radius [A]
- -
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Demchenko, Denis O. & Wang, Lin-Wang. Optical transitions and nature of Stokes shift in spherical CdSquantum dots, article, December 16, 2005; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc873079/m1/2/: accessed April 24, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.